Introduction toSTATICS DYNAMICS and Problem Book Rudra Pratap and Andy Ruina Spring 2001 c Rudra Pratap and Andy Ruina, 1994-2001. All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, or otherwise, without prior written permission of the authors. This book is a pre-release version of a book in progress for Oxford University Press. The following are amongst those who have helped with this book as editors, artists, advisors, or critics: Alexa Barnes, Joseph Burns, Jason Cortell, Ivan Dobrianov, Gabor Domokos, Thu Dong, Gail Fish, John Gibson, Saptarsi Haldar, Dave Heimstra, Theresa Howley, Herbert Hui, Michael Marder, Elaina McCartney, Arthur Ogawa, Kalpana Pratap, Richard Rand, Dane Quinn, Phoebus Rosakis, Les Schaeffer, David Shipman, Jill Startzell, Saskya van Nouhuys, Bill Zobrist. Mike Coleman worked extensively on the text, wrote many of the examples and homework problems and created many of the figures. David Ho has brought almost all of the artwork to its present state. Some of the homework problems are modifications from the Cornell’s Theoretical and Applied Mechanics archives and thus are due to T&AM faculty or their libraries in ways that we do not know how to give proper attribution. Many unlisted friends, colleagues, relatives, students, and anonymous reviewers have also made helpful suggestions. Software used to prepare this book includes TeXtures, BLUESKY’s implementation of LaTeX, Adobe Illustrator and MATLAB. Most recent text modifications on January 21, 2001. Contents Problems for Chapter 1 . Problems for Chapter 2 . Problems for Chapter 3 . Problems for Chapter 4 . Problems for Chapter 5 . Problems for Chapter 6 . Problems for Chapter 7 . Problems for Chapter 8 . Problems for Chapter 9 . Problems for Chapter 10 . Problems for Chapter 11 . Problems for Chapter 12 . Answers to *’d questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . 2 . 10 . 15 . 18 . 31 . 41 . 60 . 74 . 83 . 88 . 100 as precisely as possible. h) Angular momentum. 1. b) Dynamics. . Why are engineers still using them? 1. 1.3 This chapter says there are three “pillars” of mechanics of which the third is ‘Newton’s’ laws. not precise mathematical language): a) Statics. What are these four laws (in English.Problems for Chapter 1 1 Problems for Chapter 1 Introduction to mechanics Because no mathematical skills have been taught so far. the questions below just demonstrate the ideas and vocabulary you should have gained from the reading.9 Computation is part of modern engineering. not the standard ‘Newton’s three laws’. a problem that is not mentionned in the book but which is a mechanics problem.2 Briefly define each of the words below (using rough English. i) A rigid body.7 About how old are Newton’s laws? 1. c) (optional) Do an example of each on a computer.1 What is mechanics? 1. a) What are the three primary computer skills you will need for doing problems in this book? b) Give examples of each (different thatn the examples given). what are the other two? 1. d) Strength of materials.5 Describe. e) Force. 1. f) Motion. State which quantities are given and what is to be determined by the mechanics solution. no equations needed)? 1. 1.4 This book orgainzes the laws of mechanics into 4 basic laws numberred 0-III.6 Describe an engineering problem which is not a mechanics problem. c) Kinematics. g) Linear momentum.8 Relativity and quantum mechanics have overthrown Newton’s laws. and ˆ ˆ + k ).6) problem 2.vec1. F2 = 50 N. 2. where F1 = 20 N. i = ˆ 1 to 4. R = ˆ ˆ ˆ 10 N(cos θ ı +sin θ ) and W = −20 N . Determine the angle θ and draw the force vector R clearly showing its direction.3: f) ˆ 3 N( 1 ı + ) 3ˆ (Filename:pfigure2. Match the correct pictures into pairs.13: C (Filename:pfigure2. 1 2 B y 4 cm problem 2. β T2 . Draw the two forces.9 Three forces F = 2 Nı − 5 N .3) C z A problem 2.5) 1 60o 1 ˆ λ1 ˆ x (Filename:pfigure2. rBC = 2 ft .vec1. 2. T2 = 40 N.12 In the figure shown. where F1 = 20 Nı . (a) Find rAD in terms of the variable length .1 Vector notation and vector addition ˆ ˆ 2. Find the sum F (= F 1 + F 2 + W )? F1 F2 y d) 13 N √ 13 N 4 3 4 3 T2 60 o T1 45o x 2 3 3 W problem 2.7: (Filename:pfigure2.10 Given that R = 1 Nı +1.vec1.13 In the figure shown. B ˆ d) 2 N(-ˆ + ı √ ˆ 3 ) 2.5: (Filename:pfigure2. a) b) 2N 3N ˆ -3 Nˆ + 2 N ı ˆ ˆ ı c) 2 √ 2.4 N . is zero. and 2W still sum up to zero. respectively. 2.11 For the unit vectors below. T1 = 20 2 N.2) 2. 2. and W is such that the sum of the three forces equals zero. Given that rB/A = 3 m( 1 ı + 2 √ 3 ˆ 2 ) W problem 2.2 Which one of the following representations of the same vector F is wrong and why? r AB D ˆ λ2 ˆ k A problem 2. sum up to zero.12: (Filename:pfigure2.11) problem 2.vec1. find F4 . Find the position vector rCD = 2 ft( rAD . ˆ ˆ 2.4 Find the sum of forces F1 = 20 Nı − 1 ˆ 1 ˆ ˆ 2 N . F2 = 30 N( √ ı + √ ). ˆ ı e) ˆ 3 Nˆ + 1 N ı problem 2. ˆ ˆ 2. find 2R + 5R .3 There are exactly two representations that describe the same vector in the following pictures. rods AB and BC are each 4 cm long and lie along y and x axes.12) a) ˆ ˆ ı c) 4N 30 2N √ 2 3N o b) 4N 30 o ˆ ˆ and rC/B = 1 mı − 2 m . and W = mg.vec1. F2 = ˆ ˆ ˆ −50 N .2 CONTENTS Problems for Chapter 2 Vector skills for mechanics B r BC C r CD 2. representing them with their magnitudes.11: √ 2. find rA/C .5 In the figure shown below.vec1.1 Represent the vector r = 5 mı − 2 m in three different ways. find α and β such that α T1 . the position ˆ ˆ vectors are rAB = 3 ftk . find the scalars α ˆ ˆ ˆ α λ − 3λ = β .5 N and R = 1 2 ˆ ˆ 3.6: (Filename:pfigure2.7 Three position vectors are shown in the ˆ figure below.14 Find the magnitudes of the forces F1 = ˆ ˆ ˆ ˆ 30 Nı −40 N and F2 = 30 Nı +40 N .7) D A 4 cm 30 x o ˆ 2.2 Nı − 0. 2 2 ˆ ˆ λ and λ shown 1 2 and β such that y 2.vec1.6 The forces acting on a block of mass m = 5 kg are shown in the figure.2: (Filename:pfigure2. . F3 = 10 N(−ı + ). and F3 = 2 2 √ ˆ ˆ −20 N(−ı + 3 ). Rod CD is in the x z plane and makes an angle θ = 30o with the x-axis.vec1.13) 2. (b) Find and α such that ˆ rAD = rAB − rBC + α k .8 Given that the sum of four vectors Fi . If W is doubled. 25 Find the unit vector λ .5 m F E 1 2m 1m 2m y 1 F 2 -F 1 A F2 plate D x problem 2.8 m z E D H x (Filename:pfigure2.21 A 1 m × 1 m square board is supported by two strings AE and BF. where k = 100 lbf/ ft.17: 30 o D (Filename:pfigure2.22 The top of an L-shaped bar. P directly parallel to the center C into the page. find (a) the relative position vector rQ/P . Express this tension as a vector.20: problem 2.19 Two points A and B are located in the x y plane. y (Filename:pfigure2. x x (Filename:pfigure2. Find the magnitude and direction of F2 − F1 . directed from AB point A to point B shown in the figure. Find the magnitude of the net force.vec1. 30o z problem 2. 2. are marked on the rim of the plate. P and Q.26) x y A 0.vec1.23 A cube of side 6 inis shown in the figure.16 In Problem 2. The force in the spring is F = k rAB .27 In the structure shown in the figure. You are allowed to change the direction of the forces by changing the angles α and θ while keeping the magniitudes fixed. What is the direction of this force? 2. y 2. (b) Find the magnitude of rA and rB .8 mlong cord from a 2 mlong hoist OA. Find a unit vector ˆ λ along AB and calculate the spring force AB ˆ F = Fλ .22: x 2.18: x (Filename:pfigure2.vec1. is to be tied by strings AD and BD to the points A and B in the yz plane.25) 2.18) 1m problem 2.20) y F C G B x (Filename:pfigure2.21) (Filename:pfigure2. and k as shown in the figure.5 m A 2. (c) How far is A from B? 2.vec1.80 ) and W = −36 N act on a particle. (b) Calculate | rF |. 3 2.21: B 1m C x ˆ k ˆ ˆ ı A 12" O Q 45o 6" Q C 6" C D P F1 45o problem 2. rA/B . y y 3m 1m A B Q P α 2m θ x 1m A (Filename:pfigure2.20 In the figure shown.22) 2. 6 mm). (b) the magnitude | rQ/P |. (b) Find the distance of the ball from the origin.13.23: A 2.26: (Filename:pfigure2.17) 2. The disk is tipped 45o with the horizontal bar AC. . F1 = 100 N and F2 = 300 N. (a) Draw position vectors rA and rB .24 A circular disk of radius 6 inis mounted on axle x-x at the end an L-shaped bar as shown in the figure. The coordinates of A and B are (4 mm.vec1. and Q at the highest point above the center C. find such that the length of the position vector rAD is 6 cm. a ball is suspended with a 0. 8 mm) and (90 mm. The tension in the string BF is 20 N. h = 1.vec1.17 In the figure shown. What should be the values of α and θ if the magnitude of P + Q has to be the maximum? 2. shown in the figure. rF .25: problem 2. (a) Find the position vector of point F. respectively. AB . Taking the ˆ ˆ ˆ base vectors ı .16ı + ˆ ˆ 0.vec1.vec1. Two points.24: B y 2m ˆ 2. (c) Find rG using rF .24) problem 2.5 m 1m 3m B z problem 2.Problems for Chapter 2 ˆ 2.26 Find a unit vector along string BA and express the position vector of A with respect to B. = 2 ft.23) 2m B 45o O problem 2. 2. (a) Find the position vector rB of the ball.5 ft.vec1.18 Let two forces P and Q act in the direction shown in the figure. from the vector sum rF = rD + rC/D + rF/c .15 Two forces R = 2 N(0. Find the length of the strings AD and BD using vectors rAD and rBD . in terms of the unit vector. 1. Give examples both ways.29 Let F = 10 lbfı + 30 lbf and W = ˆ −20 lbf . (d) Is sometimes true. ˆ 2.4). ˆ ˆ 2.43: (Filename:pfigure.37 Find the angle between F1 = 2 Nı + ˆ ˆ ˆ 5 N and F2 = −2 Nı + 6 N . H 1 = ˆ ˆ ˆ ˆ (20ı +30 ) kg m2 / s and H 2 = (10ı +15 + ˆ ) kg m2 / s. Find the sum of the forces using a computer.5ı + 1 2 ˆ 0.46 From the figure shown. .8ı +0.2 The dot product of two vectors ˆ ˆ 2. 1 2 2. e.5 in − 4..27: (Filename:pfigure2.60 and λ = 0.41) 2. For each equation below state whether: (a) The equation is nonsense.41 The unit normal to a surface is given as ˆ ˆ ˆ n = 0. 2. Sketch ˆ F and λ and show what their dot product represents.blue. (c) Find the angle this vector makes with the x-y plane. and F = ˆ ˆ 10 Nk 2 3 ˆ ˆ −10 Nı − 100 Nk as a list of numbers (rows or columns). find (a) the angle between ω and 6k H 1 and (b) the angle between ω and H 2 . Find a unit vector in the direction of the net force F + W .48 Find the unit vectors e R and eθ in terms ˆ ˆ of ı and with the geometry shown in figure. find the components of vector rAB (you have to first find this position vector) along (a) the y-axis. ˆ ˆ 2. 0 in. 0 in) and (10 in. What are the x ˆ ˆ and y components of r = 3.31 If a mass slides from point A towards point B along a straight path and the coordinates of points A and B are (0 in.34 Find the dot product of two vectors F = ˆ ˆ ˆ ˆ ˆ 10 lbfı −20 lbf and λ = 0. F = −20 N + 2 Nk . ˆ λ F = 100 N 10o x (Filename:pfigure2. ˆ ˆ 2. Find the components of W along ˆ ˆ λ and n.4 y C 2.45: ˆ λ W O 30o B (Filename:pfigure2. What is the angle between the force and the z-axis? ˆ ˆ 2. 2.vec1.95 ink .2. find the angle that a 1000 N normal force makes with the direction of weight of the block. If so. ˆ ˆ 2. Find the dot product of rA with λ = √ 3ˆ 2 ı CONTENTS 2.28 Express the vector rA = 2 mı − 3 m + ˆ 5 mk in terms of its magnitude and a unit vector indicating its direction.74ı + 0.44 (a) Draw the vector r = 3.0) to point B(0. ˆ ˆ (a) Show that λ and λ are unit vectors. You may use trivial examples. λ and n are unit vectors parallel and perpendicular to the surface ˆ AB. (Hint: c = a + b .43 Use the dot product to show ‘the law of cosines’. z 2m A B 1m y 2. y a problem 2. 5 in.42 Vector algebra.866 .27) ˆ + 1.47 The net force acting on a particle is F = ˆ ˆ 2 Nı + 10 N . find the unit vector ˆ λAB directed from A to B along the path. (c) Is never true.39 A force acting on a bead of mass m is given ˆ ˆ ˆ as F = −20 lbfı + 22 lbf + 12 lbfk .36 From the figure below. c2 = a 2 + b2 + 2ab cos θ. 2. find the y. and ˆ (b) along λ. c · c = c · c ) B 30o h O problem 2.80ı + 0. i.1) ˆ ˆ 2.38 A force F is directed from point A(3. For θ = 30o.35 The position vector of a point A is rA = ˆ ˆ 30 cmı . If the x-component of the force is 120 N.vec1. Why? Give an example.30 Let λ = 0.33) 30o problem 2. A force W = −50 N acts on the block. (b) Find the angle this vector makes with the z-axis.33 Express the unit vectors n and λ in terms ˆ ˆ of ı and shown in the figure.45 In the figure shown.46: (Filename:pfigure2. Find the components of this force in another coordinate system with baˆ ˆ ˆ ˆ sis vectors ı = − cos θ ı + sin θ and = ˆ ˆ − sin θ ı − cos θ . respectively.and z-components of F .27) problem 2. respectively. find the component ˆ of force F in the direction of λ. also.67 . 1 2 (b) Is the sum of these two unit vectors also a unit vector? If not. why? (b) Is always true.2.5 ftλ? ∗ 3m 2m 30o ˆ λ x problem 2. 2. 2. 10 in). a) A + B = B + A b) A + b = b + A c) A · B = B · A d) B /C = B/C A ˆ n ˆ ˆ ı problem 2.36: ˆ 2.6 . Why? Give an example.vec1.40 Given ω = 2 rad/sı + 3 rad/s .2.42) y ˆ λ ˆ n ˆ ˆ ı θ x (Filename:efig1.33: e) b/A = b/A f) A = (A·B )B +(A·C )C +(A·D )D 2.32 Write the vectors F1 = 30 Nı + 40 N − ˆ . If the weight of a block ˆ on this surface acts in the − direction.2. 2 c b θ x 2.vec1. sketch the vector F and show its components in the two coordinate systems.5 inı + ˆ ˆ 3. ˆ ˆ ˆ ˆ ˆ 2.0 ftn − 1. and express the the net force in terms of the unit vector. then find a unit ˆ ˆ vector along the sum of λ and λ . 2.59: 2m x (Filename:pfigure2. Why? Give an example. why? (b) Is always true.5 m problem 2. Now use the program to find the components ˆ ˆ ˆ of F = (2ı + 2 − 3k ) N along the line ˆ ˆ ˆ rAB = (0. n=0 o o y F = 20 N 2.60: B A 4 problem 2.vec2. ˆ ˆ 2. Why? Give an example. moment. OA = AB = 2 m.vec2. Using a computer generate the required vectors and find the sum a) B × C = C × B b) B × C = C · B c) C · (A × B ) = B · (C × A) 5 d) A ×(B × C ) = (A · C )B −(A · B )C 2.60 Find the sum of moments of forces W and T about the origin. If F always acts normal to the arm AB. and γ using a computer.Problems for Chapter 2 ˆ What are the componets of W along e R and ˆ eθ ? 2.4) 2. ı · .1) 1. (c) Is never true. Find the moment of F about O.45) 30o 30o 4 x b b x O /2 W θ x (Filename:pfigure2. 45o O problem 2. and · k .61 Find the moment of the force x a ˆ b = 4 2 a x a) about point A b) about point O. given that W = 100 N.51 Let F1 = 30 Nı + 40 N − 10 Nk . and F3 are as given in Problem 2.55 Let rn = 1 m(cos θn ı + sin θn ).5ı − 0.2) y 2 P θ2 1 (a) y a 2 60o 4 (b) b a 3 y 105o 4 ˆ λ2 b x ˆ λ1 x (d) y 45o 5 4 2.48: (Filename:pfigure2. T = 120 N. with θ = 1 and θ0 = 45 . 2. 1 2. Give examples both ways. ˆ (b) find the component of rP alon λ .59 Find the moment of the force shown on the rod about point O. would increasing θ increase the magnitude of the moment? In particular.52 A vector equation for the sum of forces results into the following equation: √ F R ˆ ˆ ˆ ˆ ˆ (ı − 3 ) + (3ı + 6 ) = 25 Nλ 2 5 ˆ ˆ ˆ where λ = 0.2) α = 30o ˆ ˆ ˆ 2. y (c) y 4 θ1 problem 2. Solve for α.2 + 0. F = 50 N (Filename:pfigure. F2 .57 Vector algebra. = 4 m.49: a T a 45o /2 x (Filename:pfigure2. θ ˆ ˆ ı W 44 ˆ eθ ˆ eR problem 2.58 What is the moment M produced by a 20 N force F acting in the x direction with a lever ˆ arm of r = (16 mm) ? 2.3 Cross product.vec2. 2m problem 2. If the sum of all these forces must equal zero.61: A (Filename:pfigure2.2) y b (2.50 What is the distance between the point A and the diagonal BC of the parallelepiped shown? (Use vector methods.62 In the figure shown.vec1. given that θ = 45o . and F = F ı + F − ˆ −20 N + 2 Nk 3x ˆ 3y ˆ 3 ˆ F k .3) 2. Find the scalar ˆ equations parallel and perpendicular to λ.1k ) m. The force F = 40 N acts perpendicular to the arm AB.56: 2. For each equation below state whether: (a) The equation is nonsense. (d) Is sometimes true. and θ = 30o .vec1. F2 = ˆ . where θn = θ0 − n θ .blue.32.vec2. β. find the required scalar equations to solve for the components of F3 .-1) (2. If so.) (e) y b 2 2 3 3 (f) b y 3 3 2 problem 2. 3z a a= x 3ı + ˆ ˆ (-1.54 Write a computer program (or use a canned program) to find the dot product of two 3-D vectors.44) ri . You may use trivial examples. where F1 . Test the program by comˆ ˆ ˆ ˆ ˆ ˆ puting the dot products ı · ı .954 .56 Find the cross product of the two vectors shown in the figures below from the information given in the figures.50: 1 C 3 2.53 Let α F1 + β F2 + γ F3 = 0 .30ı − 0. what value of θ will give the largest moment? .49 Write the position vector of point P in ˆ ˆ terms of λ and λ and 1 2 (a) find the y-component of rP .-1) x O (Filename:pfigure2. and moment about an axis 2.3) (g) y (h) (-1.2. 73: (Filename:efig1. O problem 2.71 A force F = 20 N − 5 Nk acts through a point A with coordinates (200 mm.69 Why did the chicken cross the road? ∗ 2. ∗ z B A 1m C 1m 1m y 2.vec2. Find and draw a vector normal to the surface ACB.2. here is a function file that calculates dot products in pseudo code. L = ˆ ˆ (3.62: x (Filename:pfigure2.4k ) m. and b) the root of the pole. 2 = 0.63: B C 2 kN x (Filename:pfigure2.4 m.0 + 0.14) ˆ ˆ ˆ c) ω = (ı − 1. As a model.2.0k ) kg m/s. ∗ .73 Find a unit vector normal to the surface ABCD shown in the figure. Give your answer in terms of the variables shown in the figure. z B 45o 45o A x 60o problem 2. What is the moment M (= r × F ) of the force about the origin? 2.8 m. a triangular plate ACB.12) board 6" A 3' B 10' 1.74 If the magnitude of a force N normal to the surface ABCD in the figure is 1000 N.64: 2.2.6 y θ A F B 2. ˆ ˆ ˆ a) r = 2. 2.65: (Filename:pfigure2. if the weight is assumed to act normal to the arm OA (a good approximation when θ is very small).76: (Filename:efig1. the plate makes an angle of 60o with the x-axis. rotates about the z-axis.66 Find the sum of moments due to the two weights of the teeter-totter when the teetertotter is tipped at an angle θ from its vertical position.8) 2. Find the moment of the force about a) the point of the ring attachment to the board (point B). -100 mm). and 3 = 0.64 During a slam-dunk. ω × r =? ˆ ˆ 2.67: (Filename:pfigure2. The input to the function should be the components of the two vectors and the output should be the components of the cross product. Find a unit vector n normal to the surface. z(3)=a(3)*b(3).vec2.66: z A D 4" x 5" problem 2. write N as a vector.7) 2. r × L =? (Filename:pfigure2. CONTENTS A α α θh 2. %program definition z(1)=a(1)*b(1). r × F =? ˆ ˆ ˆ b) r = (−ı + 2.11) 2.72 Cross Product program Write a program that will calculate cross products.77 What is the distance d between the origin and the line AB shown? (You may write your solution in terms of A and B before doing any arithmetic).0 ftı + 3.vec2.63 Calculate the moment of the 2 kNpayload on the robot arm about (i) joint A.65 During weight training. attached to rod AB. A O problem 2. 300 mm.70 Carry out the following cross products in different ways and determine which method takes the least amount of time for you.1 m.0 lbfk .vec2. a basketball player pulls on the hoop with a 250 lbf at point C of the ring as shown in the figure.vec2.5' basketball hoop 2. an athelete pulls a weight of 500 Nwith his arms pulling on a hadlebar connected to a universal machine by a cable. 15o 250 lbf O problem 2.6) 2. if 1 = 0.0 ft − 1. z(2)=a(2)*b(2). At the instant shown. Find the moment of the force about the shoulder joint O in the configuration shown. F = ˆ ˆ −0. and (ii) joint B.76 In the figure. point O.9) 45o O problem 2. r = (10ı − ˆ ˆ 2 + 3k ) in.5 − 2.74: (Filename:efig1.5 ) rad/s. 2. C 1m y problem 2.vec2.3 lbfı − 1.5) B O W C OA = h AB = AC = problem 2. w=z(1)+z(2)+z(3).5 ftk .68 What do you get when you cross a vector and a scalar? ∗ 2.67 Find the percentage error in computing the moment of W about the pivot point O as a function of θ . B C 5" y y A 30o 1 2 3 W (Filename:pfigure2.75 The equation of a surface is given as z = ˆ 2x − y.10) θ W x 1m D 1m problem 2. 85 Replace the forces acting on the particle of mass m shown in the figure by a single equivalent force.81 Vector Calculations and Geometry. B. 4) 2 3 5 4 D 1 problem 2.Problems for Chapter 2 z 1 A A c) Write both F 1 and F 2 as the product of their magnitudes and unit vectors in their directions.rp1) 2.77: B (Filename:pfigure. how close is the closest point on this line to the origin?) e) Is the origin co-planar with the points A.] ∗ B (3. 3). 1) 7 x y problem 2. The 7 N force F 2 is along the line OB. ∗ 2. c) Find a unit vector perpendicular to the plane ABC. b) Find a unit vector in the direction of ˆ OA. and r C = (7. 12)m. problem 2. ∗ h) Repeat the last problem using either a different reference point on the axis DC or the line of action OB. 5.78 What is the perpendicular distance between the point A and the line BC shown? (There are at least 3 ways to do this using various vector products.2. d) How far is the infinite line defined by AB from the origin? (That is.78: ˆ ˆ ˆ 2.3. Point A has x yz coordinates (0.1.3) x A (5.rp2) ˆ problem 2. 8. Point B has x yz coordinates (4. a) Use the vector dot product to find the angle B AC (A is at the vertex of this angle). F 1 = (−3ı + 2 + 5k ) N acting at a point P whose position is given by ˆ ˆ ˆ r P/O = (4ı − 2 + 7k ) m. how many ways can you find?) y 2. b) Find the distance from this infinite plane to the point D. and C in the figure define a plane. 1) C (3. and C? 2. .4 Equivalant force systems and couples 3 B A z 5m A B F2 (Filename:pfigure. a) Find a unit normal vector to the plane. B.3.79 Given a force.84: (Filename:pfigure2. c) Find the force F which is 5N in size and is in the direction of OA. Does the solution agree? [Hint: it should. The 5 N force F 1 is along the line OA.blue. The result does not depend on which point on the axis is used or which point on the line of action of F is used. d) What is the angle between OA and OB? e) What is r B O × F ? f) What is the moment of F about a line parallel to the z axis that goes through the point B? 2. 6). 12)m. r B = (4. 7.80 Drawing vectors and computing with vectors. 5) 1 (0. 2.85: ˆ ı b) Find a unit vector in the direction OA. 4.81: (Filename:p1sp92) 8N problem 2.). 2T T T 45 o o 30 m mg (Filename:pfigure2.82 A. 5. a) Find a unit vector in the direction OB.83: (Filename:pfigure.s95q2) 2. The point O is the origin. 2. ∗ 2. ∗ d) What is the angle AOB? ∗ e) What is the component of F 1 in the x-direction? ∗ 7 c) What are the coordinates of the point on the plane closest to point D? ∗ ˆ k ˆ ı ˆ O B z 5 4 d 1 y f) What is r D O × F 1 ? ( r D O ≡ r O/D is the position of O relative to D. OB. call it λ O A . 2.86 Find the net force on the pulley due to the belt tensions shown in the figure. what is the moment about an axis through the origin O with 2 ˆ 1 ˆ ˆ direction λ = √ + √ ? 5 5 6N 4 O 4m x D F1 C 3m y 3 P 10 N ˆ ˆ ı problem 2. ∗ ∗ 2. B.2) ˆ ˆ ı C 0 2 3 x 2. b) Use the vector cross product to find the angle BC A (C is at the vertex of this angle). and C are located by position vectors r A = (1. and AB. 9).84 Find the net force on the particle shown in the figure. a) Make a neat sketch of the vectors OA. 5.83 Points A.blue.) ∗ g) What is the moment of F 2 about the axis DC? (The moment of a force about ˆ an axis parallel to the unit vector λ is ˆ ·( r ×F ) where r is defined as Mλ = λ the position of the point of application of the force relative to some point on the axis. 1. What is the magnitude of the moment Mc ? (c) F problem 2. Find an equivalent force couple system at point B.5 kg.rp9) C F R 2.88: (Filename:pfigure2.3. and Mc = 20 N·m. 4N problem 2. located at coordinates (0.0) (1. −1 m). 60 N A 0.3.4 kg 0.1.2 m 20 N 0. if the equivalent system is to act at a) point B.5.) Check your program on Problems 2.rp7) 2.97: (Filename:pfigure3.87 Replace the forces shown on the rectangular plate by a single equivalent force. 30 N 30 N 40 N 6 N·m 0. 2.92 In Fig. ∗ 2. (It is possible to write a single program for both 2-D and 3-D cases.2 m 1. 2.rp8) Locate the center of mass of the system.0. three different force-couple systems are shown acting on a square plate. Locate the center of mass of the structure.rp3) 10 N ˆ ı 50 N 2.88 Three forces act on a Z-section ABCDE as shown in the figure.91 Find an equivalent force-couple system for the forces acting on the beam in Fig.0 kg coordinates (in m) (1.8 2. Where should this equivalent force act on the plate and why? A problem 2. Find an equivalent force-couple system acting at point C. (1 m. 1 m).97 Find the center of mass of the following composite bars.rp7) . Identify which force-couple systems are equivalent.5 m Cx C ˆ problem 2.91: 2.rp5) 2. D 2m 2 kN (Filename:pfigure2.94 An otherwise massless structure is made of four point masses.90: (Filename:pfigure2. −1 m).6 m problem 2.3. It is found that if the given force-couple system is replaced by a single horizontal force of magnitude 10 N acting at point A then the net effect on the machine part is the same. CONTENTS A 30o 50 N 20 cm B MC 30 cm problem 2.cm.rp10) Cy B MC C 1. write separate programs for the two cases if that is easier for you.4 kg 0.rp4) b) point D.3. 3m and 4m.5 m B C D 40 N 100 N 0. respectively.4 kg 1.0) (2.95.86: (Filename:pfigure2.0) (1. 3 2F 0.3.3. ??.3.4 kg 0. C y = 40 N. ??. and (0.2 m 20 N 30 N 30 N 20 N (a) (b) problem 2.0.95 3-D: The following data is given for a structural system modeled with five point masses in 3-D-space: mass 0.89: (Filename:pfigure2.3. Point C lies in the middle of the vertical section BD.92: (Filename:pfigure2.93: (Filename:pfigure2. The input to the program should be a table (or matrix) containing individual masses and their coordinates. Each composite shape is made of two or more uniform bars of length 0. Find an equivalent force-couple system acting on the structure and make a sketch to show where it acts.5 Center of mass and center of gravity 2. 1 m).89 The three forces acting on the circular plate shown in the figure are equidistant from the center C.93 The force and moment acting at point C of a machine part are shown in the figure where Mc is not known.90 The forces and the moment acting on point C of the frame ABC shown in the figure are C x = 48 N.rp6) problem 2.94 and 2. (1 m.0) (2.3) 300 mm A D 200 mm 6N B C 2. 2 kN A B C 2m 1 kN problem 2.5.1.87: 5N (Filename:pfigure2. m.5 m E 2.2 m and mass 0. 2m.96 Write a computer program to find the center of mass of a point-mass-system. Problems for Chapter 2 2.] 9 (a) y m = 2 kg r = 0.98 Find the center of mass of the following two objects [Hint: set up and evaluate the needed integrals.5 m O (b) y m = 2 kg x r = 0.rp8) x 2.cm.99: (Filename:pfigure3.5 m O problem 2. (a) 200 mm x 200 mm (b) r = 100 mm 100 mm problem 2.cm.99 Find the center of mass of the following plates obtained from cutting out a small section from a uniform circular plate of mass 1 kg (prior to removing the cutout) and radius 1/4 m.98: (Filename:pfigure3.rp9) . 9a A 5a C m 6a frictionless ˆ B G ˆ ı problem 3.10 CONTENTS Problems for Chapter 3 x Free body diagrams F = 50 N x y frictionless h 3. made up of a mass m attached at the end of a rigid massless rod of length . Draw a free body diagram of the rod. Draw another that takes account of the slight drag force of the earth’s atmosphere on the satellite. The rigid body pendulum in the figure is a uniform rod of mass m.47. and evaluate rod. Draw a free body diagram of the pendulum (mass and rod system) when the pendulum is slightly away from the vertical equilibrium position. Draw a free body diagram of the satellite. Its speed is v and its distance from the center of the earth is R. Draw a free body diagram of the rod.2: (Filename:pfigure2. Evaluate the left hand side of the linear momentum balance equation ( F = m a ) as explicitly as possible. mass m problem 3.1 Free body diagrams 3. There is gravity and negligible friction.11 A pendulum.4 Draw a free body diagram of mass m at the instant shown in the figure.6) problem 3.1.5 A 1000 kg satellite is in orbit. Draw a free body diagram of the rod. hangs in the vertical plane from a hinge.7 FBD of rigid body pendulum.8 A thin rod of mass m rests against a frictionless wall and on a frictionless floor. Draw a free body diagram of the system. There is gravity. (Filename:pfig2.s94h2p5) problem 3.fbd) 3.2. Draw a free body diagram of the rod.s94h2p1) g θ uniform rigid bar.4: (Filename:pfig2.8: (Filename:ch2. Draw a free body diagram of the mass with a little bit of the cables. The groove that the disk sits in is part of an assembly that is still. Draw a free body diagram of the disk.6 The uniform rigid rod shown in the figure hangs in the vertical plane with the support of the spring shown.15 and 6.7: (Filename:pfigure2. 3. There is gravity.) problem 3.rp5) problem 3. The pendulum is attached to a spring and a dashpot on each side at a point /4 from the hinge point.5: (Filename:pfigure.5) θ L 3.2. For the simple pendulum shown the “body”— the system of interest — is the mass and a little bit of the string.3 Simple pendulum.1 How does one know what forces and moments to use in a) the statics force balance and moment balance equations? b) the dynamics linear momentum balance and angular momentum balance equations? m = 10 kg 3. . v m R 3.9: 3. set up a suitable F for the coordinate system. Identify the unknowns in the expression.rp1) B 3.2 A point mass m is attached to a piston by two inextensible cables.rod.9 A uniform rod of mass m rests in the back of a flatbed truck as shown in the figure. m k 3. A L G θ problem 3. 3.mass) /3 3.pend.10 A disc of mass m sits in a wedge shaped groove.rp1) r y θ1 x problem 3.6: (Filename:pfig2. (See also problems 4. θ2 problem 3.10: (Filename:ch2.3: (Filename:pfigure.suspended.1. m k k problem 3. Draw a free body diagram of the wheel. At an instant in time t during the motion. How many unknown forces does each equation contain? 3. a) Assuming the block remains on the floor.1.17: G A problem 3. What would be different in the expressions obtained if the wheel were slipping? 3.13 A block of mass m is sitting on a frictionless surface at points A and B and acted upon at point E by the force P.32.20 A point mass of mass m moves on a frictionless surface and is connected to a spring with constant k and unstretched length . Assume that the wheel rolls to the right without slipping.16 Assume that the wheel shown in the figure rolls without slipping.12 The left hand side of the angular momentum balance (Torque balance in statics) equation requires the evaluation of the sum of moments about some point.18 A block of mass m is sitting on a frictional surface and acted upon at point E by the horizontal force P through the center of mass.16: F = 10 N m = 20 kg c k (Filename:pfig2.15: (Filename:pfig2. At the instant of interest.12: (Filename:pfig2. b) Draw free body diagrams of the mass and spring separately at the instant of interest.rp8) problem 3. (Filename:pfig2. Draw a free body diagram of the block including the wheel. Draw a free body diagram of the rod shown in the figure and comMO as explicitly as possible.15 A small block of mass m slides down an incline with coefficient of friction µ.20: (Filename:ch2. a) Draw a free body diagram of the of the mass and spring together at the instant of interest.1.2. 3. There is gravity.11: c k P 2d A problem 3.rp4) 3. The spring is fixed to the wall on one end.2) α problem 3. Draw a free body diagram of the mass assuming it is moving to the left at the time of interest.13: C m = 20 kg x m 3.rp5) 3. There is gravity. draw a free diagram of the block.1.17 A compound wheel with inner radius r and outer radius R is pulled to the right by a 10 N force applied through a string wound around the inner wheel.14: (Filename:pfig2.21 FBD of a block. 11 2b C b D d E G m µ (Filename:pfig2. The belt moves to the right at a constant speed v0 . the mass has just been released at a distance x to the right from its position where the spring is unstretched. 2b C b E D d P 2d B (Filename:ch2.19: 1 L/2 3 C L problem 3.3. Now pute compute M C . The block is resting on sharp edge at point B and is supported by a small ideal wheel at point A. . Draw a free body diagram of the block.19 A spring-mass model of a mechanical system consists of a mass connected to three springs and a dashpot as shown in the figure. (See also problem 5. assuming the block is sliding to the right with coefficient of friction µ at point B. The coefficient of friction between the mass and the belt is µ.18: B (Filename:ch2. Draw a free body diagram of the block.rp3) m = 5 kg O r C R P problem 3. the block has speed v.2.10) k m µ problem 3.) F = 10 N r R P problem 3.1) 3. There is gravity. 3.Problems for Chapter 3 3.1. The wheels against the wall are in tracks (not shown) that do not let the wheels lift off the wall so the mass is constrained to move only in the vertical direction.14 A mass-spring system sits on a conveyer belt. The block of mass 10 kg is pulled by an inextensible cable over the pulley.rp5) 1/4 k c 3/4 m problem 3. b) Draw a free body diagram of the pulley and a little bit of the cable that rides over it.rp6) 3. Draw a free body diagram of the system. Draw a free body F and diagram of the wheel and evaluate MC .rp2) (Filename:pfig2. Ignore gravity.25 In the system shown.22: (Filename:pfigure. and c) the whole system.2.4. which you can assume to be taught..3) 3.28 FBD’s of simple pendulum and its parts. The rod is hinged at one end (with a frictionless hinge) and has no friction at the contact with the cart.pend. O θ1 massless m m k 2k k θ2 problem 3.22 A pair of falling masses.21: (Filename:pfigure2.1. c) Draw a free body diagram of the rod and pendulum bob together.block. Two frictionless blocks sit stacked on a frictionless surface.30 See also problem 11. evaluate shown in terms of the given quantities and unknown forces (if any) on the bar. g m1 F A m2 θ (Filename:pfig2. The cart rolls on massless wheels that have no bearing friction (ideal massless wheels). b) the cart.fbd) problem 3. F problem 3.27 A uniform rod rests on a cart which is being pulled to the right. k1 k2 k3 problem 3. Draw FBD’s of a) the rod.29 For the double pendulum shown in the figF and MO at the instant ure.29: m problem 3.31 For the system shown in the figure draw free body diagrams of each mass separately assuming that there is no relative slip between the two masses. massless m µ = 0. and c) the whole system. Draw free body diagrams of each rod separately.27: F problem 3. Draw free body diagrams of a) mass A with a little bit of string. There is gravity. (Filename:pfigure.rp2) B 30o A m 3. The simple pendulum in the figure is composed of a rod of negligible mass and a pendulum bob of mass m.2 k B problem 3. Repeat the procedure for the system consisting of both masses.23: x1 k4 m1 m2 x2 3. Assume that the three masses are displaced to the right by x1 . A µ = 0. b) Draw a free body diagram of the rod. 3.30: (Filename:ch2. A force F is applied to the top block.24 The figure shows a spring-mass model of a structure. a) Draw a free body diagram of the two blocks together and a free body diagram of each block separately.rp7) .1. A string. a) Draw a free body diagram of the pendulum bob.25: (Filename:pfig2.2. The vertical stiffness of the system is modeled by three springs as shown in the figure.2 3.1. Draw free body diagrams of each mass and evaluate F in each case.28: (Filename:pfigure2. b) mass B with a little bit of string.12 x x m = 10 kg y frictionless problem 3. 3. [This problem is a little tricky and there is more than one reasonable answer. Draw free body diagrams of each mass separately and then the two masses together.rp3) (Filename:pfig2.s94h2p4) (Filename:pfig2.23 A two-degree of freedom spring-mass system is shown in the figure.pulley) CONTENTS F = 50 N 3.rp6) 3.] 3.rp4) B problem 3.simp.26: m problem 3. h g ˆ ˆ ı k B A θ rigid. no relative slip).s94h2p6) x1 k5 m k1 k3 M x2 k4 m x3 k6 3.24: (Filename:pfig2.rp10) (Filename:pfig2.2.1. x2 and x3 from the static equilibrium configuration such that x1 < x2 < x3 . Draw a free body diagram of mass A and evaluate the left hand side of the linear momentum balance equation.26 Two identical rigid rods are connected together by a pin. Two masses A & B are spinning around each other and falling towards the ground. connects the two masses. assume that the two masses A and B move together (i. A snapshot of the system is shown in the figure.e.31: F k2 problem 3. fbd) nW φ a problem 3.24) 3.Problems for Chapter 3 3. b) the upper arm.48m E 1m 1m A C B 32m 1m ˆ k D ˆ ı ˆ O φ1 A problem 3. Draw a free body diagram of the plate. and wheel.1) m o massless δ 2m (Filename:pfig2. 3. blocks. Draw free body diagrams of: a) the system consisting of the box. Draw a FBD of a) the ball.pulley.35 An imagined testing machine consists of a box fastened to a wheel as shown. b) the system of box and blocks.38: (Filename:efig2. Draw a FBD of the mass just after it is released.blue. upper arm. pivot A 1 B θ 2 ˙ θ 3. Draw free body diagrams of various subsystems in a typical configuration. 3. The machine and blocks are set in motion when θ = 0o . There is gravity. forearm and a hand) that are connected with hinges. G H .34: (Filename:efig2. An arm throws a ball up. There is gravity.5 kgmass at its center. with constant ˙ θ = 3 rad/s. cord 2. nW . 3.36 Free body diagrams of a double ‘physical’ pendulum. Write the equation of angular momentum balance about the shoulder joint A. The top prism weighs W and the lower one.23) D 3. It is supported by two stretched springs. e) the box and blocks separately. At the moment of interest the shelf is in a rocket in outer space and accelerating at 10 m/s2 in the k direction.2) d) the system of box.32: (Filename:pfigure. c) the system of blocks and cords. and both sticks in arbitrary configurations. c) the fore-arm.33 In the slider crank mechanism shown. problem 3.blue. . A double pendulum is a system where one pendulum hangs from another. A and B are connected together by cords 1 and 2 as shown. b) Repeat part (a) but use the simplifying assumption that the upper bar has negligible mass.rp9) 3. A shelf with negligible mass supports a 0. At each hinge there are muscles that apply torques between the links. Draw free body diagrams of a) the system of prisms and b) each prism separately.33: 1m 30 problem 3.40 A massless triangular plate rests against a frictionless wall at point D and is rigidly attached to a massless rod supported by two ideal bearings.39 Hanging a shelf.37: (Filename:pfigure2. 3. Two identical masses. The shelf is supported at one corner with a ball and socket joint and the other three corners with strings. d) the hand. The mass is pulled down the plane by an amount δ and released. ball.28. block B.34 FBD of an arm throwing a ball.2. Draw a FBD of the shelf.3D.35: (Filename:pfigure. a) Draw a free body diagram of the lower stick. A ball of mass m is fixed to the centroid of the plate. The shelf is in the x y plane.32 Two frictionless prisms of similar right triangular sections are placed on a frictionless horizontal plane. the shoulder (fixed to the wall).37 The strings hold up the mass m = 3 kg. and e) the whole arm (all four parts) including the ball. Draw a free body diagram of the mass.38 Mass on inclined plane.39: (Filename:ch3.52.1. The floor of the box is frictionless. 1m k k 2m Crank of mass m ω θ problem 3.14) (Filename:pfigure. the upper stick.1. and a portion of cord 1 and. draw a free body diagram of the crank and F and MO as explicitly as posevaluate sible.36: φ2 problem 3.1. A block of mass m rests on a frictionless inclined plane.s94h2p3) B C problem 3. and rod as a system. A crude model of an arm is that it is made of four rigid bodies (shoulder. The box always moves so that its floor is parallel to the ground (like an empty car on a Ferris Wheel). evaluating the left-hand-side as explicitly as possible. 13 z B D C 1m 1m y A x 4m 4m 3m b φ W problem 3. c) What would be different in the free body diagram of the rod if the ball-andsocket was rusty (not ideal)? ω ball and socket O z L R C y weld x problem 3. There is gravity.41 An undriven massless disc rests on its edge on a frictional surface and is attached rigidly by a weld at point C to the end of a rod that pivots at its other end about a ball-andsocket joint at point O.14 d D z a A problem 3. a) Draw a free body diagram of the disk and rod together.40: (Filename:ch2.7) . b) Draw free body diagrams of the disc and rod separately.9) CONTENTS d=c+(1/2)b y e h G B b x c 3.41: (Filename:ch2. y + 2π z = 21.hanging. 4.9 What is the solution to the set of equations: x x x x +y+z+w −y+z−w +y−z−w +y+z−w = = = = 0 0 0 2? for the x . Assume g = 10 N/ m. ∗ 4.3: m = 2 kg L=3m (Filename:pfigure2. y and z from the following equations with r as an input variable.2 N small blocks each of mass m hang vertically as shown. a) 5x + 2r y + z = 2 b) 3x + 6y + (2r − 1)z = 3 c) 2x + (r − 1)y + 3r z = 5. F2 . and redundancy 4.a) 4.blue.8 m/s2 − ˆ a k in a rotated x y z coordinate system. F C 3m Can the student solve for F1 . rigidity.10 What is the stiffness of two springs in parallel? 4. 4.5 Write the following equations in matrix form to solve for x. and z directions. Find the solutions for r = 3.13 Optimize a triangular truss for stiffness and for strength and show that the resulting design is not the same.1 To evaluate the equation F = m a for ˆ some problem. find the scalar equations z ˆ g ˆ ı B 4m A m b) 2F1 + c) 5 F2 + 2 3F 2 2 =0 √ 2F3 = 0. Find the tension Tn in string n. and z: 2x − 3y + 5 = 0.98. a student writes F = Fx ı − ˆ in the x yz coordinate ˆ (Fy − 30 N) + 50 Nk ˆ ˆ system. y .4 Internal forces 4. x θ problem 4.5 c) 2x1 + 4x2 + 15x3 = 7.7 Write computer commands (or a program) to solve for x. ∗ 0=0 k = 5 N/m 4.Problems for Chapter 4 15 Problems for Chapter 4 Statics 4.5 Springs m n=4 n=N-2 m n=N-1 m n=N m problem 4.11 What is the stiffness of two springs in series? D=4m g = 10 m/s y 4. the following problems can be skipped. 4. = − sin 60o ı + ˆ ˆ ˆ cos 60o and k = k .3 See also problem 7. 1 x − 2y + π z − 11 = 0.12 What is the apparant stiffness of a pendulum when pushed sideways.8 An exam problem in statics has three unknown forces. y.4 What force should be applied to the end of the string over the pulley at C so that the mass at A is at rest? 1 a) F1 − 1 F2 + √ F3 = 0 2 2 4.a) 4.6 Are the following equations linearly independent? a) x1 + 2x2 + x3 = 30 b) 3x1 + 6x2 + 9x3 = 4.5. 4. If computer solution is not going to be used. m m n=1 n=2 n=3 4. 3 4.5 m/s2 ı + 1.masses) 4. but a = 2. Your program should display an error message if. If ˆ ˆ ˆ ˆ ˆ ı = cos 60o ı + sin 60o . the equations are not linearly independent. for a particular r . and F3 uniquely from these equations? ∗ 2m problem 4. .2.99.80. and 5.3 Advanced truss analysis: determinacy.2: (Filename:pfigure2. Find θ for static equilibrium. A student writes the following three equations (he knows that he needs three equations for three unknowns!) — one for the force balance in the x-direction and the other two for the moment balance about two different points.2 Elementary truss analysis The first set of problems concerns math skills that can be used to help solve truss problems. A zero length spring (relaxed length 0 = 0) with stiffness k = 5 N/m supports the pendulum shown. 4. connected by N inextensible strings.4: 3m (Filename:f92h1p1.1 Static equilibrium of one body 4. 86. Assume the gear-train is at rest. .25 m. At the output. or magnetic attraction).19 Two racks connected by three gears at rest.15 (See also problem 6. There is gravity. g.21 Assume a massless pulley is round and has outer radius R2 .17 except for the location of the connection point of the string to the reel.3.a) problem 4. The inner cylinder of the reel has radius r = 1 R. what must the tangent of the angle φ of the reaction force at C be. What is FB ? m2 60o (Filename:pulley4. and the length of each link is = 0. It slides on a shaft that has radius Ri . the line of action of the reaction force at C must pass through this point. ∗ (Filename:summer95p2. a 100 lbf force is applied to gear A. Assume the system of gears is at rest. At the input to a gear box. your solution gives = ˆ 0 and F C = Mg and for θ = π .The ˆ ˆ it gives M C negative mass ratio is impossible since mass cannot be negative and the negative normal force is impossible unless the wall or the reel or both can ‘suck’ or they can ‘stick’ to each other (that is.c) problem 4. it gives 2 m ˆ ˆ M = 2 and F C = Mg(ı + 2 ). and the mass ratio required for static equilibrium of the reel? ∗ Check that for θ = 0. g. massless rack y FB = ? x RC d E RE no slip massless rack (Filename:ch4.5. A 100 lbf force is applied to one rack. E.77 and ??. D.a) Check that for θ = 0.18 This problem is identical to problem 4.blue.15: (Filename:ch2. ∗ c) Find the corresponding force on the reel at its point of contact with the slope.5.17 See also problem 4. There are frictionless hinges at points A. m M 4.47. measured with respect to the normal to the ground.a) B r y θ1 x problem 4.2. R.19: Two racks connected by three gears. and ∗ c) the corresponding force on the reel at its point of contact with the slope. [Harder] Draw a careful sketch and find a point where the lines of action of the gravity force and string tension intersect. C. R. measured with respect to the normal to the slope? Does this answer agree with that you would obtain from your answer in part(c)? ∗ e) What is the relationship between the angle ψ of the reaction at C. a) Find the ratio of the masses so that the system is at rest.b) M g P R r massless A RA b C θ2 ˆ ˆ ı no slip G 100 lb C m θ (Filename:pfigure2. what is the compression of the spring? 4.a) 4. a) What is the relation between the two tensions when the pulley is turning? You may assume that the bearing shaft touches the hole in the pulley at only one point. R.17: no slip RA RC C A FA = 100 lb problem 4.19.18: 4. Assuming that the spring of constant k = 500 N/m is uncompressed when θ = π radians. m = −2 and F = Mg(ı − 2 ).18. There is no slip between the reel and the slope. adhesion. The central collar has mass m/4. point P.14: 4. 4. g R r G P C θ M massless ˆ ˆ ı m1 A B 30o g m (Filename:pfigure2.16: Two gears.b) problem 4. point C. The 2 slope has angle θ. and θ . and θ.6 Structures and machines 4. See also problem 7. ∗ m A θ E problem 4. m. ∗ b) the corresponding tension in the string. the machinery (not shown) applies a force of FB to the other rack. and θ. ∗ . At the output. the mass of each ball is m = 5 kg. The slope has angle θ . Using this information.47.20 In the flyball governor shown. Find the ratio of the masses m 1 and m 2 so that the system is at rest. point C. ∗ b) Find the corresponding tension in the string.3. There is gravity. in terms of M. F where the links are connected. In terms of M.47. FB = ? RB B 4. 2 There is no slip between the reel and the slope.) What are the forces on the disk due to the groove? Define any variables you need. the machinery (not shown) applies a force of FB to the output gear. A reel of mass M and outer radius R is connected by a horizontal string from point P across a pulley to a hanging object of mass m. in terms of M. provide some sort of suction.16 Two gears at rest.18 and 6.14 See also problems 6.3. Assume the two ends of the line that are wrapped around the pulley are parallel. B. The inner cylinder of the reel has radius r = 1 R. your solution gives m π ˆ M = 0 and F C = Mg and for θ = 2 . find: a) the ratio of the masses so that the system is at rest. and friction angle φ defined by µ = tan(φ).20: m/4 B θ m F k D C fixed (Filename:pg131. A reel of mass M and outer radius R is connected by an inextensible string from point P across a pulley to a hanging object of mass m.16 CONTENTS d) Another look at equilibrium. For the reel to be in static equilibrium. 4.blue. Assume there is friction between the shaft and the pulley with coefficient of friction µ. See also problems 7. What is FB ? problem 4. The shelf is in the x y plane. If you can’t.23: 4. for example. Beer and Johnston Statics section 8.8 Advanced statics 4.20. g) Instead of using a system of equations try to find a single equation which can be solved for TE H .blue. Solve it and compare to your result from before. You may assume the local gravitational constant is g = 10 m/s2 .21: (Filename:pfigure.] Must these solutions agree with (d)? Do they? G C H . You may assume the local gravitational constant is g = 10 m/s2 . Look at the force ratios from parts (a) and (b) for a reasonable value of µ. e) By taking components. turn (b) and (c) into six scalar equations in six unknowns. Find the tensions in the strings if the mass is at rest. For the three cases (a). At the moment of interest the shelf is at rest. c) Write down the equation of force equilibrium. d) Write down the moment balance equation using the center of mass as a reference point. (Filename:pfigure. It is possible to find five of the six unknown reaction components this way. There is gravity and the system is at rest What is the reaction at point D on the plate? 1m 1m y problem 4.a) z D B C 3m A x 4m 4m (Filename:f92h1p1. try moment balance about various axes as well as aforce balance in an appropriate direction. ∗ .7 Hydrostatics 4. A ball of mass m is fixed to the centroid of the plate.25 Hanging a shelf.25: A B 1m D ˆ ı 32m ˆ k ˆ FBD Ro Ri ˆ ˆ ı O T1 Forces of bearing on pulley T2 problem 4.24: 4. [ Note that in problems (b) and (c). In all cases the strings hold up the mass m = 3 kg.b) problem 4.10). in order to pull the mass up at constant rate the winches must pull in the strings at an unsteady speed. and (c).119. ∗ c) (optional) To further emphasize the point look at the relation between the two string tensions when the bearing is locked (frozen.s94h2p10. Ro and µ (or φ) to see one reason why wheels (say pulleys) are such a good idea even when the bearings are not all that well lubricated.48m E 1m 1m problem 4. a) Draw a FBD of the shelf.] ∗ b) Challenge: without doing any calculations on paper can you find one of the reaction force components or the tension in any of the cables? Give yourself a few minutes of staring to try to find this force. then come back to this question after you have done all the calculations.23 See also problem 5. welded) and the string slides on the pulley with same coefficient of friction µ (see.22 A massless triangular plate rests against a frictionless wall at point D and is rigidly attached to a massless rod supported by two ideal bearings fixed to the floor.Problems for Chapter 4 b) Plug in some reasonable numbers for Ri .22: d y e h G B b x c (Filename:ch3. find the tension in the string AB. say µ = 0. ∗ 17 (a) winch (b) winch 3m m x (c) z winch D 1m 1m y x A 4m 4m (Filename:f92h1p1) B A z m C A B 4m 2m 3m B winch C winch 3m h) Challenge: For how many of the reactions can you find one equation which will tell you that particular reaction without knowing any of the other reactions? [Hint. d=c+(1/2)b D z a A problem 4. In all cases the winches are pulling in the string so that the velocity of the mass is a constant 4 m/s ˆ upwards (in the k direction).24 The strings hold up the mass m = 3 kg.2) 4.2. ˆ Gravity acts in the−k direction. f) Solve these equations by hand or on the computer. (b). below. A uniform 5 kg shelf is supported at one corner with a ball and socket joint and the other three corners with strings.1b) 4. Write this sentence as a differential equation. Find the velocity and position of the ball as a function of time. a) What is the velocity of the mass after 2π seconds? b) What is the position of the mass after 2π seconds? c) Plot position x versus time t for the motion. ˙ Using numerical solution. (b) the mass crosses the origin once. Fd = bv. 5. a = 2 m/s2 . (b) find a constant speed motion that satisfies your differential equation. How many minutes does a bicyclist take to cover this distance if he/she maintains a constant speed of 15 mph. m. people say “F = ma“ What is a more precise statement of an equation we use here that reduces to F = ma for one-dimensional motion of a particle? velocity (which is a known constant vt ).15 A ball of mass m is dropped vertically from rest at a height h above the ground. (c) the mass crosses the origin many times.9 A sinusoidal force acts on a 1 kg mass as shown in the figure and graph below.13 A ball of mass m has an acceleration a = ˆ cv 2 ı . x(0) = v(0) = 0. of course. or. x = cx ˙ and x(0) = x0 . 5.2) 5. what is x(π/2 s)? 5. (ii) solve the equation on the computer. and x(0) = 1 ft.16 A grain of sugar falling through honey has a negative acceleration proportional to the difference between its velocity and its ‘terminal’ M problem 5.19 A ball of mass m is dropped vertically from rest at a height h above the ground. Gravity is non-negligible.18 CONTENTS Problems for Chapter 5 Unconstrained motion of particles F(t) 5N 0 2π sec t 5. and thereafter it is acted upon by a retarding force which gives it a constant acceleration ax = −10m/s 2 .7 Given that x = x/ s and x(0 s) = 1 m. and t is in seconds.12 The linear speed of a particle is given as v = v0 + at.6. Calculate the velocity and the x-coordinate of the particle when t = 8 s and when t = 12 s. 5. x F(t) 5.18 Due to gravity.. (c) pick numerical values for your constants and for the initial height.8 Let a = dv = −kv 2 and v(0) = v0 . where c = 1 s−1 and x0 = 1 m.17 The mass-dashpot system shown below is released from rest at x = 0.1) g 5. The damping coefficient of the dashpot is c. Find dt t such that v(t) = 1 v0 . Make a sketch of x versus t.5 m/s. Find the velocity as a function of position. ∗ 5. find values of A. e. Find the position of the ball as a function of velocity. x < 0) ˙ F = Ax + Bx 2 + C x Assume x(0) = 0. where v is in m/s. v0 = 20 m/s. and x(0) = 0 m.20 A force pulls a particle of mass m towards the origin according to the law (assume same equation works for x > 0. k1 = ˙ 1 ft/s. The mass is initially still. a particle falls in air with a drag force proportional to the speed squared.3 The distance between two points in a bicycle race is 10 km. For the first 5 s it has no acceleration. 5.14 A ball of mass m is dropped from rest at a height h above the ground. Solve the equation assuming some given initial velocity v0 . 5.17: x (Filename:pfigure.1 In elementary physics. Air resistance causes a drag force on the ball proportional to the speed of the ball squared. Neglect air friction.blue.blue. i.10 A motorcycle accelerates from 0 mph to 60 mph in 5 seconds. A = ˙ 0. The drag force acts in a direction opposite to the direction of motion. Find the position and velocity as a function of time. Determine an equation of motion for the particle of mass m that involves only x and x (a first-order ordinary ˙ differential equation).) 5. The drag force acts in a direction opposite to the direction of motion.2 Does linear momentum depend on reference point? (Assume all candidate points are fixed in the same Newtonian reference frame. Assume the initial speed is zero (i) set up the equation for numerical solution. Find the average acceleration in m/s2 . and find the maximum positive x coordinate reached by the particle. C. what is x(10 s)? 5. B. F = m a in terms of vari(a) Write ables you clearly define. and x0 so that (a) the mass never crosses the origin. Define appropriate dimensionless variables and write a dimensionless equation that describes the relation of v and t. k2 = 1 ft/s2 .11 A particle moves along the x-axis with an initial velocity vx = 60 m/s at the origin when t = 0. How does this acceleration compare with g. expressed slightly differently. 5. (iii) make a plot with your computer solution and show how that plot supports your answer to (b). 5. 5. 2 5. Fd = cv 2 . [hint: acceleration is the time-derivative of velocity] 5. Air resistance causes a drag force on the ball directly proportional to the speed v of the ball. ˙ find x(5 s).6 Given that x = A sin[(3 rad/s)t]. . the acceleration of an object falling near the earth’s surface? 5.151.9: (Filename:pfigure.5 Find x(3 s) given that x = x/(1 s) ˙ and x(0) = 1 m 1 kg c problem 5.1 Force and motion in 1D 5. When does the ball hit the ground? What is the velocity of the ball just before it hits? 5. defining any constants you need. Find the velocity as a function of position.4 Given that x = k1 + k2 t. 5. After the first bounce. At times t = 0 the mass is at x = d and is let go with no velocity. What is the vertical coefficient of restitution. g. Or find a related dynamics dr dt dr problem and use conservation of energy..30 Consider a mass m on frictionless rollers. dv = dt dv · dr = dv · v. Find the time in terms of the initial height h 0 . [Hint: Use the chain rule of differentiation to eliminate t.4 m. The ball strikes the ground with speed v − and after collision it rebounds vertically with reduced speed v + directly proportional to the incoming speed. The mass is held in place by a spring with stiffness k and rest length . . Assume a rider starts from rest and uses this constant power. h 2 ? Also see several problems in the harmonic oscillator section. and ¨ x(0) = 0. The gravitational constant is g. In terms of the quantities above. a) What is the initial potential and kinetic energy of the system? b) What is the potential and kinetic energy of the system as the mass passes through the static equilibrium (unstretched spring) position? h1 g h2 h0 The next set of problems concern one mass connected to one or more springs and possibly with a constant force applied.5 m from the unstretched position and released from rest. b) Do this problem again using energy balance.23. a) What is the acceleration of the block at t = 0+ ? b) What is the differential equation governing x(t)? c) What is the position of the mass at an arbitrary time t? x m problem 5. find the value of x at t = π/λ s.3 The harmonic oscillator The first set of problems are entirely about the harmonic oscillator governing differential equation. e. find the value of x at t = π/2 s. and g. c) What is the position of the mass at an arbitrary time t? d) What is the speed of the mass when it passes through the position where the spring is relaxed? 0 r 5. The mass is pulled to the right a distance x = x0 = 0. A spring with rest length 0 is attached to a mass m which slides frictionlessly on a horizontal ground as shown. v + = ev − . e.33 Reconsider the spring-mass system from problem 5.27 Given that x = −(1/s2 )x. ˙ • • • d m problem 5. the coefficient of restitution. ˙ 5. When the spring is relaxed the position of the mass is x = 0.29 Given that x + λ2 x = C0 .28 Given that x + x = 0. 5.21 A car accelerates to the right with constant acceleration starting from a stop. There is wind resistance force proportional to the square of the speed of the car.22 A ball of mass m is dropped vertically from a height h.1) problem 5. assuming it is decoupled from tangential motion? What is the height of the second bounce.23: (Filename:Danef94s1q7) 5. ¨ and x(0) = 0 find: ˙ a) x(π s) =? b) x(π s) =? ˙ 5. a) Find the potential and kinetic energy of the spring mass system as functions of time. The only force acting on the ball in its flight is gravity. (a) What is the peak speed of the cyclist? (b) Using analytic or numerical methods make a plot of speed vs. assuming that the vertical coefficient is fixed. Solve for v as a function dt of r if v(r = R) = v0 . What is the maximum height the ball reaches after one bounce.26 Given v = − g R .] a) What is the acceleration of the mass just after release? b) Find a differential equation which describes the horizontal motion of the mass.2 Energy methods in 1D 5.31 Spring and mass. no external forces act on the mass other than the spring force and gravity.Problems for Chapter 5 5.30.21: Car. and the gravitational constant.32: (Filename:ch2. Define all constants that you use.31: (Filename:s97f1) 5.25 The power available to a very strong accelerating cyclist is about 1 horsepower. x(0) = 1 m. where g and R are ˙ 2 constants and v = dr .25. (Filename:s97p1. and ¨ x(0) = 0.blue. x(0) = 1.30: (Filename:pfigure. it reaches a height of h 1 = 6. where 0 < e < 1. (c) What is the acceleration as t → ∞ in this solution? (d) What is the acceleration as t → 0 in your solution? d) What is the speed of the mass when it passes through x = 0? x d m problem 5.006 lbf/( ft/ s)2 v 2 . with no mechanics content or context.10.e. a realistic drag force of .20. Assume a mass (bike + rider) of 150 lbm. 5.a) problem 5.24 In problem 5. At the instant of release. a) What is its position as a function of time? b) What is the total force (sum of all forces) on the car as a function of time? c) How much power P is required of the engine to accelerate the car in this manner (as a function of time)? 19 5.] • • • 2 5.23 A ball is dropped from a height of h 0 = 10 m onto a hard surface. i. ∗ a) Do this problem using linear momentum balance and setting up and solving the related differential equations and “jump” conditions at collision.2) 5. x(0) = x0 . Neglect other drag forces. show that the number of bounces goes to infinity in finite time. 5.32 Reconsider the spring-mass system in problem 3. in terms of h. 5. Let m = 2 kg and k = 5 N/m. [Remember to define any coordinates or base vectors you use. time. At time t = 0 the mass is released with no initial speed with the spring stretched a distance d. what is the period of this motion? ∗ egg. keq . The trampoline is modeled as having an effective vertical undamped linear spring with stiffness k = 200 lbf/ ft. The position y(t) of the mass is measured from the fixed support. if you replace all of the springs with one spring. problem 5. a) Draw a Free Body Diagram of the mass.159. what would its stiffness have to be such that the system has the same natural frequency of vibration? ground problem 5. 0 k. taking downwards as positive. m k k 5. let all the springs have non-zero stiffness.38: (Filename:pfigure. ∗ d) Verify that one solution is that x(t) is constant at x = l0 + mg/k.149.3. Ignore gravity. m 0 5.35 A spring and mass system is shown in the figure. a) What is the period of motion if the person’s motion is so small that her feet never leave the trampoline? ∗ b) What is the maximum amplitude of motion for which her feet never leave the trampoline? ∗ c) (harder) If she repeatedly jumps so that her feet clear the trampoline by a height h = 5 ft. At the instant shown. CONTENTS l0 k x m problem 5. e) What is the meaning of that solution? (That is.. If the frame hits the ground on one of the straight sections. when no mass is attached). What is the stiffness of a single spring equivalent to the combination of k1 . what will be the frequency of vibration of the egg after impact? [Assume small oscillations and that the springs are initially stretched.3) k b) Write the equation of linear momentum balance. m 0 F(t) F(t) (c) k. We will also consider only a two-dimensional version of the winning design as shown in the figure. The static equilibrium position is ys and the relaxed length of the spring is 0 . In this problem.) ∗ f) Define a new variable x = x − (l0 + ˆ mg/k).2 ft/s2 . k3 . Draw a ˙ free body diagram of the mass at the instant of interest and evaluate the left hand side of the energy balance equation (P = E˙K ).3) . y F(t) m x problem 5. The person is modeled as a rigid mass m = 150 lbm. find the equation of motion of the mass in each case. the position of the mass. k2 . What is the motion of the mass? ∗ h) What is the period of oscillation of this oscillating mass? ∗ i) Why might this solution not make physical sense for a long. k2 .rp1) m 5. and k3 equal zero and k4 be nonzero.38 One of the winners in the egg-drop contest sponsored by a local chapter of ASME each spring.blue. e. we will consider the simpler case of the egg to be a particle of mass m and the springs to be linear devices of spring constant k. All springs are massless and are shown in their relaxed states.1) k y (a) k.39 A person jumps on a trampoline. That is. of all of the springs together. as a review. ∗ ˆ g) Assume that the mass is released from an an initial position of x = D. Substitute x = x +(l0 +mg/k) ˆ into your differential equation and note that the equation is simpler in terms of the variable x. ∗ c) Reduce this equation to a standard differential equation in x. Are the potential and kinetic energy ever equal at the same time? If so. describe in words what is going on. at what position x(t)? c) Make a plot of kinetic energy versus potential energy.) ∗ 5. k4 ? What is the frequency of oscillation of mass M? c) What is the equivalent stiffness. let k1 . What is the natural frequency of this system? b) Now. directed downwards.34: (Filename:summer95f. soft spring if D > 0 + 2mg/k)? ∗ 5. was a structure in which rubber bands held the egg at the center of it.] (b) k. the position of the mass is y and its velocity y .36: (Filename:pfig2. What is the phase relationship between the kinetic and potential energy? 5.37 Mass hanging from a spring. 0 The following problem concerns simple harmonic motion for part of the motion. It only moves vertically. It involves pasting together solutions.34 For the three spring-mass systems shown in the figure.1) k1 k2 k4 problem 5.blue. a) First.37: ys (Filename:pg141.35: x k3 M (Filename:pfigure. g = 32.20 b) Using the computer. make a plot of the potential and kinetic energy as a function of time for several periods of oscillation. 0 k. A mass m is hanging from a spring with constant k which has the length l0 when it is relaxed (i.36 The mass shown in the figure oscillates in the vertical direction once set in motion by displacing it from its static equilibrium position. (In problem (c) assume vertical motion. 125? (d) How much additional thickness of concrete should be added to the footing to reduce the damped steady-state amplitude by 50%? (The diameter must be held constant. find x(t). ¨ ˙ x(0) = 0. These first problems are just math problems. 5. There is gravity.5 rad/s and ξ ∗ = 0. but this linear case is prevalant in the analytic study of dynamical systems. x1 = x3 ˙ x2 = x4 ˙ ∗ x 3 + 5 2 x 1 − 4 2 x 2 = 2 2 v1 ˙ ∗ x 4 − 4 2 x 1 + 5 2 x 2 = − 2 v1 ˙ 5.4 More on vibrations: damping 5.42 A 3 kg mass is suspended by a spring (k = 10 N/m) and forced by a 5 N sinusoidally oscillating force with a period of 1 s.47 Solve for the constants A and B in Problem 5.40 If x + c x + kx = 0. what would the steady-state displacement be? (c) What is the steady-state displacement given that d = 0. where A and B are constants to be determined from initial conditions. 000. the mass is moving to the right and the spring is stretched a distance x from its position where the spring is unstretched.41: (Filename:ch2..51 Write each of the following equations as a system of first order ODE’s.39: A person jumps on a trampoline. ∗ . ¨ a) θ + λ2 θ = cos t. ξ (0) = 0. In general things are not so simple. 5.trampoline) 5. What is the amplitude of the steady-state oscillations (ignore the “homogeneous” solution) ¨ 5. if ξ(0) = ˙ 0. x1 + x1 ¨ x2 + x2 ¨ = = x2 sin t ˙ x1 cos t ˙ x k problem 5. yy . ˙ ˙ and θ (0) = θ0 .45 Write the following set of coupled second order ODE’s as a system of first order ODE’s. x1 ¨ x2 ¨ = = k2 (x2 − x1 ) − k1 x1 k3 x2 − k2 (x2 − x1 ) 5. a) Draw a free body diagram of the mass at the instant of interest.41 A mass moves on a frictionless surface. (a) What is the natural frequency of the system? (b) If the system were undamped. Experts note: normal modes are coverred in the vibrations chapter.46 using the matrix form. It is connected to a dashpot with damping coefficient b to its right and a spring with constant k and rest length to its left. The machine operates at a frequency of ω = 4 Hz. The solution of a set of a second order differential equations is: ξ(t) = A sin ωt + B cos ωt + ξ ∗ ˙ ξ (t) = Aω cos ωt − Bω sin ωt. (Filename:pfigure3.46 See also problem 5.43 Given that θ + k 2 θ = β sin ωt. 5.47. where Q 0 = 5000N . using some of the skills that are needed for the later problems.5 Forced oscillations and resonance 5. ˙ x(0) = x0 . problem 5. and 5.) 5.44 A machine produces a steady-state vibration due to a forcing function described by Q(t) = Q 0 sin ωt. The equivalent spring constant of the half-space is k = 2.49 Write the following pair of coupled ODE’s as a set of first order ODE’s.11) m b 5.125. 5. b) Evaluate the left hand side of the equation of linear momentum balance as explicitly as possible. elastic half-space.. x1 . The foundation rests on an isotropic.5. ¨ ˙ . ω = 0. Assume A and B are the only unknowns and write the equations in matrix form to solve for A and B in terms of ˙ ξ(0) and ξ (0).50 The following set of differential equations can not only be written in first order form but ˙ in matrix form x = [A]x + c .Problems for Chapter 5 21 yy . The machine rests on a circular concrete foundation.2. b) x + 2 p x + kx = 0. At the instant of interest.48 A set of first order linear differential equations is given: x1 = x2 ˙ x2 + kx1 + cx2 = 0 ˙ ˙ Write these equations in the form x = [A]x . find θ(t) . 000 N·m and has a damping ratio d = c/cc = 0. 5. θ(0) = 0.6 Coupled motions in 1D The primary emphasis of this section is setting up correct differential equations (without sign errors) and solving these equations on the computer. where x = x2 5. s94h5p1) x2 x1 k1 m1 F k2 m2 c k3 5. Make up appropriate initial conditions.55 Normal Modes.54 Two equal masses. Getting the signs right is important. m A . ¨ ˆ find the acceleration of mass B. k2 . and t. B. m and k. x2 k2 m2 k3 k1 m1 problem 5.22 c) x + 2c x + k sin x = 0. the constant force F. are on an air track. xC . Three equal springs (k) hold two equal masses (m) in place. say. k3 .k3 .s95f1) x1 k m problem 5. and c are constants. ˙ ˙ ˙ c1 .56: (Filename:pfigure. What is x2 (t)? (The answer is not unique. c. c) Write the total energy of the system. One mass is connected by a spring to the end of the track. m. idealized as a point mass. The other mass is connected by a spring to the first mass. m D . is walking at an absolute absolute velocˆ ity u ı .59: xB k2 c1 B mB k3 D xD mD mA k k k m x1 problem 5. x1 (t) is shown. a) Draw a free body diagram for each mass. m B . x1 and x2 are the displacements of the masses from their equilibrium positions. In the rest configuration (springs are relaxed) the masses are a distance apart. d) Pick parameter values and initial conditions of your choice and simulate a motion of this system. m D . Assume that all the springs are relaxed when x A = x B = x D = 0. (Filename:pfigure. x A . The force F acts on mass 2. b) Write the kinetic energy of the system at the same time t in terms of x1 . a) How many independent normal modes of vibration are there for this system? ∗ b) Assume the system is in a normal mode of vibration and it is observed that x1 = A sin(ct) + B cos(ct) where A. c) For constants and initial conditions of your choosing. Some candidates for x2 (t) are shown. You may express your answer in terms of any of A.f. Motion of the two masses x1 and x2 is measured relative to this configuration.58 The springs shown are relaxed when x A = x B = x D = 0. ∗ b) Does each mass undergo simple harmonic motion? ∗ 5.60 Equations of motion. x D . and k1 . Given k1 . 5.f96. k2 . where u > v. B. A passenger. a B = x B ı . x A . Make a plot of the motion of.f93f2) (Filename:pfigure. a) Write the governing equations in a neat first order form. k2 . x A . In terms of some or all ˙ ˙ ˙ of m A . Neglect friction. In the rest (springs are relaxed) configuration. m B . and x3 .s95q3) 5.59 A system of three masses.57 For the three-mass system shown. k4 . x A . Your equations should be in terms of any or all of the constants m 1 .52 A train is moving at constant absolute veˆ locity v ı . Which of the x 2 (t) could possibly be associated with a normal mode vibration of the structure? Answer “could” or “could not” next to each . x2 . m 2 . L k m L k x1 m L k x2 m L k x3 (Filename:s92f1p1) problem 5.61 x1 (t) and x2 (t) are measured positions on two points of a vibrating structure. C. The governing equations for the system shown can be writen in first order form if we ˙ ˙ define v1 ≡ x1 and v2 ≡ x2 .3) problem 5. e) Explain how your plot does or does not make sense in terms of your understanding of this system. and F. Write the spring forces in terms of the displacements x1 . ∗ xA k4 k1 A problem 5.56 A two degree of freedom spring-mass system. x2 . each denoted by the letter m. ∗ mA mD 5. made up of two unequal masses m 1 and m 2 and three springs with unequal stiffnesses k1 . x B . The displacements x1 and x2 are defined so that x1 = x2 = 0 when the springs are unstretched. x D .twomassenergy) x2 m 5.58: c1 k2 B mB k3 D (Filename:pfigure. draw a free body diagram of each mass. A two degree of freedom mass-spring system. c1 . one of the masses vs time. a) Derive the equations of motion for the two masses. The other mass is connected by a spring to the first mass. One mass is connected by a spring to the end of the track. ∗ xA xB xD k4 k1 A problem 5.57: 5. The two spring constants are equal and represented by the letter k.s94f1p4) x1 5. There is no friction.53: x2 k m 5. find the acceleration of block B. each denoted by the letter m. ˙ ˙ and k. ¨ ˙ CONTENTS x1 k k m problem 5. k2 and k3 . b) Write computer commands to find and plot v1 (t) for 10 units of time. All three springs are relaxed in the configuration shown. and one damper are connected as shown. c) Write the equations as a system of first order ODEs.55: m x2 (Filename:pfigure. Two masses are connected to fixed supports and each other with the three springs and dashpot shown. a) Write the potential energy of the system for arbitrary displacements x1 and x2 at some time t. x B . x B . k3 . and x D . The two spring constants are equal and represented by the letter k. plot x1 vs t for enough time so that decaying erratic oscillations can be observed. b) Write the equation of linear momentum balance for each mass. The ground is frictionless. are on an air track. Is the initial motion in the right direction? Are the solutions periodic? Bounded? etc. the masses are a distance apart. is shown in the figure. k1 . four springs. What is the velocity of the passenger relative to the train? • • • 5.54: (Filename:pfigure. x B .53 Two equal masses. Motion of the two masses x1 and x2 is measured relative to this configuration. ) ∗ c) Find all of the frequencies of normalmode-vibration for this system in terms of m and k. k4 .60: (Filename:p. ) a) Determine the natural frequencies and associated mode shapes for the system.158. a system of three masses.65: ˆ ı k m1 x1 k k m2 x2 k m3 k x3 k m 2m m x3 x2 problem 5. In the special case when c1 = 0. four springs. then you should have also obtained the mode shape (0. it is. 1)T and (1. then by “symmetry”.1) 5. find another normal mode of vibration.64: (Filename:pfigure. four springs. problem 5. All three masses have m = 1 kg and all 6 springs are k = 1 N/ m. circular hoop.68 Normal modes. ∗ b) In the same special case as in (a) above. and one damper are connected as shown. find the normal modes of vibration. (Hint: Note that in this mode of vibration the middle mass does not move.67: mA problem 5. −1.) b) If your calculations in (a) are correct.68: (Filename:pfigure3. Assume x1 = x2 = x3 = 0 when all the springs are fully relaxed. one of the normal modes is described with the eigenvector (1. and m A = m B = m D = m. a system of three masses. a) What is the angular frequency ω for this mode? Answer in terms of L .63: (Filename:pfigure. c) Draw free body diagrams for the beads and use Newton’s second law to derive the equations for motion for the system. a) Find as many different initial conditions as you can for which normal mode vibrations result.) ∗ b) Make a neat plot of x2 versus x1 for one cycle of vibration with this mode. e) Show that the center of mass must either remain at rest or move at constant velocity. wound around the hoop as shown and equally spaced when the springs are unstretched (the strings are unstretched when θ1 = θ2 = θ3 = 0. a) Write expressions for the total kinetic and potential energies. (Filename:pfigure.s96.66 As in problem 5. 2m.63 The three beads of masses m.64 The system shown below comprises three identical beads of mass m that can slide frictionlessly on the rigid. m. and g. Explain how we can have three mode shapes associated with the same frequency.161. find a normal mode of vibration.1) m problem 5. k.1) 5. Assume that all the springs are relaxed when x A = x B = x D = 0.blue.59. Assume the initial velocities are zero. ∗ a) X2 ? X2 ? X2 ? X2 ? X2 ? b) c) xA k4 k1 A problem 5.) b) For the initial condition [x0 ] = [ 0. b) Write an expression for the total linear momentum. 1. In each case.1) m 5. (Hint: you should be able to deduce a ‘rigid-body’ mode by inspection.s95f1a) 5. For the special case that C = 0 and F0 = 0 clearly define two different set of initial conditions that lead to normal mode vibrations of this system. (we will call two initial conditions [v] and [w] different if there is no constant c so that [v1 v2 v3 ] = c[w1 w2 w3 ].s95q3a) 5. 0. 1.143. 0)T are also mode shapes (draw a picture). and m connected by massless linear springs of constant k slide freely on a straight rod. −1)T . [x0 ] = [ 0 2 m/s 0 ] ˙ what is the initial (immediately after the start) acceleration of mass 2? F0 sin(λt) k1 c M1 x1 M2 k2 m k x2 (Filename:p. c1 = 0.65 Equations of motion.67 As in problem 5. -1). and one damper are connected as shown. xA k4 xB c1 k2 B mB k3 D xD mD L k m L x1 k m L x2 k m L x3 k m θ2 k θ1 g=0 k1 A m problem 5. Write down the most general set of initial conditions so that the ensuing motion of the system is simple harmonic in that mode shape. c) Since (0. find the associated natural frequency. compare the frequencies of the constrained system to those of the unconstrained system. obtained in (a). (−1. −1)T is a mode shape. The beads are connected by three identical linear springs of stiffness k.62 For the three-mass system shown. Let xi denote the displacement of the i th bead from its equilibrium position at rest.61: (Filename:pfigure.f95p2p2) . Two masses are connected to fixed supports and each other with the two springs and dashpot shown. (If a curve looks like it is meant to be a sine/cosine curve. d) Verify that total energy and linear momentum are both conserved. The displacements x1 and x2 are defined so that x1 = x2 = 0 when both springs are unstretched. 5. a) In the special case when k1 = k2 = k3 = k4 = k. Define it in any clear way and explain or show why it is a normal mode in any clear way. d) Without doing any calculations.p3.blue.1 m 0 0 ]. immobile.66: xB k2 c1 B mB k3 D xD mD d) e) mA problem 5.blue.62: (Filename:pfigure.2) k θ3 problem 5.) X1(t) 23 5.160.Problems for Chapter 5 choice. The system is at rest when x1 = x2 = x3 = 0. 0.blue. f) What can you say about vibratory (sinusoidal) motions of the system? Static equilibrium configuration Displaced configuration x1 k 2m R=1 k (Filename:pfigure.144. 2 kg is thrown from a height of h = 20 m above the ground ˆ ˆ with velocity v = 120 km/hı − 120 km/h − ˆ . 5. and acceleraˆ ˆ tion a = −2 m/s2 ı + 9 m/s2 . ˆ ˆ R cos t + gt k If r (0) = R ı .91 A ball of mass m = 0. ˆ ˆ ˆ 5. and rate of change of speed of the particle as functions of time? What kind of path is the particle on? What are the distance of the particle from the origin and its velocity and acceleration when x = 3 m? 5.95 A bullet of mass 50 g travels with a velocˆ ˆ ity v = 0. For each position of a point P defined below.77 Let r = v0 cos α ı + v0 sin α + ˆ (v0 tan θ − gt)k .83 A particle travels on an elliptical path given by y 2 = b2 (1 − x 2 ) with constant a speed v.6 km/s . α. What are the velocity and acceleration t 3 s of the particle in cartesian coordinates when t = (π ) 3 s? 5.74 The velocity of a particle of mass m on a frictionless surface is given as v = ˆ ˆ (0. What is the kinetic energy of the ball at its release? 5.85 A particle starts at the origin in the x yplane.70 Given r (t) = A sin(ωt)ı + Bt + C k . find (a) v (t) (b) a (t) (c) r (t) × a (t). v2 is a constant ˆ 5 m/s. θ. velocity and acceleration 5. t ˆ ˆ ˆ r = (sin ts ) mı +(cos ts ) m +te s m/sk . (a) What is the linear momentum of the bullet? (Answer in consistent units.81 The position of a particle is given by t ˆ ˆ r (t) = (t 2 m/s2 ı + e s m ). At t = 3 s.0 mı + 2. respectively. ˆ ˆ a) r P = 4 mı + 7 m . Find the velocity of the particle when x = a/2 and y > 0 in terms of a.24 CONTENTS 5. Find (a) the distance of the particle from the origin and (b) a unit vector in the direction of r . find (a) the distance traveled by the mass in 2 seconds and (b) a unit vector along the displacement. where v0 .50 ft)ı − (0. y0 = 0) and travels only in the positive x y quadrant. find r ((π/3) s). the power of all external forces acting on a particle. 5.5 m . b. compute (a) the linear momentum L = m v and (b) the angular momentum (H O = r/O × (m v )). h = 2 m. Its speed and x coordinate are known to be v(t) = 1 + ( 4 )t 2 m/s 2 s 2 1 5. What is the rate of change of speed of the particle? What angle does the velocity vector make with the positive x axis when t = 3 s? 5.79 Draw unit vectors along .94 A particle A has velocity v A and mass m A .80 What is the angle between the x-axis and ˆ ˆ ˆ the vector v = (0. ˆ ˆ ω = (4. acceleration.86 What symbols do we use for the following and that v1 is a constant 4 m/s. from the forces F i and the velocity v of the particle? 5. A particle B has velocity v B = 2 v A and mass equal to the other m B = m A .93 How do you calculate P. At t = 5 s. and r is measured from the origin. 5.90 What is the relation between the dynamics ‘Angular Momentum Balance’ equation and the statics ‘Moment Balance’ equation? 5. and . find the particle’s a) velocity and b) acceleration. If r (0) = 0 .5 kg is r (t) = sin(ωt)ı + h .50 rad/s) and r = ˆ ˆ (0.33 rad/s)ı + (2. 5.) ˆ ˆ 5.84 A particle travels on a path in the x y-plane x given by y(x) = (1 − e− m ) m. and v. ˙ ˆ ˆ 5. d) r P = 0 5. What is the kinetic energy of the 10 km/hk ball at its release? 5. and c is a constant 4 s−1 .88 Does kinetic energy depend on reference point? (Assume that all candidate points are fixed in the same Newtonian reference frame.71 A particle of mass =3 kg travels in space with its position known as a function of time. the angular momentum of the particle with respect to the point P.5 m/s) . ˆ ˆ velocity v = 6 m/sı − 3 m/s .92 A ball of mass m = 0.89 What is the relation between the dynamics ‘Linear Momentum Balance’ equation and the statics ‘Force Balance’ equation? 5. find H P . and g are constants.98 The position of a particle of mass m = ˆ ˆ 0. find r (t) = ˆ ˆ x(t)ı + y(t) . It is known that the x coordinate of the particle is given by x(t) = t 2 m/s2 .87 ft) and find the angle between the two unit vectors. Make a plot of the path.97 The position vector of a particle of mass ˆ ˆ 1 kg at an instant t is r = 2 mı − 0. find r (t). If the velocity of the particle at this instant is ˆ ˆ v = −4 m/sı + 3 m/s . 5.2k ) m/s? 5. Assume ı and ˆ are constant.76 Find r (5 s) given that 5.75 For v = vx ı + v y + vz k and a = ˆ 2 ı − (3 m/s2 − 1 m/s3 t) − 5 m/s4 t 2 k . s3 find the particle’s a) velocity and ∗ b) acceleration. where ω = 2 rad/s. 5. where x(t) = 3 ( m ).) 5. What are the velocity and acceleration of the particle? ∗ 5. ˆ ˆ b) r P = −2 mı + 7 m . ˆ 2 m/s ˆ write the vector equation v = a dt as three scalar equations.87 Does angular momentum depend on reference point? (Assume that all candidate points are fixed in the same Newtonian reference frame. ˆ ˆ ˆ 5.0 + 2.7 Time derivative of a vector: position.5 m/s)ı − (1. What is r (t) in cartesian coordinates? What are the velocity. ∗ ˙ ˆ ˆ 5.72 A particle of mass m = 2 kg travels in the x y-plane with its position known as a function ˆ ˆ of time.3ı − 2. and ˆ ˆ c) r P = 0 mı + 7 m .73 If r = (u 0 sin t)ı + v0 and r (0) = ˆ ˆ x0 ı + y0 .8 km/sı + 0. b) H A/C and H B/C . (x0 = 0.96 A particle has position r = 4 mı + 7 m .78 On a smooth circular helical path the ˙ ˆ velocity of a particle is r = −R sin t ı + ˆ.8 Spatial dynamics of a particle quantities? What are the definitions of these quantities? Which are vectors and which are scalars? What are the SI and US standard units for the following quantities? a) linear momentum b) rate of change of linear momentum c) angular momentum d) rate of change of angular momentum e) kinetic energy f) rate of change of kinetic energy g) moment h) work i) power ˙ ˆ ˆ ˆ ˆ r = v1 sin(ct)ı +v2 and r (0) = 2 mı +3 m5.1 kg is thrown from a height of h = 10 m above the ground with ˆ ˆ velocity v = 120 km/hı − 120 km/h . 5. v0 . with u 0 .5 m . and c) E KA and E KB ? 5. If the displacement is given by r = v t. What is the relationship between: a) LA and LB . ∗ 5. = 2 m.) 5.82 A particle travels on a path in the x y-plane x given by y(x) = sin2 ( m ) m. r = 3t 2 m/s2 ı + 4t 3 m . and x(t) = t m/s.69 The position vector of a particle in the x yˆ ˆ plane is given as r = 3. Find the ˙ linear momentum L and its rate of change L of the the potato’s center of mass. respectively.1.part.rp3) Particle FBD F problem 5. ˆ ˆ ˆ and a = 5 m/s2 ı + 11 m/s2 − 9 m/s2 k . b) rate of change of linear momentum L. H P .103 A 2 kg particle moves so that its position r is given by ˆ ˆ ˆ r (t) = [5 sin(at)ı + bt 2 + ct k ] m where a = π/ sec.fbdb) 5. respectively.104 A particle A has mass m A and velocity v A . ˙ b) rate of change of linear momentum L. and acceleration a = 1 m/s2 ı − ˆ ˆ 8 m/s2 + 3 m/s2 k . velocity. . At the instant of interest find its: a) linear momentum L. and a = −10 m/s2 ı + ˆ ˆ 24 m/s2 + 32 m/s2 k . and f) rate of change of kinetic energy E˙K . velocity. v = 3 m/sı − 7 m/sk . What is the translational kinetic energy of the wad at the instant of interest? 5. respectively. b) the net moment on the particle about MO due to the applied the origin forces.part. a) What is the linear momentum of the particle at t = 1 sec? b) What is the force acting on the particle at t = 1 sec? 5. ˆ ˆ ˆ b) r P = 6 mı − 4 m + 8 mk . c) angular momentum about the origin H O. and acceleration of its cenˆ ˆ ˆ ter of mass are r = 3 mı + 3 m + 6 mk . velocity. ˆ ˆ ˆ c) r P = −9 mı + 6 m − 12 mk . and acceleration of its center of mass ˆ ˆ ˆ ˆ are r = 3 mı +3 m +6 mk . and a = −5 m/s2 ı + 12 m/s2 + ˆ ˆ 10 m/sk ˆ 16 m/s2 k .100 Consider a projectile of mass m at some instant in time t during its flight. b) H A/C and H B/C . 2 Why does it follow that E˙K = m v · a ? [hint: write v 2 as v · v and then use the product rule of differentiation.1. ˆ ˆ ˆ a) r P = 3 mı − 2 m + 4 mk . d) rate of change of angular momentum ˙ about the origin H O . v b) the net moment on the particle about MO due to the applied the origin forces. and acceleration r = 2 m .101 A 10 gm wad of paper is tossed in the air (in a strong turbulent wind).102 A 10 gm wad of paper is tossed into the air. and acceleration at a particular instant of interest are ˆ ˆ ˆ ˆ r = 2 mı + 3 m . v = −9 m/sı + ˆ ˆ ˆ 24 m/s + 30 m/sk . What is the relationship between: 5. find its: a) linear momentum L.107 A particle of mass m = 5 kg has poˆ sition. A particle B at the same location has mass m B = 2 m A and velocity equal to the other v B = v A . and ˙ i) rate of change of work W = P done on the particle by the applied forces. respectively. of these forces (P = a) LA and LB . and f) rate of change of kinetic energy E˙K . The drag force is proportional to the square of the speed (speed≡ | v | = v) and acts in the opposite direction. Find an expression for the net power F · v ) on the particle. find: a) the net force F on the particle. a drag force acts on the projectile. respectively. and c) E KA and E KB ? 5. velocity. For this particle at the instant of interest. Its position. 5. For this a = 3 m/s ˆ particle at the instant of interest. velocity v = 2 m/sı −3 m/s + ˆ ˆ 7 m/sk . Point C is a reference point. v = −3 m/sı + 8 m/s .Problems for Chapter 5 a) Find the kinetic energy of the particle at t = 0 s and t = 5 s. e) kinetic energy E K . In addition to the force of gravity. The position. and c) the power P of the applied forces. and ˆ 2 ı −4 m/s2 k .fbd) 5. velocity.108 A particle of mass m = 6 kg is moving in space. b = . find the rate of change of angular momentum of the particle with re˙ spect to the point P. and ˆ ˆ v = −9 m/sı + 24 m/s + 30 m/sk ˆ ˆ ˆ a = −10 m/s2 ı + 24 m/s2 + 32 m/s2 k .105 A particle of mass m = 3 kg moves in space. At a particular instant of interest. For this particle at the instant of interest. at a particular instant of interest.106 A particle has position r = 3 mı − ˆ ˆ ˆ ˆ 2 m +4 mk .] 5.part.3. and d) r P = 0 5. respectively. and ˙ c) rate of change of work W = P done on the particle by the applied forces. and acceleration at a particular instant in time are r = ˆ ˆ ˆ ˆ ˆ 2 mı +3 m +5 mk . Let v be the velocity of the projectile at this instant (see the figure). the position. Its position.109: FBD of the particle (Filename:pfigure1. and acceleration at a particular instant of interest are ˆ ˆ ˆ ˆ r = 4 mı + 2 mk .fbda) F on the particle. b) Find the rate of change of kinetic energy at t = 0 s and t = 5 s. c = 2/ sec . Its position.25/ sec2 . find its: a) the net force F on the particle. and a = −7 m/s2 ı . find its: a) linear momentum L. respectively. ˆ ˆ and a = −5 m/s2 ı + 12 m/s2 . 5.100: (Filename:pfig2.1. v = 3 m/sı +4 m/s −7 m/sk . velocity. e) kinetic energy E K . g) the net force ˙ 25 Particle FBD F problem 5. v = −3 m/sı +8 m/s + ˆ .109 A particle of mass m = 3 kg is moving in the x z-plane. c) angular momentum about the origin H O. and acceleration ˆ at a particular instant in time are r = 1 mı − ˆ ˆ ˆ ˆ ˆ 2 m +4 mk . θ problem 5.110 A particle of mass m = 3 kg is moving in the x y-plane. Its position. ˆ 5. ˆ ˆ v = 3 m/sı . velocity. h) the net moment on the particle about MO due to the applied the origin forces. For this particle at the instant of interest.99 For a particle 1 EK = m v2 .107: FBD of the particle (Filename:pfigure1. d) rate of change of angular momentum ˙ about the origin H O .108: FBD of the particle (Filename:pfigure1. 5. ˆ . For each position of a point P defined below. Particle FBD F problem 5. 117 Neglecting gravity. The ˆ velocity of the particle is given by v = v 2 where v 2 = v0 − 2ah. 5.2m A problem 5.4. and F3 . ˙ b) rate of change of linear momentum L c) angular momentum about the origin H O. write the equations of linear momentum balance for the mass. F 1 and F 2 act on a mass P as shown in the figure. neglecting gravity.2 m/s2 + 1. 5. 75 Nk 2 2x ˆ 2y ˆ 3 3 ˙ F = L. b) Write the equation in scalar form (use any method you like to get two scalar equations in the two unknowns F1 and F2 ). Note also that 32 + 42 + 122 = 132 . and F 3 = F3 λ. velocity.2m B T2 1m z 1. and f) rate of change of kinetic energy E˙K . and f) rate of change of kinetic energy E˙K . ˆ and a 1 = 6 m/s2 . Given that the tensions T1 = 1200 N and T2 = 1500 N.2m 1. ∗ .5 + 1 k ) m/s2 .5ı + 2. b) Find the angular momentum of the particle about the origin at the outset of motion (h = 0).2. and lbf/ lbm = g. v0 = 100 m/s. (Filename:pfigure. and F3 . v 1 = −4 m/s . [ a) Given the free body diagram in the figure. c) Write the equation in matrix form. d) Find F1 = |F 1 | and F2 = |F 2 | by the following methods: (a) from the scalar equations using hand algebra.63a) 1.112 A particle of mass m = 250 gm is shot straight up (parallel to the y-axis) from the xaxis at a distance d = 2 m from the origin. and ˆ a 2 = −4 m/s2 . F2y . 5. The acceleration of the mass is somehow measured to be a = ˆ ˆ −2 ft/s2 ı + 5 ft/s2 . Use the dimensions and quantities given.2m 1m y y P 30o F1 1. The forces actˆ ˆ ing on the particle are F 1 = 25 Nı + 32 N + ˆ ˆ . and z directions.117: m 12' 4' 3' y F = m a . respectively.115 The rate of change of linear momentum of a particle is known in two directions: ˙ ˙ L x = 20 kg m/s2 . y.114 Three forces. separate the vector equation Using into scalar equations in the x. respectively. d) rate of change of angular momentum ˙ about the origin H O . velocity. ˆ ˆ and acceleration r 1 = 3 mı .116: (Filename:efig1. lbm is a pound mass. and T3 .2m B 1. F = F ı +F and F = −F k .2m 1. c) angular momentum about the origin H O. They have unknown tensions T1 . Solve these equations (maybe with the help of a computer) to find F2x .2m x ˆ T1 λ1 ˆ T2 λ 2 C mg 1m 1m y problem 5. Two forces 5. The acceleration of the mass is ˆ ˆ 3ˆ given as a = (−0. a particle of mass m 1 = 5 kg has position. find its a) linear momentum L.7) ˆ 5. CONTENTS z A T1 1. For the system of particles. The acceleration of the body is ˆ ˆ ˆ a = −0. act on a body with 2 mass 2 kg. Use g = 32 ft/s2 . a) Write the equation F = ma in vector form (evaluating each side as much as possible). and (c) from the vector equation using a cross product. the only force acting on the mass shown in the figure is from the string. a = 10 m/s2 and h is the height of the particle from the x-axis. a) Find the linear momentum of the particle at the outset of motion (h = 0).5 lbm x problem 5. (b) from the matrix equation using a computer.111 At a particular instant of interest. F 2 = F2x ı √ F2z k . d) rate of change of angular momentum ˙ about the origin H O .2m C x FBD F2 45o x (Filename:h1. P has mass 2 lbm.blue.26 ˙ b) rate of change of linear momentum L. Recall that lbf is a pound force. e) kinetic energy E K . e) kinetic energy E K . T2 . and a particle of mass m 2 = 5 kg has position.113 For a particle. ˆ ˆ ˆ where λ = 1 ı + 23 . 5. There is no gravity. find the magnitude and direction of the F = ma ] acceleration of the block.7 m/s2 k . F2z . b) Find the tension T3 .118 Three strings are tied to the mass shown with the directions indicated in the figure.1) 5. d) Find the angular momentum of the particle about the origin when the particle is 20 m above the x-axis. F 1 = 20 Nı − ˆ ˆ ˆ+ ˆ 5 N .113: 1. Find the acceleration of the mass. v 2 = 5 m/s . F = m a as scalar equaWrite the equation tions and solve them (most conveniently on a computer) for F2x . and ˆ ˆ acceleration r 2 = −6 mı . L y = −18 kg m/s2 and unknown in the z direction.2 m/s2 ı + 2.116 A block of mass 100 kg is pulled with two strings AC and BC. z a=? string pulley 3 lbf ˆ k ˆ ˆ ı m = 2. 5. c) Find the linear momentum of the particle when the particle is 20 m above the x-axis. The collision is completely elastic (the coefficient of restitution e = 1). c) Find the three tensions T1 .s94f1p3) ˆ λ ˆ n v0 θ problem 5. Neglect gravity.124 Particle moves on a strange path. the plot that results from running these commands.6ı − 0. a) Accurately plot the trajectory of the particle. Defend your answer. Take O to be the origin of an x y coˆ ordinate system aligned with the unit vectors ı ˆ and a) Assume you are given the position of ˆ ˆ the person r = x ı +y and the velocity ˙ˆ ˙ ˆ of the person v = x ı + y .2) 5.23. a) Draw as accurately as you can.128 Bungy Jumping.121 Use a computer to draw a square with corners at (1. ˆ ˆ ˆ λ = − cos θ ı + sin θ ˆ ı= ˆ = ˆ ˆ sin θ n + − cos θ λ ˆ ˆ cos θ n + sin θ λ b) For what value of θ would the vertical component of the speed be maximized? z T2 T1 m x 1.Problems for Chapter 5 ˆ λ3 = 1 ˆ ı 13 (−3ˆ + 12 27 ˆ + 4k) T3 = ? a) You can program the circular shape any way that you think is fun (or any other way if you don’t feel like having a good time). and friction. more complicated motions will be more difficult. and no computer is available at the moment. ˆ ˆ ˆ ˆ θ. assigning numbers etc. 5. You may express your anwhere x and y are the horizontal distance in meters and t is measured in seconds. ı . and the other parameters given.77 s obeying problem 5. ˆ n= ˆ ˆ sin θ ı + cos θ 5. You may find these tensions by using hand algebra with the scalar equations.120 A small object (mass= 2 kg) is being pulled by three strings as shown. 0). 5. b) See if you can guess a mechanical situation that is described by this program.127: ball 4m 2m 1m 2m T3 y x (Filename:ballramp) y 5.2 + 2. Sketch the system and define the variables to make the script file agree with the problem stated. For the situation shown the spring AB has rest length 0 = 2 m and a stiffness of k = 200 N/ m.119 For part(c) of problem 4. Given that a particle moves in the x y plane for 1.125: (Filename:pfigure. 5. One can draw a circle quite well with a compass or with simple drawing programs. they should be within 10% of each other (mark the dimensions by hand on your drawing). or by using a cross product on the vector equation.118: (Filename:pfigure. A simple example is that of a particle going in circles at a constant rate. Draw a circle on the computer and label the drawing (using computer generated lettering) with your name and the date. 0). mass.123 What curve is defined by x = cos(t)2 and y = sin(t) ∗ cos(t) for 0 ≤ t ≤ π ? Try to figure it out without a computer. (0. n. Use λ’s as unit vectors along the strings. b) A good solution will clearly document and explain the computer methodology. −1).blue. and T3 at the instant shown. unfortunately. In a new safer bungy jumping system. Use the numbers given.29.122 Draw a Circle on the Computer. Measure its length and width with a ruler.126 A particle is blown out through the uniform spiral tube shown. 1). T2 . ˆ ˆ r = (5 m) cos2 (t 2 / s2 )ı +(5 m) sin(t 2 / s2 ) cos(t 2 / s2 ) collision. people jump up from the ground while suspended from a rope that runs over a pulley at O and is connected to a stretched spring anchored at B. a) Draw a free body diagram of the mass.5 + 1 k) s 2 ı problem 5. using a computer with the matrix equation. . 5.127 A ball going to the left with speed v0 bounces against a frictionless rigid ramp which is sloped at an angle θ from the horizontal. v0 .5ˆ + 2.5m C problem 5. which lies flat on a horizontal frictionless table. The inextensible massless rope from A to P has length r = 8 m. and λ.blue. Explain how you know these points are the fast and slow places. But. a) Find the velocity of the ball after the swer in terms of any combination of m.1) ˆ k ˆ ı ˆ m = 6 kg z ˆ T3 λ3 T2 T1 y x ˆ 3ˆ m a = (−.125 COMPUTER QUESTION: What’s the plot? What’s the mechanics question? Shown are shown some pseudo computer commands that are not commented adequately. This must be done with scientific software and not with a purely graphics program. her velocity. The pulley has negligible size. Find her acceleration in terms of some or all of her position.126: (Filename:pfigure. (−1.120: (Filename:pfigure. Here begins a series of problems concerning the motion of particles over noninfinitesimal time. We will be interested in keeping track of the motions of systems.s94h2p9) 5. (0. 5.1.0k m/s2 . the person has a mass of 100 kg. Make a computer plot. Draw the particle’s path after it is expelled from the tube. where supplied. The acceleration of the object at the position shown is ˆ ˆ ˆ a = −0. assume now that the mass at A has non-zero acceleration of ˆ ˆ ˆ (1m/s 2 )ı + (2m/s 2 ) + (3m/s 2 )k . 5. . Find the tension in the three ropes at the instant shown. b) Mark on your plot where the particle is going fast and where it is going slow. z2=1} Solve ODEs with ICs from t=0 to t=5 plot z2 and z1 vs t on the same plot problem 5. b) Write the equation of linear momentum balance for the mass. Your circle should be round. in your final answer. ODEs = {z1dot = z2 z2dot = 0} ICs = {z1 =1. 135 A satellite is put into an elliptical orbit around the earth (that is. The cannon ball is launched at 100 m/s at a 45 degree angle. The speed of the ball is given ˙ ˙ by v 2 = x 2 + y 2 .1) 5. 45◦ from horizontal. The player controls the angle θ at which the ball is ejected and the initial velocity vo . conserved? ground. • Write linear momentum balance in vector form. The coordinates of A and B are (2 . Rewrite these equations as a system of 4 first order ODE’s suitable for computer solution. solve for the x and y coordinates of the ball and make a plots its trajectory for various initial angles θ0 . modelled as a point. ˆ O 10 m A k B ˆ ı g = 10 m/s2 m P problem 5. where is your favorite length unit. Such descriptions come later in the text. The force on the missile from the earth’s gravity is F = mg R 2 /r 2 and is directed towards the center of the earth. . Break this equation into x and y components. the object of the game is to propel the small ball from the ejector device at O in such a way that is passes through the small aperture at A and strikes the contact point at B. Write appropriate initial conditions for the ODE’s. as follows.128: (Filename:s97p1.133 In the arcade game shown. • Solve the equations on the computer and plot the trajectory.] vP RE A vA problem 5.blue. 5. you can assume the orbit is closed) and has a velocity v P at position P.9 Central force motion and celestial mechanics Experts note that none of these problems use polar or other fancy coordinates.blue.129 A softball pitcher releases a ball of mass m upwards from her hand with speed v0 and angle θ0 from the horizontal. you can think of this problem as two-dimensional in the plane shown. and v0 = 9000 m/s r (t) is the distance of the missile from the center of the earth. ∗ 5. respectively. 5. 2 ) and (3 . The only external forces acting on the ball after its release are gravity and air resistance. A missile. which may or may not be vertical. respectively. • Draw a free body diagram of the mass. Runge-Kutta.130 Find the trajectory of a not-verticallyfired cannon ball assuming the air drag is proportional to the speed. r A and r P . d) Use your simulation to find the initial angle that maximizes the distance of travel for ball. a) Draw a free body diagram of the missile. after its release. a) What is the equation of motion for the ball after its release? b) What are the position. c) Find the answer to part (b) with pencil and paper (a final numerical answer is desired).134 Under what circumstances is the angular momentum of a system. That is.136 The mechanics of nuclear war. The only external force acting on the ball after its release is gravity.133: (Filename:pfigure. R = 6. solve the equations numerically. [Hint: both total energy and angular momentum are conserved. 000 m is the radius of the earth. velocity. what is a qualitative description for the curve described by the path of the ball? 5. F d = −bv 2 et .132 The equations of motion from problem 5.32.3) 5. and initial speed v0 . For the purposes of this calculation ignore the earth’s rotation. no contact after jump off 5. Make a computer simulation of the flight of the baseball. The radii to A and P are. the drag proportionality constant is C = 5 N/( m/s).131 are nonlinear and cannot be solved in closed form for the position of the baseball.132. A baseball pitching machine releases a baseball of mass m from its barrel with speed v0 and angle θ0 from the horizontal. • Solve the equations by hand and then use the computer to plot your solution. find the equation of motion for the ball. is launched on a ballistic trajectory from the surface of the earth. ). Assume the mass is 10 kg. Instead. At this point we want to lay out the basic equations and the qualitative features that can be found by numerical integration of the equations. If you need numbers. Find an expression for the velocity v A at position A. Write the linear momentum balance equation. and acceleration of the ball? c) What is its maximum height? d) At what distance does the ball return to the elevation of release? e) What kind of path does the ball follow and what is its equation y as a function of x? 5. or other suitable method to numerically integrate the system of equations.131 See also problem 5. with and without air resistance. 400. When it is launched from the equator it has speed v0 and in the direction shown. c) Use Euler’s. g = 10 m/s2 is earth’s gravitational constant at the earth’s surface. CONTENTS 5. calculated relative to a point C which is fixed in a Newtonian frame. e) If the air resistance is very high.2) rA rP P y θ A B O ejector device x problem 5.135: (Filename:pfigure. a) Convert the equation of motion into a system of first order differential equations.64. the coefficient of resistance b. Taking into account air resistance on the ball proportional to its speed ˆ squared. Find the value of θ that gives success. The trajectory is confined to the frictionless x y-plane. in cartesian coordinates. g = 10 m/s. ∗ b) Pick values for the gravitational constant g. use the following values: m = 1000 kg is the mass of the missile.28 b) Given that bungy jumper’s initial posiˆ ˆ tion and velocity are r0 = 1 mı − 5 m and v0 = 0 write MATLAB commands √ to find her position at t = π/ 2 s. a) Is the total linear momentum conserved? Justify your answer. 29 ˙ b) rate of change of linear momentum L.140 A particle of mass m 1 = 5 kg and a particle of mass m 2 = 10 kg are moving in space.blue. Express your answer in terms of v1 .1) a 2 c) angular momentum about the origin H O. velocity. ˆ ˆ and a 1 = 10 m/s2 ı − 24 m/s2 . and acceleration ˆ ˆ ˆ ˆ r 1 = 3 mı +2 m .137: 5. a) Find the linear momentum L and its ˙ rate of change L of each particle at the instant of interest. d) rate of change of angular momentum ˙ about the origin H O . They are connected by a linear elastic cord whose center is stationary in space. b) Write the differential equations that describe the motion. particle 1 has position.141 Two particles each of mass m are connected by a massless elastic spring of spring constant k and unextended length 2R. [Use a much shorter time when debugging your program.72. What does this mean? What is F ? What is m? What is a ? 5. break into x and y components. [hint: Write F = m a .139 A particle of mass m 1 = 6 kg and a particle of mass m 2 = 10 kg are moving in the x yplane. y 45o x problem 5. so that no net external forces act. and particle 2 has position. respectively.blue. where k = 200 N/m and r is the position of the particle relative to the force center. and a 2 = 5 m/s2 ı − 16 m/s2 . The system slides without friction on a horizontal table. v 2 = 8 m/sı + ˆ ˆ ˆ 4 m/s . e) kinetic energy E K . and accelˆ ˆ ˆ eration r 2 = −6 mı − 4 m . velocity.141: (Filename:pfigure. particle 1 has position. and particle 2 has position. velocity. b a O b 1 v1 (Filename:pfigure. √ b) ω = k/m . Neglect all other forces. and acceleration 5. a) Show that circular trajectories are possible.10 Coupled motions of particles in space problem 5.8 m. c) Find the center of mass of the system at the instant of interest. We wish to consider the motion of one of the skaters by modeling her as a mass m held by a cord that exerts k Newtons for each meter it is extended from the central position. b) Can the center of mass accelerate? Justify your answer. and acceleration r2 v2 a2 = = = ˆ 2 mı ˆ 1 m/sk ˆ 1 m/s2 respectively. √ c) ω > k/m. as shown. the velocity of the particle at point 1 is observed to be perpendicular to the radial direction. d) Find the velocity and acceleration of the center of mass. and f) rate of change of kinetic energy. v 1 = −16 m/sı +6 m/s .] On the same plot draw a (round) circle for the earth. c) Draw free body diagrams for each mass.136: (Filename:pfigure.65. velocity. respectively.] b) It turns out that trajectories are in general elliptical. d) Derive the equations of motion for each mass in terms of cartesian coordinates. 5. y R m θ R m problem 5.137 A particle of mass 2 kg moves in the horizontal x y-plane under the influence of a central force F = −k r (attraction force proportional to distance from the origin).138 Linear momentum balance for general systems with multiple interacting parts moving more or less independently reduces to F = m a if you interpret the terms correctly. pick fortuitous values for the free constants in your solutions. a) linear momentum L. as depicted in the diagram. When the particle reaches point 2.2) x r1 v1 a1 = = = ˆ ˆ 1 mı + 1 m ˆ 2 m/s 2ˆ 3 m/s k 5. e) What are the total kinetic and potential energies of the system? f) For constant values and initial conditions of your choosing plot the trajectories of the two particles and of the center of mass (on the same plot). find its . Determine the speed increment v which would have to be added (instantaneously) to the particle’s speed at point 2 to transfer it to the circular trajectory through point 2 (the dotted curve). For a particular elliptical trajectory with a = 1 m and b = 0. For the system of particles at the instant of interest. solve the separate scalar equations. with magnitude v1 . its velocity is again perpendicular to the radial direction. a) Draw a free body diagram showing the forces that act on the mass is at an arbitrary position. respectively. and determine the relation between speed v and circular radius ro which must hold on a circular trajectory. 5. At a particular instant of interest.142 Two ice skaters whirl around one another.Problems for Chapter 5 b) Using the computer (or any other means) plot the trajectory of the rocket after it is launched for a time of 6670 seconds. b) Find the linear momentum L and its ˙ rate of change L of the system of the two particles at the instant of interest. c) Describe in physical and mathematical terms the nature of the motion for the three cases √ a) ω < k/m . At a particular instant of interest.s94q12p1) 5. 5.67m B v B − 0. and x y CONTENTS ice skater 1 r 1 ice skater 2 problem 5. v0 . respectively. 5.146 Set up the following equations in matrix form and solve for v A and v B .153 Reconsider problem 5. and v Bx : b) the velocity of the bullet as it travels from block A to block B.142: (Filename:pfigure. of v A and v B . v B .150. m 0 v0 = 24. Use any computer program.3 m/s m A − 0.143 Theory question.145.2) 5.144 The equation ( v 1 − v 2 ) · n = e( v 2 − ˆ v 1 ) · n relates relative velocities of two point masses before and after frictionless impact ˆ in the normal direction n of the impact. 5.145 See also problem 5. and vC are the only unknowns. find the initial velocity of puck B. m A = x 1.1) 0 (fixed) problem 5. γ γ (VB)i problem 5.0 m/s m A + 0. where O is not at the center of mass? If so.5 m/s. (V A )i = 0. v 2 = 0 .32.148 See also problem 5. how and why? If not. Given that (V B )i = 1. ∗ ˆ 5. The bullet causes A and B to start Write the equations in matrix form set up to moving with speed of v A and v B respectively. write the following equations in matrix form set up to solve for v Ax and v Ay : sin θv Ax + cos θv Ay = ev0 cos θ cos θv Ax − sin θv Ay = v0 sin θ.0 m/s m a − 0. m A = 2 kg.67m B v B − 0.148 taking θ = 50o . If ˆ ˆ ˆ v 1 = v1x ı + v1 y . v 2 = −v0 ı .154: 5. and (V A ) f = 0. then give a reason and/or a counter example. The following a) the initial speed v0 of the bullet in terms three equations are obtained to solve for v Ax . the blocks fly apart and the spring falls to the ground.blue.blue.and ∗ (v − v ) cos θ = v sin θ − 10 m/s Bx Ax Ay v Ax sin θ = v Ay cos θ − 36 m/s m B v Bx + m A v Ax = (−60 m/s) m A . Assuming θ. e = ˆ − 5 ft/s . Puck A is initially at rest.blue. solve for v Ax and v Ay if θ = 20o and v0 = 5 ft/s.149. Determine 5.0 m/s.) Set up these equations in matrix form.5.152 Two frictionless pucks sliding on a plane collide as shown in the figure. block A of mass m and becomes embedded in A block B of mass m B . v0 m mA A mB B (Filename:pfigure. v A .30 (You are not asked to solve the equation of motion.155 A massless spring with constant k is held compressed a distance δ from its relaxed length by a thread connecting blocks A and B which are still on a frictionless table. the position.6 m/s. respectively. find the approach angle φ and rebound angle γ . and (V A ) f = 0.151 Two frictionless masses m A = 2 kg and mass m B = 5 kg travel on straight collinear paths with speeds V A = 5 m/s and V B = 1 m/s. by thin wires.8. Find the speed of block A as it slides away.9. ∗ v Ay . e = 0. and the acceleration of the center of mass of a system of particles can you find the angular momentum H O of the system. and e to be known quantities.5 m/s.5 m B and m B = 0.8 kg.33m B v B + 0.58m C vC 0 = 36.152.149 Solve for the unknowns v A . The coefficient of restitution is e = 0. The coefficient of restitution is e = 0. (VA)f (VB)f (VA)i=0 A φ 5. 5.1) θ v B in problem 5. k mA A VA problem 5. Given instead that γ = 30◦ . the velocity. spring. ∗ 5. and m B = 500g: m A v0 = m A v A + m B v B −ev0 = v A − v B . v 1 = 2 ft/sı 1 ˆ ˆ ˆ n = √ (ı + ). The masses collide since V A > V B .5. Find the amount of energy lost in the collision assuming normal motion is decoupled from tangential motion.154.154 A bullet of mass m with initial speed v0 is fired in the horizontal direction through 0 = 23.147 The following three equations are obtained by applying the principle of conservation of linear momentum on some system. 5. ∗ . find the scalar equation relat2 ing the velocities in the normal direction. If you are given the total mass.58m C vC . c) the energy loss due to friction as the bullet (1) moves through block A and (2) penetrates block B. The thread is suddenly but gently cut.150 Using the matrix form of equations in Problem 5. (V A )i = 0.23. The blocks have mass m A and m B . and ˆ 0. solve for the unknowns. Each block is suspended Assume v0 .3m C vC 5.155: (Filename:pfigure. if v0 = 2.152: B (Filename:Danef94s2q8) 5.151: B VB mB thread (Filename:Danef94s3q7) problem 5. 27. the string. are connected together across their tops by a massless string of length S.3. b) Write separate equations for linear momentum balance for each block.Problems for Chapter 6 31 Problems for Chapter 6 Constrained straight line motion d) What is the force in the string? e) What is the speed of the center of mass for the two blocks after they have traveled a distance d down the slope.8: (Filename:pfig2. At some instant in time t.blue. f) How would your solutions to parts (a) and (c) differ in the following two variations: i.) if the string were replaced by a massless rod? Qualitative responses to this part are sufficient. and the system of the two blocks and string. the box. What is the tension in the string AB? B A m problem 6. The string can only withstand a tension Tcr . 6. They slide down a slope of angle θ .1 Write the following equations in matrix form for a B . the string. Two masses connected by an inextensible string hang from an ideal pulley.rp8) 6.newtrain) 6. Do not neglect gravity but do neglect friction. Assume there is no resistance to rolling for all of the cars. a) Draw separate free body diagrams of each block. and the system of blocks and string. 2 m s m 6.9 Pulley and masses.2: µ m2 (Filename:Danef94s3q5) 6.] 6. d) Find the displacement of the cart at the time when the box has moved a distance along the cart. Assume the cars are connected with rigid links. The power of the engine is Pt and its speed is vt . having started from rest. the blocks’ dimensions are small compared to S. and the system of the two blocks and string.6 A cart of mass M. b) Write the equation of linear momentum balance for the cart. c) What is the acceleration of the center of mass of the two blocks? d) What is the force in the string? e) What is the speed of the center of mass for the two blocks after they have traveled a distance d down the slope. having started from rest.28.2 The two blocks.1) 6. a small box of mass m slides along the cart from the front to the rear. and the system of blocks and string. the point of application A of the force moves twice as fast as the mass. There is gravity. Find the input power to the system at time t. Find the maximum value of the applied force P so that the string does not break. When t = 0.5: (Filename:pfigure. aC . and the system of cart and box. the string. During the acceleration of M by the force F. a) Draw free body diagrams of the cart.blue. a force F is applied as shown to the cart. the blocks’ dimensions are small compared to S.27. the speed of the mass is x to ˙ the left.12. each of mass m. ∗ n=1 n=2 n=N-2 n=N-1 n=N m m problem 6.1a) problem 6.blue.5 Two blocks.8 For the mass and pulley system shown in the figure. m 1 = m 2 = m.2) 6.4: (Filename:pfigure. m M problem 6. the box. are connected by an inextensible string AB. [Hint: You need to dot your momentum balance equations with a unit vector along the ramp in order to reduce this problem to a problem in one dimensional mechanics. c) What is the acceleration of the center of mass of the two blocks? 6. each of mass m. The coefficient of friction between the cart and box is µ. and T : 4a B + aC = 0 2T = −(25 kg) a B 64 kg m/s = T + (10 kg) ac . They slide down a slope of angle θ. θ problem 6. Find the tension Tn between car n and car n+1. g B P m1 problem 6. A m s F m m θ problem 6. and the cart and box together. .) If the two blocks were interchanged with the slippery one on top or ii.6: m F no friction (Filename:pfigure. a) Draw separate free body diagrams of each block. The materials are such that the coefficient of dynamic friction on the top block is µ and on the bottom block is µ/2.1) c) Show that the equations of motion for the cart and box can be combined to give the equation of motion of the mass center of the system of two bodies. are connected together across their tops by a massless string of length S.blue.4 Two blocks.3: m m m (Filename:pfigure.3 A train engine of mass m pulls and accelerates on level ground N cars each of mass m. can move horizontally along a frictionless track. and it is assumed that the acceleration of the cart is sufficient to cause sliding.7 A motor at B allows the block of mass m = 3 kg shown in the figure to accelerate downwards at 2 m/s2 . initially at rest.1 1-D constrained motion and pulleys 6.7: (Filename:pfigure. b) Write separate equations for linear momentum balance for each block. the string. The sliding coefficient of friction between the blocks and the ground is µ. find the acceleration of the mass A and the point B shown using balance of linear momentum ( F = m a ). coordinates or sign conventions that you need to do your calculations and to define your solution. Make reasonable assumptions about strings.19. Find the acceleration of points A and B in terms of F and m.f93q4) 6.29. ı and a0 . ∗ . coordinates. Point B is attached to the cart and point A is attached to the rope.13: 6. coordinates. or sign conventions that you need to do your calculations and to define your solution. 6.11 What is the acceleration of block A? Use g = 10 m/s2 .12 For the system shown in problem 6.13 find the acceleration of the ˙ mass using energy balance (P = E K ).9. round. coordinates.19 For the various situations pictured in problem 6. 32 kg B m problem 6. modeled as a rigid body is sitting on a cart of mass M > m and pulling the massless inextensible string towards herself. or sign conventions that you need to do your calculations and to define your solution. coordinates or sign conventions that you need to do your calculations and to define your solution. Define any variables.10 The blocks shown are released from rest. and have frictionless bearings.10: (Filename:pfigure. Define any variables.16 See also problem 6. As a check.9: (Filename:pfigure3.18 find the acceleration of the mass us˙ ing energy balance (P = E K ). and the present velocities of the blocks. a) What is the acceleration of block A at t = 0+ (just after release)? b) What is the speed of block B after it has fallen 2 meters? problem 6. a A/B ≡ a A −a B = −a0 ı ) and that she is not slipping relative to the cart.18 See also problem 6. find the acceleration of the mass A and the point B shown using balance of linear momentum 6. g. ∗ A B 30o g m2 60o A (c) A B m A = mB = m B problem 6. 6.p3) B F 6. ∗ ( F = m a ).f95p1p2) A B F A F B (a) b) ∗ A m B F F (d) m A B (b) m1 (Filename:pulley1) 6. your answer should give a B = g when m A = 0 and a B = 0 when m A = m B .s94q5p1) . m B . Define any variables. ∗ CONTENTS 6. For the various situations pictured. µ. and have wheels of negligible mass. (Answer in terms of ˆ some or all of m. pulleys.14 For each of the various situations pictured in problem 6. 6. g. As a check.p1. 6. find the acceleration of mass B using energy bal˙ ance (P = E K ).) A 2 kg problem 6. m A B m F problem 6. a) If you are given that she is pulling rope in with acceleration a0 relative to herˆ self (that is.15: (Filename:sum95.13 For the various situations pictured. the strings are inextensible. and gravity. a) ∗ (a) m A B F (b) m g mA (c) m mB problem 6.17 For the situation pictured in problem 6.18: (Filename:pulley4) A m F 6. or sign conventions that you need to do your calculations and to define your solution.2) m F massless problem 6. The coefficient of friction between her seat and the cart is µ. Answer in terms of any or all of m A . and the mass and force the same. the pulleys massless. Assume the string is massless and that the pulleys are massless. Define any variables. M. find a A . b) Find the tension in the string.16 find the accelerations of the two masses using ˙ energy balance (P = E K ).15 What is the ratio of the acceleration of point A to that of point B in each configuration? In both cases. Define any variables. All wheels and pulleys are massless and frictionless.17.20 A person of mass m. have good (frictionless) bearings. string lengths.11: ˆ ˆ ı (Filename:pfigure.blue. your answer should give T = m B g = m A g when m A = m B and T = 0 when m A = 0. Assume that the carts stay on the ground.16: (Filename:pfigure.32 a) Find the downward acceleration of mass B. Assume a massless pulley with good bearings.25: (Filename:pfigure. Define any variables. do the following. All rotating parts have negligible mass. find the acceleration of point A using energy ˙ balance (P = E K ). carefully keep track of string length to figure spring stretch. b) If ω = 0.151. You may assume her legs get out of the way if she slips backwards. find the acceleration of the mass A and the points B and C shown using balance of linear momentum ( F = m a ). pulleys and wheels. as shown in the figure.s95q4) δ(t) g A 6. [Hint: the situation with point C is tricky and the answer is subtle. Define any variables.) c) If instead. A C B ˆ ı 6.27 The block of mass m hanging on the spring with constant k and a string shown in the figure is forced (by an unseen agent) with the force F = A sin(ωt). a) Derive the equation of motion for the block in terms of the displacement x from the static equilibrium position. check to see if the pulley is always in contact with the cable (ignore the transient solution). k m y ˆ g m ˆ ı ideal pulley m M B problem 6.) ∗ 6. Only vertical motion is of interest.) 33 6. M.25 Pulley and spring. What is the natural frequency of vibration of this system? ∗ problem 6.21: (Filename:p. m A . [Hint: Draw FBD. (Filename:s97p2.23: (Filename:pulley2) ˆ F m m problem 6. Include gravity.s96. (Do not neglect gravity). is pulled to the right a distance d from the position it would have if the spring were relaxed. g and µ. for what values of ω would the string go slack at some point in the cyclical motion? (You should neglect the homogeneous solution to the differential equation. a) What is the differential equation governing the motion of the block? You may assume that the only motion is vertical motion.1) m problem 6.23.blue. Assuming that the cable is inextensible and massless and that the pulley is massless.blue. and b) the tension in the line.20: Pulley. There is gravity. The spring has constant k.26: x (Filename:pfigure. g and µ.22 The pulleys are massless and frictionless.22: (Filename:pfigure. For the situation pictured. as shown in the figure. You may assume her legs get out of the way if she slips backwards. a) What is the acceleration of block A just after it is released (in terms of k.p1.] k m F = A sin(ωt) k g 6. k = 10 lb/ f t) is excited by moving the free end of the cable vertically according to δ(t) = 4 sin(ωt) in.] Static equilibrium position at δ = 0 and x = 0 m problem 6. Two equal masses are stacked and tied together by the pulley as shown. The massless string is inextensible. set up equations of motion and solve them. or sign conventions that you need to do your calculations and to define your solution.1) 6. what is the largest possible value of a0 without the person slipping off the cart? (Answer in terms of some or all of m.9 rad/s.26 The spring-mass system shown (m = 10 slugs (≡ lb·sec2 / f t). For the mass hanging at the right.155. It is then released from rest.23 See also problem 6. find the period of oscillation. ∗ b) Given A. problem 6. with mass m A . M.24 For situation pictured in problem 6. y measures the vertical position of the upper mass from equilibrium.24.27: (Filename:pfigure.2) x problem 6. All bearings are frictionless. x measures the vertical position of the lower mass from equilibrium.s94h4p4) 6. Find a) the acceleration of point A.21 Two blocks and a pulley. m and k. coordinates or sign conventions that you need to do your calculations and to define your solution. Assume ideal string.Problems for Chapter 6 b) Find the largest possible value of a0 without the person slipping off the cart? (Answer in terms of some or all of m. coordinates. m < M.1) m A C B F 6.28 Block A. The pulley is massless. and d)? ∗ . Neglect air friction. Let the accel¨ˆ eration of the block be a = x ı . Consider the mass at B (2 kg) supported by two strings in the back of a truck which has acceleration of 3 m/s2 . The piston has upˆ wards acceleration a y .s94h2p8) ˆ ˆ ı 4 C F2 k.34 A point mass of mass m moves on a frictional surface with coefficient of friction µ and is connected to a spring with constant k and unstretched length .28: ˆ ı x (Filename:pfigure. m problem 6.2D.a) 6.30: (Filename:pulley3) G C E 2a x 6.37: (Filename:ch2. ∗ (Filename:pulley3. What is the tension T AB in the string AB in Newtons? 1m A 1m B C y D E G x problem 6.10a) problem 6. The mass of D and forces F1 and F2 are unknown. The cart has accelˆ eration ax ı due to a force F. Define any variables. b) At the instant of interest. The cart has acceleration ax ı due to a force not shown.29 What is the static displacement of the mass from the position where the spring is just relaxed? B m 3 3 4 A F1 D problem 6. F1 and F2 are applied so that the system shown accelerˆ ates to the right at 5 m/s2 (i. The dimensions are as shown.30 For the two situations pictured.33 A point mass m is attached to a piston by two inextensible cables. the mass is at a distance x to the right from its position where the spring is unstretched and is moving with x > 0 to the ˙ right. In terms of some or all of m. Assuming both masses are deflected an equal distance from the position where the spring is just relaxed.guyed) B ˆ problem 6. (a) L m A K. a) Draw a free body diagram of the plate. CONTENTS 6.e.2 2D and 3D forces even though the motionis straight 6.1) 6. There is gravity. Define any variables. a = 5 m/s2 ı + ˆ 0 ) and has no rotation. (Filename:tfigure3.L0 A A 3a 3a a D 2a B ˆ ˆ ı problem 6. find the acceleration of the ˙ mass using energy balance (P = E K ).36 Guyed plate on a cart A uniform rectangular plate ABC D of mass m is supported by a rod D E and a hinge joint at point B. d. find the acceleration of point A shown using balance of linear momentum ( F = m a ). write the equation of linear momentum balance for the block evaluating the left hand side as explicitly as possible.12) .35: (Filename:pfigure. There is gravity.f93q5) 6. There is gravity.13) g ˆ ı 6.34: (Filename:ch2. g. Use your favorite value for the gravitational constant.31 For each of the various situations pictured in problem 6.32 Mass pulled by two strings.32: (Filename:pg9. c) Write the equation for angular momentum balance about point E and evaluate the left hand side as explicitly as possible.35 Find the tension in two strings. What must the acceleration of the cart be in order for the rod D E to be in tension? ∗ 3 A 2 B D C 4 ˆ E ˆ ı F A 9d 5d C 6d cart 6. (b) m d K. coordinates. coordinates or sign conventions that you need to do your calculations and to define your solution. or sign conventions that you need to do your calculations and to define your solution. What is the tension in string AB? B C A 6.36: Uniform plate supported by a hinge and a cable on an accelerating cart. b) Write the equation of linear momentum balance for the plate and evaluate the left hand side as explicitly as possible.34 b) What is the speed of the mass when the mass passes through the position where the spring is relaxed? ∗ .a. and a y find the tension in cable AB. L0 d A m problem 6. coordinate system origins.. At the instant of interest.33: (Filename:ch3.29: g problem 6. problem 6. There is gravity.L0 6. how much smaller or bigger is the acceleration of block (b) than of block(a). a) Draw a free body diagram of the mass at the instant of interest.37 A uniform rectangular plate of mass m is supported by two inextensible cables AB and C D and by a hinge at point E on the cart as ˆ shown.30. 9) P 2d 6. m r .42: problem 6.40: (Filename:ch3.5 m/s2 ı .41 Reconsider the block in problem 6.44 Forces of rod on a cart. what are the forces on the disk due to the groove? . b) The force on the rod from the cart at point A. A block of mass m is sitting on a frictionless surface and acted upon at point E by the horizontal force P through the center of mass. Find the tension in cable C D.46 The box shown in the figure is dragged in the x-direction with a constant acceleraˆ tion a = 0. the mass is moving to the right at speed v = 3 m/s. This time. Find the acceleration of the block and reactions on the block at points A and B.8 m/sı . θ and F. 2b C b D d E G A problem 6. Find the tension in cable AB. Find: a) The force on the rod from the cart at point B.45 At the instant shown. A uniform rectangular plate of mass m is supported by an inextensible cable C D and a hinge joint at point E on the cart as shown. (What’s ‘wrong’ with this problem? What if instead point B was at the bottom left hand corner of the plate?) ∗ G A problem 6.40.s94h3p3) F 3 A B ˆ ˆ ı 3 D 2 G C E 2 6.mom.Problems for Chapter 6 6.46: 30o O x (Filename:pfigure3. and d.41. The cart is rolling on wheels that are modeled as having no mass and no bearing friction (ideal massless wheels). 6. Are the acceleration and the reactions on the block different from those found if P is applied at point E? 35 A θ B problem 6.43: y m = 2 kg 2m 6. θ.b) P D E d 2d B (Filename:ch3.45: (Filename:pfig2. find the acceleration of the box and the reactions at A and B in terms of W .12) C 6. There is gravity.38: (Filename:ch3. 6. The coefficient of sliding friction between the floor and the points of contact A and B is µ.40 See also 6. Find the rate of work done on the mass. find the acceleration of the block and the reactions on the block at points A and B. The block is resting on a sharp edge at point B and is supported by an ideal wheel at point A. The hinge joint is attached to a rigid column welded to the floor of the cart.rp9) A 3 3 C G E ˆ ı D 2 2 ˆ B 6.11a) d) Find the rate of change of angular momentum of the box about the contact point O. Assuming that the box slides when FC is applied.3.47 (See also problem 4. Neglecting gravity. 6. b) Find the rate of change of linear momentum of the box. There is gravity. Draw a free body diagram of the block. The cart is at rest. A problem 6.43 A force FC is applied to the corner C of a box of weight W with dimensions and center of gravity at G as shown in the figure.39: (Filename:ch3.44: (Filename:pfigure.67. At the instant shown. A uniform rod with mass m r rests on a cart (mass m c ) which is being pulled to the right. F = 50 N x m = 10 kg h frictionless 2b b problem 6. c) Find the angular momentum of the box about the contact point O. Answer in terms of g.15. There is gravity. a = a .11. 1m problem 6. b. find the acceleration of the block and the reactions at A and B if the force P is applied instead at point D.) The groove and ˆ disk accelerate upwards.rp1) B 6.38 See also problem 6. the velocity of (every point on) the box is ˆ v = 0. The rod is hinged at one end (with a frictionless hinge) and has no friction at the contact with the cart.39 A uniform rectangular plate of mass m is supported by an inextensible cable AB and a hinge joint at point E on the cart as shown. FC ˆ a = ax ı θ C 2b b d G b B (Filename:Mikes92p3) problem 6. The hinge joint is attached to a rigid column welded to the floor of the cart. m c . FC .42 A block of mass m is sitting on a frictional surface and acted upon at point D by the horizontal force P. Assuming the block is sliding with coefficient of friction µ at point B. The cart has ˆ acceleration ax ı . There is gravity. a) Find the linear momentum of the box. . In the problems below you should express your solutions in terms of the variables given in the figure. The acceleration a(t) of the conveyer belt gets extremely large sometimes due to an erratic over-powered motor. Does the block tip over? Assuming the block does not tip over. There are a few puzzles in dynamics concerning the differences between front and rear braking of a car. There is gravity. what is the total force that the truck exerts on the box? d) Assuming the box does slide. h. braking inadequately as is the case for rear wheel braking. h. Note that the advantageous points to use for angular momentum balance are now different. ∗ e) Repeat steps (a) to (d) for front-wheel skidding.) b) Write down the equation of linear momentum balance. The bumper is at a height h b above the ground. the gravity force. The box has sharp feet at the front and back ends so the only place it contacts the truck is at the feet. g.47: (Filename:ch2.51 Car braking: front brakes versus rear brakes versus all four brakes. The pickup truck is accelerating forward at an acceleration of at . lr . e. length and depth w (into the paper. and the coordinate directions.ba) g ˆ ı m cm h A B y x at vt θ2 µ problem 6.50 After failure of her normal brakes. l f . but assume the center of mass height h is above the ground. This action locks the rear wheels (friction coefficient = µ) but leaves the well lubricated and light front wheels spinning freely. Label points of interest that you will use in your momentum balance equations. and that there is no friction at the front wheels). g. what is the total force that the truck exerts on the box (i.22. the point at the height of the center of mass.) motor problem 6.48: (Filename:pfigure. here are some steps to follow.e. the coefficient of friction between rubber and road varies between about . The box has height h. should be less than half the wheel base w. d) Solve the momentum balance equations for the wheel contact forces and the deceleration of the car.] c) Assuming the box does slide. that the friction force Fr = µNr . c) Write down the equation of angular momentum balance relative to a point of your choosing. Assume that the car. The car bumper is modeled here as a linear spring (constant = k. h b . m.7 and 1. m. hits a stiff and slippery phone pole which compresses the car bumper. Treat this problem as two-dimensional problem. excepting the bumper.. the sum of the reactions at A and B)? b) Assuming the box does not slide what are the reactions at A and B? [Note: You cannot find both of them without additional assumptions. k.49: (Filename:pfigure. i.5. and vt . a driver pulls the emergency brake of her old car. rest length = l0 .49 A collection of uniform boxes with various heights h and widths w and masses m sit on a horizontal conveyer belt. what is the maximum acceleration of the truck for which the box will not tip over (hint: just at that critical acceleration what is the vertical reaction at B?)? f) What is the maximum acceleration of the truck for which the block will not slide? g) The truck hits a brick wall and stops instantly. l f = lr = w/2).e. present length = ls ). a) Assuming the box does not slide. It is observed that some boxes never tip over.s94q4p1) hb ls .50: (Filename:pfigure. behind the rear wheel that makes a 45 degree angle line down to the rear wheel ground contact point. What is true about µ. What is the peak deceleration of a car when you apply: the front brakes till they skid.. and all four brakes till they skid? Assume that the coefficient of friction between rubber and road is µ = 1 (about right. The height h. Assume the boxes touch the belt at their left and right edges only and that the coefficient of friction there is µ. Further assume that the C M is halfway between the front and back wheels (i. Assume also that the car has a stiff suspension so the car does not move up or down or tip during braking. If you have used any or all of the recommendations from part (c) you will have the pleasure of only solving one equation in one unknown at a time. i. the car does not rotate in the x y-plane. To organize your work. at .) Its mass is m.f93q3) 6. h cm . the distance between the front and rear wheel. and ls (and any other parameters if needed)? y x hcm C lr lf D problem 6. Does a car r y θ1 x problem 6. Neglect the mass of the rotating wheels in the linear and angular momentum balance equations. the bumper is compliant but the suspension is stiff). the car is symmetric left to right.48 The following problems concern a box that is in the back of a pickup truck. Some particularly useful points to use are: the point above the front wheel and at the height of the center of mass. and m for the boxes that always maintain contact at both the right and left bottom edges? (Write an inequality that involves some or all of these variables. If any variables do not enter the expressions comment on why they do not. Pick the dimensions and mass of the car. The car is still traveling forward at the moment of interest. • What is the acceleration of the car in terms of g.. The FBD should show the dimensions. Here is one you can deal with now. the rear brakes till they skid. The friction coefficient between the truck and the box edges is µ. and that the left and right wheels carry the same loads. In all cases you may assume that the box does not rotate (though it might be on the verge of doing so).e. what you know a priori about the forces on the wheels from the ground (i. The car. how far does it slide on the truck before stopping (assume the bed of the truck is sufficiently long)? 6. The center of mass of the box is at the geometric center of the box. is a non-rotating rigid body and that the wheels remain on the ground (that is. what are the reactions at A and B? e) Assuming the box does not slide.36 ˆ CONTENTS 6. e. µ. w.1) 6. .3) and that g = 10 m/s2 (2% error). l0 .blue. a) Draw a FBD of the car assuming rear wheel is skidding. and the point on the ground straight under the front wheel that is as deep as the wheel base is long. The truck’s speed is vt . µ. (Hint: also draw a free body diagram of the rear wheel. does not turn left or right. 9. the mass of the car m.54 Block sliding on a ramp with friction. What is the coefficient of friction µ between the coin and the ramp. and the gravitational constant g. ∗ g) Does the deceleration in (f) equal the sum of the deceleration in (d) and (e)? Why or why not? ∗ h) What peculiarity occurs in the solution for front-wheel skidding if the wheel base is twice the height of the CM above ground and µ = 1? ∗ i) What impossibility does the solution predict if the wheel base is shorter than twice the CM height? What wrong assumption gives rise to this impossibility? What would really happen if one tried to skid a car this way? ∗ the same altitude. where = length of the spring and 0 is the ’rest’ length.51: (Filename:pfigure.54: (Filename:pfigure3. a) What is the force on the block from the ramp at point A? Answer in terms of ˆ ˆ any or all of θ . can the box slide this way or would it tip over? Why? ∗ 6.57 Two blocks A and B are pushed up a frictionless inclined plane by an external force F as shown in the figure. would give the correct answer. The coefficient of friction between the block and slope is µ. a) Given x and x find the acceleration of ˙ the bead (in terms of some or all of ˙ D.57: (Filename:summer95f. . The spring obeys the equation F = k ( − o ). ∗ 6. A square box is sliding down a ramp of angle θ ˆ with instantaneous velocity v ı . m.58 A bead slides on a frictionless rod. . k and any base vectors that you define). At the instant of release. g.52 At time t = 0. m. see if you can write a set of computer commands that.2 µ=0 B A F y x C lr problem 6. the wheel base . (Do not try to solve this somewhat ugly nonlinear equation.56: A car skidding downhill on a slope of angle θ (Filename:ch8. Answer in terms of some or all of m.58: 6. b) If the bead is allowed to move. 0 .53: g θ ˆ ˆ ı problem 6.2) φ problem 6. Find the magnitude of the maximum allowable force such that no relative slip occurs between the two blocks. v. It takes twice as long to slide down as it does to slide up.56 A skidding car. µ=0. the block of mass m is released at rest on the slope of angle φ.52: (Filename:Danef94s1q5) problem 6. g. As a check. a) What is the acceleration of the block for µ > 0? ∗ b) What is the acceleration of the block for µ = 0? ∗ c) Find the position and velocity of the block as a function of time for µ > 0. It is assumed to not tip over. µ. µ. What is the braking acceleration of the front-wheel braked car as it slides down hill. x. x. Use µ = 1. x(t = 0) = 0.p1. k.2.f95p1p1) d) Find the position and velocity of the block as a function of time for µ = 0. ı .Problems for Chapter 6 stop faster or slower or the same by skidding the front instead of the rear wheels? Would your solution to (e) be different if the center of mass of the car was at ground level(h=0)? ∗ f) Repeat steps (a) to (d) for all-wheel skidding. The spring has constant k and rest length 0 . if θ.53 A small block of mass m 1 is released from rest at altitude h on a frictionless slope of angle α. There are some shortcuts here. . The bead has mass m. your answer should reduce ˆ to mg when θ = µ = 0. m. You determine the car deceleration without ever knowing the wheel reactions (or using angular momentum balance) if you look at the linear momentum balance equations carefully.s94h3p6) 30 o problem 6. Express your answer as a function of any or all of the following variables: the slope θ of the hill. The coefficient of friction between the two blocks is µ = 0.car) 6. φ and the initial sliding velocity v. ∗ m µ g 6. ∗ 2 b) In addition to solving the problem by hand.1) h lf D ˆ ˆ ı µ 6.55 A coin is given a sliding shove up a ramp with angle φ with the horizontal.s96. D 0 m x (Filename:p. and . another small block of mass m 2 is dropped vertically from rest at 6.59 A bead oscillates on a straight frictionless wire.) c) In the special case that 0 = 0∗ find how long it takes for the block to return to its starting position after release with no initial velocity at x = x0 . The masses of the two blocks are m A = 5 kg and m B = 2 kg. ∗ 6. m A θ G B µ ˆ ˆ ı problem 6. Assume ˙ x(t = 0) = x0 . What is the velocity of the first block relative to the second block after t seconds have passed? 37 /4 /2 cm d m2 h m1 t=0 α problem 6.7) (Filename:pfigure3. v and g were specified. as constrained by the slippery rod and the spring. find a differential equation that must be satisfied by the variable x. c) Assuming θ = 80◦ and µ = 0. The second block does not interact with the ramp. A very compact bicycler (modeled as a point mass M at the bicycle seat C with height h.64 A 320 lbm mass is attached at the corner C of a light rigid piece of pipe bent as shown.) ∗ 6. g and d. d) For this special case ( o = 0) solve the equation in (a) and show the result agrees with (b) in this special case. There is gravity.60 A cart on a springy leash. Assume all dimensions and masses are known (you have to define them carefully with a sketch and words). Thus. and a . The person is a rigid body.63: D m x problem 6. e) What are the forces on the motorcycle from the rider? M C g a b B (Filename:f. D. b) Draw a Free Body Diagram of the rider. The pipe is supported by ball-and-socket joints at A and D and by cable E F. Points A. The spring has unstretched length 0 and spring constant k. [Hint: find θ first.1) 6. and G H . 0 . a) Assuming that the cart never leaves the floor. The rider applies a force Fp to the pedal perpendicular to the pedal crank (with length L c ). in terms of m. ∗ y E B F 6' ball & socket z 12' problem 6. what is the tension or compression in rod B D? ∗ b) If the truck’s acceleration is a cm = ˆ ˆ (5 ft/s2 ) + (6 ft/s2 )k . One end of an inextensible string is attached to the the cart. what is the maximum possible forward acceleration of the bicycle. modeled as a rigid body. what is the tension or compression in rod G H ? ∗ 6.63 Acceleration of a bicycle on level ground. You may use numbers and/or variables to describe the quantities of interest.60: B g 6. d) Find all forces on the rider from the motorcycle (i. When the cart is at point O. and D are fixed to the truck but the truck is not touching the plate at any other points. but is free to rock in the plane of the motorcycle(as if there is a hinge connecting the motorcycle to the rider at the seat). The person’s feet are off the pegs and the legs are sticking down and not touching anything. The acceleration of the truck is a = ˆ 5 ft/s2 k . A cart B (mass m) rolls on a frictionless level floor. The gravitational constant is g. b) What is x when x = 0? [hint: Don’t ˙ try to solve the equation in (a)!] c) Note the simplification in (a) if o = 0 (spring is then a so-called “zero– length” spring).59: (Filename:pfigure. The entire system is attached to a truck which is moving with acceleration a T . Find the tension in rod B D. a) Draw a clear sketch of the problem showing needed dimension information and the coordinate system you will use..] .3) problem 6.65 A 5 ft by 8 ft rectangular plate of uniform density has mass m = 10 lbm and is supported by a ball-and-socket joint at point A and the light rods C E.B D. Assume the forward acceleration of the motorcycle is known.61: (Filename:Danef94s3q6) CONTENTS O A θ h = 2Rf LC Rf A 6. The cart is pulled a horizontal distance d from the center of the room (at O) and released. what is the speed of the cart when it passes through the center of the room. Find the tension in cable E F. C. The points A. The spring and room height are such that the spring would be relaxed if the end of the cart B was in the air at the ceiling pulley. The ball has mass m.62 Consider a person. The ceiling height is h. c) Write the equations of linear and angular momentum balance for the rider. at the hands and the seat). what is the forward acceleration of the bicycle? [ Hint: draw a FBD of the front wheel and crank. The person’s arms are like cables (they are massless and only carry tension). and E are fastened to the floor and vertical sidewall of a pick-up truck which is accelerating in the zdirection. riding an accelerating motorcycle (2-D). Dimensions are as shown.64: (Filename:mikef91p1) C 15' D ball & socket A 12' x 6' A k h O m d problem 6. Assuming the ball is stationary with respect to the cart find the distance from O to A in terms of k. rides a very light old-fashioned bicycle (all components have negligible mass) that is well maintained (all bearings have no frictional torque) and streamlined (neglect air resistance). 2-D . a) If the truck’s acceleration is acm = (5 ft/s2 )k . h. ∗ (Filename:cartosc) 6. and distance b behind the front wheel contact).e. The motorcycle is accelerating. point A. The person is sitting on the seat and cannot slide fore or aft.blue. ∗ b) Does the cart undergo simple harmonic motion for small or large oscillations (specify which if either)? (Simple harmonic motion is when position is sinewave and/or cosine-wave function of time. and another FBD of the whole bicycle-rider system.60.61 The cart moves to the right with constant acceleration a. a) Assuming no slip. the center of mass acceleration of the plate is the same as the truck’s. The string wraps around a pulley at point A and the other end is attached to a spring with constant k.38 a) Write a differential equation satisfied by x(t). problem 6. The plate is moving without rotation or angular acceleration relative to the truck. No force is applied to the other pedal. it is in static equilibrium. The radius of the front wheel is R f .] ∗ b) (Harder) Assuming the rider can push arbitrarily hard but that µ = 1. 1m B (Filename:pfigure. ∗ h) Challenge: For how many of the reactions can you find one equation which will tell you that particular reaction without knowing any of the other reactions? [Hint.38. The shelf is in the x y plane. C.48m E 1m 1m C B 1m D ˆ ı A 32m ˆ k ˆ z (Filename:pfigure.11) 6. (Filename:ch3. m. • T = the thrust of one engine. In terms of some or all of the following parameters. ∗ e) By taking components.67 See also problem 6. There is gravity.70 Towing a bicycle. There is no gravity. . G. and E are all accelerating in the x-direction with acceleration a = 3 m/s2 i . and. then come back to this question after you have done all the calculations. A ball of mass m is fixed to the centroid of the plate. α. ∗ d) Write down the angular momentum balance equation using the center of mass as a reference point. turn (b) and (c) into six scalar equations in six unknowns. CB. • FD = the drag force on each wing.s94h2p10) The pick-up truck skids across a road with acˆ ˆ celeration a = ax ı +az k . b) Challenge: without doing any calculations on paper can you find one of the reaction force components or the tension in any of the cables? Give yourself a few minutes of staring to try this approach. Solve it and compare to your result from before. ∗ c) Write down the linear momentum balance equation (a vector equation). The hinge joint is attached to a rigid column welded to the floor of ˆ the cart. The tow force F applied at C has no z component and makes an angle α with the x axis.68: G F 1m C y 6.69 A massless triangular plate rests against a frictionless wall of a pick-up truck at point D and is rigidly attached to a massless rod supported by two ideal bearings fixed to the floor of the pick-up truck. φ. The shelf is supported at one corner with a ball and socket joint and the other three corners with strings. The support points A. The cart has acceleration ax ı . g and (Note: F should not appear in your final answer.71 An airplane in flight. The towing force F is just the magnitude needed to keep the bike accelerating in a straight line without tipping any more or less than the angle φ.Problems for Chapter 6 y 8' 2' 4' C 3' A H z G problem 6.s94q3p1) • g= gravitational constant. An airplane is in straight level flight but is accelerating in the forward direction.) F φ z π/2-α α 6. • m tot ≡ the total mass of the plane (including the wings). A bicycle on the level x y plane is steered straight ahead and is being towed by a rope. a) Draw a FBD of the shelf. • D= the drag force on the fuselage. CF. The bike is tipped an angle φ from the vertical. h.67: (Filename:ch3.66: G B b x c 6. ∗ f) Solve these equations by hand or on the computer. The rolling wheel contacts are at A and B. ˆ a) What is { F } · ı for the forces acting on the plate? b) What is the tension in bar CB? φ C y B b h A x problem 6. a) What is the lift on each wing FL ? ∗ b) What is the acceleration of the plane a P? ∗ 6. There is gravity. ED. What is the acceleration of the bicycle? Answer in terms ˆ of some or all of b.] Must these solutions agree with (d)? Do they? 3 A B ˆ ˆ ı C E D 2 G 2 ˆ a = ax ı problem 6. Find the tension in cable C D. It is possible to find five of the six unknown reaction components this way.70: (Filename:s97f3) z H E D 1m A x problem 6. If you can’t. ∗ g) Instead of using a system of equations try to find a single equation which can be solved for TE H .1a) 3 6. What is the reaction at point D on the plate? d=c+(1/2)b D e h a A problem 6.65: (Filename:Mikesp92p1) 39 G D 4' H .66 Hanging a shelf. GH.69: d y 6' E 2' B 5' x problem 6. At the moment of interest the shelf is in a rocket in outer space and accelerating at 10 m/s2 in the k direction.68 The uniform 2 kg plate DBFH is held by six massless rods (AF. The bicycle and rider are modeled as a uniform plate with mass m (for the convenience of the artist). try angular momentum balance about various axes as well as linear momentum balance in an appropriate direction. A uniform rectangular plate of mass m is supported by an inextensible cable C D and a hinge joint at point E on the cart as shown. and EH) which are hinged at their ends. A shelf with negligible mass supports a 0.5 kg mass at its center. The only brake which is working is that of the right rear wheel at C which slides on the ground with friction coefficient µ.72 A rear-wheel drive car on level ground. Assume that the left rear wheel and the front wheel have negligible mass. F = Fx ı + Fy + Fz k ˆ? ∗ ˆ ˆ and M = Mx ı + M y + Mz k CONTENTS above the rear axle. in terms of m tot . such as when µ = 0 or when h = 0.74: (Filename:p. [Hint: check your answer against special cases for which you might guess the answer. What.73 A somewhat crippled car slams on the brakes.1) problem 6. B. The suspension springs at A.) Suspension at left rear wheel keeps normal force equal to ND G D C Right rear wheel locks with friction µ Front wheels roll without slip rolling contact D A B C rolling contact g D b G h B C branch caught in spokes.b. FL . . Dimensions are shown in the lower sketch. ∗ w C w/2 problem 6. Find the acceleration of the tricycle (in terms of ˆ ˆ some or all of . µ. M z c a mwg FD FL T b 6. Wheels A. b. My FL Fx . and N D . skidding ˆ k A ˆ ˆ ı problem 6. ˆ and k ). and are the reactions at the base of the wing (where it is attached ˆ ˆ ˆ to the plane). w. Dimensions are as shown. and C are frozen and keep the car level and at constant height. h. a) Find the acceleration of the car in terms ˆ of some or all of m. b) From the information given could you also find all of the reaction forces at all of the wheels? If so. Dimensions are as shown and the car has mass m .73: (Filename:s97p2. [I cm ]. m w . The negligible-mass front right wheel at B is steered straight ahead and rolls without slip. B. A scared-stiff tricyclist riding on level ground gets a branch stuck in the right rear wheel so the wheel skids with friction coefficient µ. FD . w. thus constraining the overall motion of the tricycle. . a free body diagram of the plane.f. µ.40 c) A free body diagram of one wing is shown. h. a. g. Credit depends on clear explanation.] ice m C w/2 h B b ˆ k w ˆ h C w/2 (Filename:pfigure3. Assume that the center of mass of the tricycle-person system is directly . The mass of one wing is m w . .f96.74 Speeding tricycle gets a branch caught in the right rear wheel. m. g. What is the sideways force from the ground on the right front wheel at B? Answer in terms of any or ˆ all of m. b.airplane) A B g ˆ k h ˆ ˆ ı B A b D 6.71: An airplane in flight. ı . M x Fz . and D roll freely without slip. what can’t you find and why? (No credit for correct answer.72: T mtotg D FBD of wing FD Fy . and a free body diagram of one wing (Filename:pfigure3.f95p1p3) ˆ ı cartoon to show dimensions j i B b ˆ k ˆ ı ˆ FBD of airplane FD FL T k w problem 6. b. The right rear wheel at C also rolls without slip and drives the car forward with velocity v = ˆ ˆ v and acceleration a = a . and have sufficient friction that ˆ they roll in the direction without slip. good bearings. The two left wheels are on perfectly slippery ice. why? If not.3) 6. and ı . a. The normal force at D is kept equal to N D by the only working suspension spring which is on the left rear wheel at D. g. c. . g. The right wheels are on dry pavement. A bead on a circular path of radius R in the x y-plane has rate of change of angular speed α = bt 2 . What is the angular position and speed of the bead as a function of time? [Hint: this problem has 2 answers! (one of which you can find with a quick guess.Problems for Chapter 7 41 Problems for Chapter 7 7. The particle’s motion is tracked from the instant when θ = 0. and for ˆ ˆ ˆ ˆ θ. θ = ω. is given by F = mω2 r . ∗ 7. Draw the path and show the position of the particle at each instant. Let θ = α. θ(0) = π/2.9.2 Dynamics of a particle in circular motion skills 7.5 m.12 Solve θ = α. b) the distance traveled by the particle in 5 s.15 Two runners run on a circular track sideby-side at the same constant angular rate ω about the center of the track. 1 m · sin((5 rad/s) · t). 7. at t = 0. If the particle has not yet completed one revolution between the two instants. and λ so that the maximum acceleration in this motion occurs at θ = 0. a) What is the bead’s angular position θ (measured from the positive x-axis) and angular speed ω as a function of time ? b) What is the angular speed as function of angular position? 7. its ˆ ˆ velocity is v(t0 ) = −3 m/sı +4 m/s and after √ ˆ 5 s the velocity is v(t0 + 5 s) = 5/ 2 m/s(ı + ˆ ).2 The motion of a particle is described by the following equations: x(t) y(t) = = 1 m · cos((5 rad/s) · t).3 Find a planar motion of a particle where the velocity and acceleration are always orthogonal. 7. a) Show that the speed of the particle is constant.7 A particle undergoes constant rate circular motion in the x y-plane. a) What is the angular momentum of the bead about the origin at t = t1 ? b) What is the kinetic energy of the bead at t = t1 . b) There are two points marked on the path of the particle: P with coordinates (0. The bead starts from rest at θ = 0. etc.5 s and b) t = 15 s.)] 7. where α is the angular acceleration.1 If a particle moves along a circle at constant ˙ rate (constant θ) following the equation ˙ ˆ ˙ ˆ r (t) = R cos(θ t)ı + R sin(θ t) which of these things are true and why? If not true. “Acceleration” here means the magnitude of the acceleration vector. ˙ 7. moving with a constant angular speed ω on a circular path with radius r .21 A bead of mass m on a circular path of radius R in the x y-plane has rate of change of angular speed α = ct 3 . What is the velocity of the outside runner relative to the inside runner in polar coordinates? 7. 0).18 The sum of forces acting on a mass m = ˆ ˆ 10 lbm is F = 100 lbfı − 120 lbf . Using F = m a .5 A 200 mm diameter gear rotates at a constant speed of 100 rpm.17 Force on a person standing on the equator. and ˙ θ (0) = 0. and r = −r ı .e. 7. b) Find the magnitude of acceleration of the bead.11 Solve ω = α. Give your solution in both pounds ( lbf) and Newtons ( N).19 The acceleration of a particle in planar ˆ ˆ circular motion is given by a = αr eθ −ω2 r er . Neglect the motion of the earth around the sun and of the sun around the solar system.8 See also problem 7. ing 7. θ(0) = ω0 .13 Given ω = d θ = α (a constant). It makes around the track in exactly 1 s.51 but assume that the moment of interest is at position B with θ = 60◦ . 7. What are the conditions on R. Using Fx and Fy in terms of α. find a) the angular speed of the particle. 7. and c) the acceleration of the particle at the two instants. find the expressions path. 7.6 A particle executes circular motion in ˙ the x y-plane at a constant angular speed θ = 2 rad/s. θ = 0. . given that er = cos θ ı + sin θ and eθ = ˆ ˆ − sin θ ı + cos θ . θ0 . At some instant t0 . The total force acting on an object of mass m. The particle is going in circles at constant rate with ˆ ˆ ˆ r = 18 in and er = cos 30o ı + sin 30o . where er = ˆ ˆ ˆ −ı and eθ = k .1 Kinematics of a particle in circular motion skills 7. ¨ 7.16 A particle oscillates on the arc of a circle with radius R according to the equation θ = θ0 cos(λt).4 A bead goes around a circular track of radius 1 ft at a constant speed.9 Reconsider the bead from problem 7. The inside runner is in a lane of radius ri and the outside runner ˙ is in a lane of radius ro . a) What is the speed of a peripheral point on the gear? b) If no point on the gear is to exceed the centripetal acceleration of 25 m/s2 . and α a constant. find the maximum allowable angular speed (in rpm) of the gear. The bead starts from rest at θ = 0. How much time does the particle take to go from P to Q? c) What is the acceleration of the particle at point Q? 7. 1 m) and Q with coordinates (1 m. The radius of the earth is 3963 mi. ∗ 7. i. given ω(0) = ω0 and α is a ˙ constant. a) Find the speed of the bead. explain why. find an ˙ dt expression for ω as a function of θ if ω(θ = 0) = w0 . find the value of θ at t = 1 s. find v. UsF = m a . ω. Find the velocity and acceleration of the particle at a) t = 0.10 Repeat problem 7. 7..14 Given that θ − λ2 θ = 0. The bead starts from rest at θ = 0. 7. This time. Find the magnitude of the total force acting on a 150 lbm person standing on the equator. given θ(0) = θ0 . express Fr and Fθ in terms of α. ˆ ˆ ˆ Let F = Fr er + Fθ eθ . α = cθ 3/2 . r .20 Consider a particle with mass m in circular ¨ ˙ ˆ motion. 7. r .8. (a) v = 0 (b) v = constant (c) | v | = constant (d) a = 0 (e) a = constant (f) | a | = constant (g) v ⊥ a 7. ˙ ¨ 7. ω is the angular speed and r is the radius of the circular F = m a . and m. The radius of the circular path is 0. it has rate of change of angular speed proportional to angular position rather than time. ω. ∗ c) Find H O for both masses at positions A and B. b) Find the net force on the particle at time t. It takes 7. s s a) What is the path of the particle? Show how you know what the path is. Arrows representing e R . d) After how much time does the force on the particle reverse its direction.24 A particle moves on a counter-clockwise. 2. is tension in the rods enough to keep the two motions going? Explain. a) With what speed should the mass be launched in the track so that it keeps going at a constant speed? b) If the spring is replaced by another spring of same relaxed length but twice the stiffness. Another identical mass is going in circles of the same radius but at a non-constant rate. a) Find and draw the accelerations of the two masses (call them I and II) at position A. 7. and a at each of these points. ˙ b) Find H O for both masses at position A. What applies this force on the car? Does the driver have any control over this force? 7. The bead slides without friction in the tube shown. 0 r A problem 7.5 m/s. and 210◦ . b) What is the angular velocity of the particle? Is it constant? Show how you know if it is constant or not.28 A small mass m is connected to one end of a spring.25 The velocity and acceleration of a 1 kg particle. c) Take center of the circle to be the origin of a x y-coordinate system.s94h8p2) 7. v . The radius of the circle is r . A 100 gm mass is going in circles of radius R = 20 cm ˙ at a constant rate θ = 3 rad/s. Find the elapsed time t − t ∗ . A dot on the path when the particle is at θ = 0◦ . if this motion was imposed by the tension in a string. The radius of the track is R. Find the net force on the particle when it is at θ = 30o from the x-axis. the mass of the particle is m. c) At some later time t ∗ . There is no gravity. The other end of the spring is fixed to the center of a circular track. what is Fx . and the spring constant is k. the x component of the force in the string. Do the changes in H O between the two positions reflect (qualitatively) the results obtained in (b)? ∗ d) If the masses are pinned to the center O by massless rigid rods. what will be the new required launch speed of the particle? 7.42 7. 7. what would that tension be? ∗ d) Is radial tension enough to maintain this motion or is another force needed to keep the motion going (assuming no friction) ? ∗ e) Again. where θ is measured from the x-axis (positive counter-clockwise). are known at some instant t: ˆ ˆ v = −10 m/s(ı + ). The second mass is ¨ accelerating at θ = 2 rad/s2 and at position A.27: (Filename:pfigure. ∗ A' ω0 m a) Write the position of the particle at time ˆ ˆ t using e R and eθ base vectors. It goes around the track once in three minutes. ˆ ˆ a = 2 m/s2 (ı − ). b) Find the magnitude of acceleration of the particle. Find the magnitude of the centripetal force on the car. the unstretched length of the spring is l0 (< R). a) Neatly draw the following things: 1. Find the radial position r of the bead if it is stationary with respect to the rotating tube.29: (Filename:Danef94s2q6) . 7.22 A 200 gm particle goes in circles about a fixed center at a constant speed v = 1. f) What is the angular momentum of the particle at t = 3 s about point O located at the origin of the coordinate system that the particle is referenced to? CONTENTS (I) y B R 45o O A x ¨ θ =0 (II) y B R 45o O A x ¨ θ = 2 rad/s2 problem 7. eθ . it happens to have the same angular speed as the first mass. The path of the particle. (represent vector quantities in terms of ˆ ˆ the cartesian base vectors ı and ). when θ = 210◦ ? Ignore gravity. The tube is driven at a constant angular rate ω0 about axis A A by a motor (not pictured).27 Going in circles — a comparison of constant and nonconstant rate motion. a) Find the angular speed of the particle. origin-centered circular path in the x y-plane at a constant rate. the net force on ˆ the particle is in the − direction. c) What is the velocity of the particle in polar coordinates? d) What is the speed of the particle at t = 3 s? e) What net force does it exert on its surroundings at t = 0 s? Assume the x and y axis are attached to a Newtonian frame. 7. ∗ c) If this motion was imposed by the tension in a string. undergoing constant rate circular motion. The unstretched spring length is r0 . and the particle completes one revolution in time τ .23 A race car cruises on a circular track at a constant speed of 120 mph. ˆ ˆ 3.26 A particle of mass 3 kg moves in the x yplane so that its position is given by 2π t 2π t ˆ ˆ r (t) = 4 m[cos( )ı + sin( ) ]. b) Calculate all of the quantities in part (3) above at the points defined in part (2). 90◦ .5 s to go around the circle once.29 A bead of mass m is attached to a spring with constant k. blue.5 kg. ∗ . by pulling the string with a force F through a hole in the table.38 Pendula using energy methods. ∗ θ(t) O problem 7. Neglect gravity. If the radius of the circle is reduced to r . y ω x r m problem 7. A F y 3/4 m k B d y ˆ ˆ ı motor O ω x x problem 7. the string is quickly and cleanly cut. What is the position of the mass 1 sec later? Make a sketch.30: (Filename:pfigure. Find M A (the moment on the rod AB from its support point at A) in terms of ˙ ˙ ¨ θ and θ . what will the particle’s angular velocity be? Is kinetic energy is conserved? Why or why not? 43 M T b) Find the ratio Tmax where T = Tmax − Tmin .blue. b) What is the tension in the string (in Newtons)? ∗ c) What is the angular momentum of the mass about O? ∗ d) When the string makes a 45◦ angle with the positive x and y axis on the plane.Problems for Chapter 7 7.e. The rod can only withstand a tension of Tcr before breaking. moves in a circular path in the vertical ˆ plane at a constant rate. . The rod is initially lined up with the positive x-axis.1) 7. Find the maximum angular speed of the ball so that the rod does not break assuming a) there is no gravity. The rotation rate is 2π rev/sec. when Tmax < 0.37: (Filename:pfigure. and frictionless bar is made up of two uniform segments of length = 0. The structure rotates clockwise at a constant rate ω = 2 rad/s. a distance d from the hinge. attached to a spring at one end. a) Find the difference between the maximum and minimum tension in the rod.36: (Filename:pfigure. The particle rotates counter-clockwise in circles on a frictionless horizontal plane. m r O ˆ ˆ ı g F problem 7. A force F is applied at point A at the end of the bar. massless.30 A particle of mass m is restrained by a string to move with a constant angular speed ω around a circle of radius R on a horizontal frictionless table.mike.94 using energy methods. Gravity acts in the − direction.32 A massless rigid rod with length attached to a ball of mass M spins at a constant angular rate ω which is maintained by a motor (not shown) at the hinge point. find the relaxed length of the spring. problem 7.4 m each. ∗ 7. ∗ .34 A ball of mass M fixed to an inextensible rod of length and negligible mass rotates about a frictionless hinge as shown in the figure. A collar of mass m = 0.31 An ‘L’ shaped rigid. For a given length r of the rod. A motor (not shown) at the hinge point accelerates the mass-rod system from rest by applying a constant torque M O .31: Forces in constant rate circular motion. There is no gravity. A mass m is glued to the bar at point B. If the collar is steady at a distance 3 = 0.3 m away from the elbow of the 4 ‘L’. 7. Find the equations of motion for the pendula in problem 7. What is the sign of M A if θ = 0 and ¨ ¨ ˙ θ > 0? What is the sign if θ = 0 and θ > 0? A R (Filename:pfigure5.84. The spring is fixed to the elbow of the ‘L’ and has a spring constant k = 6 N/m. and b) there is gravity (neglect bending stresses). Assume an x y-coordinate system in the plane with its origin at O.1) M problem 7.blue. (Filename:pfigure. What is the acceleration of point A at the instant shown? Assume the angular velocity is initially zero.35: (Filename:pfigure4. tied to one end of a rod whose other end is fixed at point O to a motor.sum95) B m 7.37 The mass m is attached rigidly to the rotating disk by the light rod AB of length .2) 7. At what time will the rod break and after how many revolutions? Include gravity if you like.32: (Filename:Danef94s2q5) c) For ω = 300 rpm. ∗ .02. problem 7.33 A 1 m long massless string has a particle of 10 grams mass at one end and is tied to a stationary point O at the other end. what would be the length of the rod for the condition in part (b) to be satisfied? ∗ . a) Make a clear sketch of the system.rph) 7.Lbar) y ω x 7.35 A particle of mass m. A criterion for ignoring gravity might be if the variation in tension is less than 2% of the maximum T tension. slides frictionlessly on one of the arms of the ‘L’.. The rod can only withstand a tension of Tcr before breaking.10.44.34: 7. i. find the rotation rate ω for which this condition is met.36 A massless rigid bar of length L is hinged at the bottom. 0 . 42 Pendulum.42: 7. How hard does she have to hang on compared. A point mass M = 1 kg hangs on a string of length L = 1 m. Make a single plot. how does v depend on θ. where g = 10 m/s2 .1) m 7. one should be able to find what the position and speed of the pendulum is as a function of time. Using any computer and any method you like. to her own weight? You are to find the solution two ways. find: ˙ θ(t). the angle between the pendulum and the straight-down position is θ(t). Gravity pulls down on the mass with force Mg. This equation is the governing differential equation.s94h10p1) . ∗ e) Numerical solution. and m. ˙ b) Use conservation of energy to find θ at θ = 0. The coefficient of friction between the bead and the wire is µ. & µ. Use the same m. Use any ways that please you. g and any base vectors you define clearly. When the rope passes through vertical the tension in the rope is higher (it is hard to hang on). ∗ b) Tension. Using the results from (b) and (c) one can also find the reaction components. you can plot the tension as a function of time as the mass falls. Solve them nu˙ merically. ¨ a) Find θ as a function of g. find θ. Assuming that ˙ ¨ you know both θ and θ. It can be changed to a pair of first order equations by defin˙ ing a new variable α ≡ θ.40 Simply the simple pendulum. The pendulum lies in a vertical plane. A simple pendulum of length 2 m with mass 3 kg is released from rest at an initial angle of 60◦ from the vertically down position. m. g. Assuming you ˙ know θ and θ. (Filename:spend) g x θ problem 7.44 7. Given the initial conditions θ(t = 0) = π/2 and ˙ α(t = 0) = θ(t = 0) = 0. m. a) Find the acceleration (a vector) of the mass just after it is released at θ = π/2 in terms of . based on the (inappropriate-to-use in this case) small angle approximation sin θ = θ ? 7. Some questions: Does the solution to (f) depend on the length of the string? Is the solution to (f) exactly 30 or just a number near 30? How does the period found in (g) compare to the period found by solving ¨ the linear equation θ + (g/l)θ = 0. R. If you like. Assuming that you know θ ˙ and θ. How long does it take to make one oscillation? h) Other observations.44 Bead on a hoop with friction. find the maximum value of the tension in the rod as the mass swings. The equation that you found in (a) is a nonlinear second order ordinary differential equation. extended version. Make any observations that you think are interesting about this problem. A bead slides on a rigid. as many different ways as you can. y v m O R θ x problem 7. Write it as a system of first order equations.41: 7.s94h8p5) 7. Solving these equations is equivalent to solving the original second order equation. θ(t) & T (t). Once you know θ at the time the rope is vertical you can use other mechanics relations to find the tension. c) Reaction components. There are several ways to do this problem. Find the nonlinear governing differential equation for a simple pendulum ¨ θ =− g sin θ l m (Filename:pfigure. The swing starts from rest at an angle θ = 90◦ . find the tension T in the string. find the x and y components of the force that the hinge support causes on the pendulum .39 Tension in a simple pendulum string. ∗ (Filename:s97p2. a) Given v. Using your numerical solutions. Model a swinging person as a point mass. a) What is the tension in the string just after the pendulum is released? b) What is the tension in the string when the pendulum has reached 30◦ from the vertical? CONTENTS d) Reduction to first order equations.41 Tension in a rope-swing rope. m. m and g). stationary. At any time t. There is no air friction. A person wants to know ahead of time if she is strong enough to hold on. say. b) Find the acceleration (a vector) of the mass when the pendulum passes through the vertical at θ = 0 in terms of . or three vertically aligned plots. Write the equation from (a) as a pair of first order equations. circular wire. of these variables for one full oscillation of the pendulum. θ . Assume gravity is negligible. what is v? ∗ ˙ b) If v(θ = 0) = v0 .44: (Filename:pfigure. µ. Then use other mechanics relations to find the tension. v0 and m? ∗ A θ L B problem 7. θ y problem 7. A pendulum with a negligible-mass rod and point mass m is realeased from rest at the horizontal position θ = π/2. Define your coordinate directions sensibly. a) Equation of motion.40: A simple pendulum. The bead is loose on the wire (not a tight fit but not so loose that you have to worry about rattling). and L for both solutions. f) Maximum tension.43 Simple pendulum. g and any base vectors you define clearly c) Find the string tension when the pendulum passes through the vertical at θ = 0 (in terms of . L . ∗ g) Period of oscillation. blue. R and µ [Hint: Draw FBD. Disc A rotates at 100 rpm and disc B rotates at 10 rad/s.toygun) g 7.] b) Find the particle speed at the bottom of the chute if R = .46: (Filename:s93q4Ruina) 7. b) What is the force on the collar from the ring when it passes point A? Solve in terms of R. it is unstretched when its length is .46 Due to a push which happened in the past. express as ODE.1) 7. m. k. if ω = 1.3 Kinematics of a rigid body in planar circular motion skills ˆ 7. g and any base vectors you define. which points up in the air. Its mass is m and it slides with coefficient of friction µ (Actually it slides.49: (Filename:pfigure. the collar with mass m is sliding up at speed v0 on the circular ring when it passes through the point A.65.blue.2. Compute the magnitude of the speed v such that the car just leaves the ground at the top of the hill. The spring is placed in a straight section of length . The disc lies in the x yplane and its angular displacement θ is measured (positive counter-clockwise) from the xaxis. A spring of constant k and unstretched length R is also pulling on the collar.54 Two discs A and B rotate at constant speeds about their centers. The car starts at a point on the hill at point O. a) Find a differential equation that is satisfied by θ that governs the speed of the rice as it slides down the hoop. . One of a million non-interacting rice grains is sliding in a circular chute with radius R.) Gravity g acts downwards.2 as well as the initial values of θ0 = 0 and its initial downward speed is v0 = 10 m/s. Find. and µ = . b) What is the force on the block from the ramp just after it gets onto the ramp at point A? Solve in terms of R. m and g. and label the tangential and normal acceleration a) of the particle at the location A. [Hint: you are probably best off seeking a numerical solution. with ˆ angular velocity ω = 5 rad/sk and angular ˆ acceleration α = 2 rad/s2 k . a) What is the acceleration of the collar at A. force is a vector. a) What is the speed of the block when it reaches point B? Solve in terms of R.47 A toy used to shoot pellets is made out of a thin tube which has a spring of spring constant k on one end.48 A block with mass m is moving to the right at speed v0 when it reaches a circular frictionless portion of the ramp. g = 10 m/s2 . what is the pellet’s velocity v as it leaves the tube? ∗ b) What force acts on the pellet just prior to its departure from the tube? What about just after? ∗ 45 g O R problem 7.1) 7.45 Particle in a chute. draw.p3. Remember.45: (Filename:p. m = . k R θ R m µ problem 7. Which is rotating faster? problem 7. m and g. v0 . ∗ b) of the particle at the location B. at the moment of interest. problem 7.51: R B (Filename:efig1. a) A pellet of mass m is placed in the device and the spring is pulled to the left by an amount . The straight part is attached to a (quarter) circular tube of radius R.5m. g.18) v0 g R A collar problem 7.48: A (Filename:s93q4Sachse) 7.51 A particle goes around a circle shown in the figure. y B 7. v0 .] 7.53 A disc is rotating at 15 rpm. v0 . Solve in terms of R. Parameters in this equation can be m.f96.50 Find v = ω × r . write eqs of mechanics. Ignoring friction along the travel path. What is the angular displacement θ (t) of ˙ the disc if it starts at θ(0) = θ0 and θ (0) = ω? What are the velocity and acceleration of a ˆ ˆ point P at position r = x ı + y ? 7.5 rad/sk and ˆ ˆ r = 2 mı − 3 m .47: (Filename:pfigure. k. m. rolls and tumbles — µ is just the effective coefficient of friction from all of these interactions.1 grams. g and any base vectors you define.49 A car moves with speed v along the surface of the hill shown which can be approximated as a circle of radius R. v0 .Problems for Chapter 7 7. θ A x g r = 10 in problem 7. How many seconds does it take to rotate by 180 degrees? What is the angular speed of the disc in rad/s? 7.52 A motor turns a uniform disc of radius R counter-clockwise about its mass center at a constant rate ω. The ring is frictionless. the velocity ˆ of the center of mass is v G = −1. b) What is the angular velocity of the disk. CONTENTS vA B aB θ O R A 7. 7. point P is on the x-axis of an x y-coordinate system centered at point G. ωC ? d) What is the velocity v P of point P? e) What is the acceleration a P of point P? 7.65 A horizontal disk D of diameter d = 500 mm is driven at a constant speed of 100 rpm.60: (Filename:pfigure4.62: (Filename:pfigure. G. the velocity of end B is v B = π ˆ −3 m/s . The data stream comes out at a constant rate 4. the disk rotates at 200 rpm. The angular speed of the disk remains constant as long as the head is positioned over a particular track.5(−d ı + c ) on the disk? (c2 + d 2 < R 2 .61 The circular disc of radius R = 100 mm rotates about its center O. which is fixed. point A on the disk has a velocity v A = 0.60 A uniform disc of radius r = 200 mm is mounted eccentrically on a motor shaft at point O.60 m/s and the magnitude of its centripetal acceleration is 0.rp9) 2 r 3 O x 7.rp4) c) find the numbers of bits on the innermost track. If the rod has not completed one revolution during this period. is the point you locate unique? If not.61: (Filename:pfigure. b) Is the given information enough to locate the center of rotation of the disk? c) If the acceleration of the center has no ˆ component in the direction at the moment of interest.5.DH.58 A circular disc of radius r = 250 mm rotates in the x y-plane about a point which is at a distance d = 2r away from the center of the disk. a) Draw a neat diagram showing the disk.3. A Compact Disk (CD) has bits of data etched on concentric circular tracks. a) find the angular velocity of the rod. positive counter-clockwise. A point P is marked on the disk at a radius of one meter. b) Find the point with the highest linear speed on the disc.32 × 106 bits/second.57 A 0. At the moment of interest. The data from a track is read by a beam of light from a head that is positioned under the track.6. Disk 7. for which r = 56 mm. the particle.64 Bit-stream kinematics of a CD. A small disk C can be positioned anywhere between r = 10 mm and r = 240 mm on disk D by sliding it along the overhead shaft and then fixing it at the desired position with a set screw (see the figure). a) Find the rotational speed of the disk. ωC ? c) What is the angular acceleration of the ˙ disk. so that the linear speed at any track is always the same.8 m/s in the direction shown.blue. the acceleration of all points in a given radial line are parallel. a) What is the number of bits of data on the outermost track. At the same instant. It rotates counterclockwise at a constant angular speed about a peg whose location is not known. What are the velocity and acceleration of a point ˆ ˆ P at position r P = cı + d relative to the velocity and acceleration of a point Q at poˆ ˆ sition r Q = 0.46 7. the velocity of end B is ˆ v B = −3 m/sı . 7.59 A disc C spins at a constant rate of two revolutions per second counter-clockwise about its geometric center.) C ˆ ˆ ı r problem 7.56 A motor turns a uniform disc of radius R counter-clockwise about its mass center at a constant rate ω.43. At the instant of interest. what other information is required to make the point unique? G r problem 7. for non-constant rate circular motion.63 A motor turns a uniform disc of radius R about its mass center at a variable angular rate ω with rate of change ω. What is its velocity? a O problem 7. What are the velocity and acceleration of ˆ ˆ a point P at position r P = cı + d relative to the velocity and acceleration of a point Q ˆ ˆ at position r Q = 0. counter˙ clockwise. and the coordinate axes.1a) B ˆ ˆ ı 7. the angular speed of the disk changes. The disc lies in the x y-plane and its angular displacement θ is measured from the x-axis. After 20 s. When the head is positioned on the outermost track.2) problem 7. and hour hands of a clock. 7. and b) find the location of the peg along the length of the rod. can you locate the center of rotation? If yes.3. Compute the angular acceleration α of the disc. a) Find the angular velocity of the disc. At the instant shown. As the head moves to the next track. At a given instant. the tangent of the angle θ made by the total acceleration vector of any point B with its radial line to O is 0.5(−d ı + y ) on the disk? (c2 + d 2 < R 2 . the linear speed of the center C is 0.) 7.4 m long rod AB has many holes along its length such that it can be pegged at any of the various locations. The motor rotates the disc at a constant angular speed. b) find the angular speed of the disk when the head is on the innermost track (r = 22 mm). The disc lies in the x yplane and its angular displacement θ is measured (positive counter-clockwise) from the xaxis.58: (Filename:pfigure4.72 m/s2 . .5 m/s .3. and y A problem 7.62 Show that. minute. At some instant t.57: (Filename:pfigure4.rp8) 7.55 Find the angular velocities of the second. its rotational speed can be varied by varying the position of disk C . and ω A ? .1) set screw D ¨ 7. a) What is the total force and moment required to hold it in place (use the origin as the reference point of angular momentum and torque).3) 7.71 Neglecting gravity. The motor delivers a constant power P to the disk. calculate α = ω = θ ˙ at the instant shown for the system in the figure. r C θ 5 cm rigid massless tubing 4 cm 4 cm problem 7. illustrate your results with two numerical examples ˆ ˆ (in consistent units): i) rA = 1ı . mass = m disk.1.41. This gear system is called brush gearing.blue.69 A thin uniform circular disc of mass M and radius R rotates in the x y plane about its center of mass point O.72 A disk of mass M and radius R is attached to an electric motor as shown. and λ B = −1ı (thus rC = 0 ). Find the maximum and minimum rotational speeds of the overhead shaft.70 A uniform circular disc rotates at constant angular speed ω about the origin. with the center of the coin a distance r from the center of the disk. problem 7. The structure rotates counterclockwise at a constant rate of 60 rpm. which is also the center of the disc. a) Describe in detail what equations must be satisfied by the point rC . Determine the initial angular acceleration and the reaction forces at the pin O. ˆ ˆ [λ A ] and [λ B ] and gives as output the ˆ pair of numbers [λ ].) 7. At the instant shown. . C 30 problem 7. The overhead shaft rotates with disk C and. indicated by unit vectors as ˆ ˆ λ A and λ B . A coin of mass m rests on the disk. 47 y ω R O x m = 0. therefore. ∗ Side View C massless rods such that the whole structure behaves like a rigid body. For each of the parts below.66 Two points A and B are on the same machine part that is hinged at an as yet unkown location C. It’s total mass is M. Assume that m M.65: (Filename:pfigure4.45.73 2-D constant rate gear train.71: (Filename:pfigure.70: (Filename:pfigure.blue.67: (Filename:pfig4.4 Dynamics of a rigid body in planar circular motion skills 7. IG = 1 M R 2 2 problem 7.5 kg m M bearing 1m r 45o problem 7.Problems for Chapter 7 C rolls without slip on disk D . find the force in each rod. a) What is the angular velocity of the disk as a function of time? b) What is its angular acceleration? c) At what time does the coin begin to slip off the disk? (It will suffice here to give the equation for t that must be solved. b) Write a computer program that takes as input the 4 pairs of numbers [ rA ].67 The structure shown in the figure consists of two point masses connected by three rigid. [ rB ].rpc) 30 R o P 7. mass = M r R G ω g c) Find a formula of the form rC = .20. What is the angular velocity ωoutput = ωC of the output shaft and the speed of a point on the outer edge of disc C. Assume you are given that points at positions rA and rB are supposed to move in given directions. rB = 1 . The disk starts from rest when the motor is turned on at t = 0.68 The hinged disk of mass m (uniformly distributed) is acted upon by a force P shown in the figure.rp10) 2 kg pt.blue.68: o (Filename:pfigure. that explicitly gives the position vector for point C in terms of the 4 given vectors.1) 7. and ii) a more complex example of your choosing.1) g D O 100 rpm problem 7.blue. ˆ ˆ ˆ ˆ λ A = 1 . It’s radius is R. The disc starts from rest at θ = 0. 7.87a. and that the coefficient of friction between the coin and the disk is µ. RC . ωinput = ω A . . in terms of R A .72: (Filename:pfigure. it has rate of change of angular speed proportional to angular position. mass F=10N 1 cm Top View 7. Driven by a motor. R B . a) What is the rate of change of angular momentum about the origin at θ = π rad? 3 b) What is the torque of the motor at θ = π rad? 3 c) What is the total kinetic energy of the disk at θ = π rad? 3 7. b) What is the total kinetic energy of the disk? z coin. The angular velocity of the input shaft (driven by a motor not shown) is a constant. α = dθ 5/2 . A motor (not shown) rotates the disks at constant rate ω = 2 rad/s.77 Two gears rotating at constant rate. At the input to a gear box a 100 lbf force is applied to gear A. Gear A is connected to a motor(not shown) and gear B.78 Two racks connected by a gear.5 m and negligible mass.79 Constant rate rack and pinion. (Filename:pg131.0 m from the center.16 and ??. The two gears shown are welded together and spin on a frictionless bearing. The outer disk has 1 m radius and a uniformly distributed mass of 0. would that increase or decrease FB to achieve the same constant rate? ωA RA no slip RC ω 6 in uniform disk m = 10 lbm ωC RB A B C y (Filename:pfigure.3) no slip RA no slip A FA = 100 lb (Filename:pg45. At the output the machinery (not shown) applies a force FB to the other rack. See also problems 4. The two gears shown are welded together and spin on a frictionless bearing. Assume the bearings and contacts are frictionless. RC C RB B problem 7. the machinery (not shown) applies a force of FB to the output gear.76: G1 A ωout G2 G4 G3 ωin B (Filename:pfigure. a) What is the angular speed of the right gear? b) What is the velocity of point P? c) What is FB ? d) If the gear bearings had friction.5 m and negligible mass.2 kg. At the time of interest. What is FB ? b) If the gear bearing had friction. The inner gear has radius 0.74 2-D constant speed gear train. At the output.48 no slip RC CONTENTS uniform disk m = 2kg e) If instead of applying a 100 lbf to the left gear it is driven by a motor (not shown) at constant angular speed ω. At the time of interest.2 kg.78: Two racks connected by a gear. The gears and the rack have teeth that are not shown in the figure. which is welded to gear C.3) 7.s94h6p5) x ωout C G5 G6 ωin G1 G2 G3 G4 Front view C G5 MA problem 7. Gear G 5 meshes with G 2 which is welded to shaft A. The point P is on the outer gear a distance 1. The outer gear has 1 m radius and a uniformly distributed mass of 0. a) What is the speed of point P? b) What is the velocity of point P? c) What is the angular acceleration α of the gear? d) What is the acceleration of point P? e) What is the magnitude of the acceleration of point P? f) What is the rate of increase of the speed of point P? problem 7. G 3 meshes with gear G 6 . The gears drive the massless rack which is held in place by massless frictionless rollers as shown. is connected to a taffy-pulling mechanism. At the time of interest the angular velocity is ω = 2 rad/s (though ω is not constant). The input torque Minput = 500 N·m and the spin rate ωinput = 150 rev/min. Shaft A and shaft B spin independently. the torque that gear C applies to its surroundings in the clockwise direction? ion 7. a) What is the speed of point P? ∗ b) What is the velocity of point P? ∗ c) What is the acceleration of point P? ∗ d) What is the velocity of the rack v r ? ∗ e) What is the force on the rack due to its contact with the inner gear? ∗ . A 100 lbf force is applied to one rack. a) What is the input power? ∗ b) What is the output power? ∗ c) What is the angular velocity ωoutput of the output shaft? ∗ d) What is the output torque Moutput ? ∗ x problem 7.76 A 2-D constant speed gear train. point P is on the positive y axis. Assume you know the torque Minput = M A and angular velocity ωinput = ω A of the input shaft. point P is on the positive y axis.73: A set of gears turning at constant rate.2) FB = ? problem 7. They are loaded as shown with the force F = 20 N on the massless rack which is held in place by massless frictionless rollers. Assume the bearings and contacts are frictionless.. Shaft B is rigidly connected to gears G 4 and G 3 .5m ω RB A B C massless inner disk y massless rack (Filename:ch4. a) Assume the gear is spinning at constant rate and is frictionless.75 Accelerating rack and pinion. clockwise. Gear A rotates at constant angular rate ω = 2 rad/s.74: D 100 lbf z G6 D A B massless racks FB = ? 7. Gears G 6 and G 5 are both rigidly attached to shaft CD.77: Two gears.s94h6p6) 7. would FB have to be larger or smaller in order to achieve the same constant velocity? 7. a) What is the input power? b) What is the output power? c) What is the output torque Moutput = MC . what is the angular speed of the right gear? ωA RA ωC F=20N P 1m .75: Accelerating rack and pin7. The inner gear has radius 0. (Filename:ch4. The point P is on the disk a distance 1 m from the center.4) RG5 = 3RG2 3RG3 = 5RG6 problem 7. ∗ c) Find the tension in the part of the string between the block and the overhead pulley. c) Find the output torque at B.81: 7. is used to transmit power between two shafts that are perpendicular to each other. Two parallel shafts. a) Find the input power. . b) Find the tension in strings AB and C D.83 A bevel-type gear system.83: (Filename:pfigure4.) Draw a neat diagram of the pulleys and the belt-drive system and find the angle of lap.rpa) A 50 kg (Filename:pfigure4. are connected by a belt passing over the pulleys A and B fixed to the two shafts. AB and C D.) The power transmitted by the belt is given by power = net tension × belt speed. of the belt must be different.rpd) 7. driving gear problem 7. b) Find the rotational speed of the driven pulley. a) Find the velocity of the mass at B. Two uniform disks A and B of non-negligible masses 10 kg and 5 kg respectively.19.85: problem 7.5 m apart. ∗ b) Find the acceleration of block C. The mean radius of the driven gear is 80 mm and the driven shaft is expected to deliver a torque of Mout = 25 N·m. are used as pulleys to hoist a block of mass 20 kg as shown in the figure. What is FB ? ∗ b) If the gear bearings had friction would that increase or decrease FB to achieve the same constant rate? c) If instead of applying a 100 lbf to the left rack it is driven by a motor (not shown) at constant speed v. When the motor is running at a constant 100 rpm. what is the speed of the right rack? ∗ MO C D driven pulley (Filename:pfigure4.5m 10 kg A 60 cm y no slip x problem 7. The driver pulley A rotates at a constant 200 rpm..e.80 Belt drives are used to transmit power between parallel shafts. The centers of the pulleys are located 2.86 Two racks connected by three constant rate gears. ∗ d) The belt in use has a 15 mm×5 mm rectangular cross-section. Use g = 10 m/s2 . the machinery (not shown) applies a force of FB to the other rack.102.2. i. shown in the figure. where v is the linear speed of the belt. ∗ b) Find the rotational speed of the driven pulley B ∗ . Which part has a greater tension? Does your conclusion about unequal tension depend on whether the pulley is massless or not? Assume any dimensions you need. The block is pulled up by applying a force F = 310 N at one end of the string. the contact angle θ. c) (See the figure in problem 7. motor r = 150 mm ˆ ˆ ı r = 250 mm spindle r = 300 mm C D B problem 7.81 In the belt drive system shown. If the motor applies a constant torque M O on the driver pulley. 3 m apart. show that the tensions in the two parts. problem 7. [Hint: T1 µθ (see problem 7. a) Assume the gear-train is spinning at constant rate and is frictionless. Assume no loss of power between the two shafts. A 100 lbf force is applied to one rack. find the input torque supplied by the driving shaft 7.79: Constant rate rack and pinion (Filename:ch4. At the output.s94h8p4) 7.2) C B 20 kg 40 cm 5 kg F = 310 N (Filename:pfigure.rpg) A B 7. Find the maximum tensile stress in the belt.5. assume that the driver pulley rotates at a constant angular speed ω. of the belt on the driver pulley. ∗ 7. Assuming no power loss. Assume the string to be massless but ‘frictionful’ enough to not slide on the pulleys. Find the maximum tension in the belt. Assume that the pulley is to be of negligible mass.] T =e 2 P 1m . The driving gear has a mean radius of 50 mm and rotates at a constant speed ω = 150 rpm. The coefficient of friction between the belt and pulleys is µ = 0.84 Disk pulleys.82 A belt drive is required to transmit 15 kW power from a 750 mm diameter pulley rotating at a constant 300 rpm to a 500 mm diameter pulley. See also problem 4. P = (T1 − T2 )v.Problems for Chapter 7 vr 49 ω a) (See problem 7.102).85 A spindle and pulley arrangement is used to hoist a 50 kg mass as shown in the figure. The speed ratio between the pulleys A and B is 1:2.84: A (driver) (driven) B r = 100 mm 3m 7.102.80: (Filename:pfigure4.rpe) driver pulley problem 7. The input torque is 350 N·m. a) Find the angular acceleration of pulley B. When the bar is horizontal. the machinery (not shown) applies a force of FB to the other rack.92 A physical pendulum.5) CONTENTS FB = ? B ωout x C G5 A G6 D y G1 G2 RG5 = 3 RG2 3 RG3 = 5 RG 6 G3 G4 B ωin z A ωout G2 G3 G4 G1 ωin B C G5 G6 D 7.87 Two racks connected by three accelerating gears. and ˆ is to the right. at this position. A motor at the hinge drives the disk/point mass assembly with constant angular acceleration α.88 3-D accelerating gear train. c) Evaluate the left-hand-side as explicitly as possible in terms of the forces showing on your Free Body Diagram. a) Draw a free body diagram of the system. At the output. (Filename:3dgeartrain) problem 7. θ is small enough so that y ≈ L θ. Take the ‘body’. Gear G 5 meshes with G 2 which is welded to shaft A. The dumbbell rad oscillates about the horizontal position with small amplitude θ.90 The asymmetric dumbbell shown in the figure is pivoted in the center and also attached to a spring at one quarter of its length from the bigger mass. G 5 meshes with gear G 6 . the bar is at an angle θ from the horizontal.90: (Filename:pfig2.89 A uniform disk of mass M and radius R rotates about a hinge O in the x y-plane. 2 If. Shaft A and shaft B spin independently. ı is vertically down. A point mass m is fixed to the disk at a distance R/2 from the hinge. At an instant when the an˙ˆ gular velocity of the bar is θ k . Gears G 6 and G 5 are both rigidly attached to shaft AD. A 100 lbf force is applied to one rack. (b).rp4) ˆ ı problem 7. a) Assume the gear-train is spinning at constant rate and is frictionless.91 The dumbbell shown in the figure has a torsional spring with spring constant k (torsional stiffness units are kN· m ). the compression in the spring is ys .86: Two racks connected by three gears. Assume the bearings and contacts are frictionless. evaluate the power term ( F · v ) in the energy balance equation. You will have to set up and evaluate an integral. At the instant of interest.87: Two racks connected by three gears. 7.2) problem 7. the velocity of ˙ˆ the left mass is −L θ and that of the right mass ˙ˆ is L θ .91: 7. what is FB ? b) If the gear bearings had friction would that increase or decrease FB to achieve the same constant rate? c) If the angular velocity of the gear is increasing at rate α does this increase or decrease FB at the given ω. Assume you know the torque Minput . L 3m m k O A RA b C d E RE massless θ no slip 100 lb ˆ L no slip massless rack (Filename:pg131. You may use the following facts: A RA b C Front view no slip 100 lb problem 7. Shaft B is rigidly connected to gears G 4 and G 5 .3. the velocity of mass ‘m’ is ˆ ˆ v and that of mass 3m is −v .50 massless rack y x RC d E RE no slip massless rack (Filename:ch4.88: A 3-D set of gears turning at constant rate. d) Evaluate the right hand side as completely as possible.rp10) problem 7. v a = = ˙ ˙ ˆ ˆ l θ cos θ + −l θ sin θ ı 2 ˙ ˆ ˆ −l θ cos θ ı + sin θ ¨ ˆ ˆ +l θ cos θ − sin θ ı massless rack y x RC FB = ? B where is the distance along the pendulum from the top. This is really a 2-D problem. Find the expression for the power P of the spring on the dumbbell at the instant of interest. θ is the angle by which the pendulum is displaced counter-clockwise from the vertically ˆ down position.92: (Filename:ppend) ˆ ˆ ı 2m L torsional spring constant k m L (Filename:pfig2. What torque at the hinge does the motor supply to the system? 7. 7. 7. the system of interest. problem 7. to be the whole stick. and (c) are two force members? .93 Which of (a). angular velocity ωinput and the angular acceleration αinput of the input shaft.3. a) What is the input power? b) What is the output power? c) What is the angular velocity ωoutput of the output shaft? d) What is the output torque Moutput ? 7. A swinging stick is sometimes called a ‘physical’ pendulum. b) Write the equation of angular momentum balance for this system about point O. each gear turns in a different parallel plane. φ. b) Using numerical integration of the nonlinear differential equation for an inverted pendulum find φ at t = 1s and t = 7s. φ and any unit vectors you may need to define.759 kg point mass at the other end.2) rigid.blue. the other end sticking up. massless bar rigid. slowest to fastest) Assume small angles of oscillation. [Hint: use balance of angular momentum about the point 0]. You are holding a stick upside down. Idealize the pendulum as a point mass attached to a rigid massless rod of length 1.94 For the pendula in the figure : a) Without doing any calculations. mass m g uniform rigid bar. not a numerical integrator.) ∗ [Hint: use vectors (otherwise it’s hard)] [Hint: For the special case. Also. A zero length spring (relaxed length 0 = 0) with stiffness k = 5 N/m supports the pendulum shown. Use g = 10m s−2 . but calculations are not necessary. ∗ ¨ ˙ b) Find θ as a function of θ and θ (and k. Assume 7. measured along the rod from a frictionless hinge located at a point A. φ. b) Calculate the period of small oscillations.80. m. b) What is the force of the hinge on the ˙ ¨ rod? Solve in terms of m.94: 7.95 A massless 10 meter long bar is supported by a frictionless hinge at one end and has a 3. The static equilibrium position is θ = 0 (pendulum hanging vertically). Use a calculator. respectively. c) Would you get the same answers if you put a mass 2m at 2. Include on the same plot the angle versus time from the approximate linear solution from part (a).e. There is gravity.twomasspend) 7. the order.97 A rigid massless rod has two equal masses m B and m C (m B = m C = m) attached to it at distances 2L and 3 . massless bar O (d) O 2m θ M 2m (e) O θ k 2m M x problem 7. Let φ denote the angle of the rod measured from the vertical. imagine that it is only a two-dimensional problem (either you can ignore one direction of falling for simplicity or imagine wire guides that keep the stick from moving in and out of the plane of the paper on which you draw the problem). The horizontal damping force applied at B is given by FD = −c y B ˙ The dimensions and parameters are as follows: r B/0 r A/0 k c = = = = 2ft = 3ft 2 lbf/ ft 0.3 lbf · s/ ft problem 7.98 See also problem 4. φ ˙ and φ. You note that if you model your holding the stick as just having a stationary hinge then you . To simplify things. g.3. the solution simplifies greatly.96: (Filename:pfigure.] 0=0 k = 5 N/m D=4m g = 10 m/s y g M 1m (c) O θ rigid. . 51 ˙ that φ and φ are known at the moment of interest.5 ? Why or why not? g θ uniform rigid bar.96 A spring-mass-damper system is depicted in the figure. mass m (c) Massless swinging rod m rigid. ∗ c) Rank the relative duration of oscillations and compare to your intuitive solution in part (a). so I O = m 2 .1n) problem 7.157.02 radians measured from vertically upright position (hinge at the bottom). c) Make a plot of the angle versus time for your numerical solution. Your graph should show the correct qualitative behavior. g.99 Robotics problem: Simplest balancing of an inverted pendulum. ¨ a) What is φ? Solve in terms of m.97: 2 φ m rigid massless bar m (Filename:pfigure. and g. Also use the “small angle approximation” where appropriate. think of the stick as massless but with a point mass at the upper end. 7.Problems for Chapter 7 (a) Swinging rod with mass (b) Stationary rod with mass a) Using a small angle approximation and the solution to the resulting linear differential equation. It is released at t = 0 from a tip angle of φ = .98: m = 2 kg =3m (Filename:pfigure. a) Determine the natural circular frequency of small oscillations about equilibrium for the pendulum shown. ¨ ˙ a) Find θ assuming θ = 2 rad/s. assume that sin(θ ) ≈ θ and cos(θ ) ≈ 1. b) Sketch a graph of θ as a function of t (t ≥ 0) if the pendulum is released from rest at position θ = 0.2) B θ c x θ problem 7. . The rod swings freely from the hinge. .93: (Filename:pfigure2.two. massless bar problem 7.2 rad when t = 0. 7. θ = π/2. try to figure out the relative durations of the periods of oscillation for the five pendula (i. and explain in words why things work the way they do.force) g θ d) Comment on the similarities and differences in your plots.blue. find the angle of tip at t = 1s and t = 7s. massless bar y mA g M (Filename:pg144. so the spring is at its rest point in this position. k D = mg. 7. one end is in your hand. (a) O θ M rigid. massless bar (b) 1m 1m M uniform bar M = 1kg O θ 2m For small θ . Three masses.s94h6p2) 7. and if the stick tips. and see if the solution of the differential equation has exponentially growing (i. d) Test your guess the following way: plug it into the equation of motion from part (b). Assuming small angles. a) Draw a picture and a FBD b) Write the equation for angular momentum balance about the hinge point. . How can you prevent this falling? One way to do keep the stick from falling over is to firmly grab it with your hand. the radial distance from the center of rotation to any desired location on the rope. where T1 2 and T2 are the tensions in the lower and the upper segments of the belt. m 2 = 150 grams and m 3 = 100 grams. a) Write the equations of linear momentum balance for section ab of the belt ˆ ˆ in the ı and directions. unstable) solutions. Let the angle of contact between the belt and pulley surface be θ. There is no gravity. Use initial conditions both close to and far from the upright position and plot φ versus time. A wire frame structure is made of four concentric loops of massless and rigid wires. point mass m. The stick might be tipped an angle φ from the vertical. a) Find the tension in the rope as a function of r . respectively. This corrective torque is (roughly) how your ankles keep you balanced when you stand upright. 7. and a hinge at the bottom with a motor that can apply a torque Tm . linearize the equation. The structure rotates counter-clockwise at a constant rate ˙ θ = 5 rad/s.99: (Filename:pfigure. m 1 = 200 grams. are glued to the structure as shown in the figure. f) If you are ambitious. ∗ CONTENTS c) Show that the solution to the equation T in part (b) satisfies T1 = eµθ . a) Find the net force exerted by the structure on the support at the instant shown. Ignore gravity.100 Balancing a system of rotating particles.e. ∗ b) Where does the maximum tension occur in the rope? ∗ . That is you can make ˙ Tm any function of φ and φ that you want. ˆ ˆ ı a dN T+dT dθ/2 (b) problem 7. this l hinge leads to exponentially growing solutions. e) Pick numbers and model your system on a computer using the full non-linear equations.rpf) b T dFf = µdN y ω m1 r4=40 cm r3=30 cm r2=20 cm r1=10 cm x 45 o m2 30o m3 dθ problem 7. etc).101 A rope of length and total mass m is held fixed at one end and whirled around in circular motion at a constant rate ω in the horizontal plane.102: (Filename:pfigure4. Your task in this assignment is to design a robot that keeps an inverted pendulum balanced by applying appropriate torque. connected to each other by four rigid wires presently coincident with the x and y axes. assume the disturbing force is zero. a b dθ ω θ (a) T2 T1 b) You are to put a mass m at an appropriate location on the third loop so that the net force on the support is zero.100: (Filename:pfigure. 7. ∗ c) Imagine that your robot can sense the angle of tip φ and its rate of change ˙ φ and can apply a torque in response to that sensing.s94h9p1) b) Eliminate the normal force N from the two equations in part (a) and get a differential equation for the tension T in terms of the coefficient of friction µ and The contact angle θ .52 ¨ get φ = g sin φ. A horizontal disturbing force F(t) is applied to the mass (representing wind. The free body diagram of an infinitesimal section ab of the belt is shown in figure(b). pick a non-zero forcing function F(t) (say a sine wave of some frequency and amplitude) and see how that affects the stability of the solution in your simulations. Go back to (c) if it does and find a control strategy that works. Assume that the belt is massless and that the condition of impending slip exists between the pulley and the belt. Can you find a function that will make the pendulum stay upright? Make a guess (you will test it below). Upside-down sticks fall over. c) At what distance from the center of rotation does the tension drop to half its maximum value? ∗ .102 Assume that the pulley shown in figure(a) rotates at a constant speed ω. apply a torque in order to right it. length . ˆ eθ y g φ m C ˆ er A x problem 7. Find the appropriate mass and the location on the loop. annoying friends. Your model is: Inverted pendulum. ∗ x O y y 7. Point C is the center of mass of the system. one at each end of the bar. or equal to the length of the rod? ∗ (Filename:pfigure.Problems for Chapter 7 53 A m 7.4 m in the x y-plane.4. so that one mass is along the x-axis.rp1) 7.4. b) What is the natural frequency of the system for small oscillations θ? 7.112 A small particle of mass m is attached to the end of a thin rod of mass M (uniformly distributed). calculate later. a) About which point A. ∗ m = 1 kg 20 mm y 100 mm x 5 mm y C x problem 7. a) About which point on the dumbbell is its polar moment of inertia Izz a minimum and what is this minimum value? ∗ m 120o m m O 120o x h problem 7.rp5) d) Is the radius of gyration of the system greater.blue.111 A short rod of mass m and length h hangs from an inextensible string of length .103 A point mass m = 0.105 Two identical point masses are attached to the two ends of a rigid massless bar of length (one mass at each end). is 1 m 2 . O c) Find the percent error in Izz in treating the bar as a point mass by comparing the expressions in parts (a) and (b). I yy .] 7.5 Polar moment of inertia: cm O Izz and Izz Skills 3 7. O a) Find the moment of inertia Izz of the rod. B. a) What is value of Izz of the ball about the center of rotation? ∗ b) How much must you shorten the string to reduce the moment of inertia of the ball by half? ∗ B 2m problem 7. is the polar moment of inertia Izz of the system a minimum? ∗ b) About which point is Izz a maximum? ∗ A B c) What is the ratio of Izz and Izz ? ∗ 7.112: cm b) by computing Izz first and then using the parallel axis theorem. The ball is to execute circular motion in the x y-plane with the string fully extended. Find the moment of inertia of the mass about the z-axis. a) Obtain the equation of motion governing the rotation θ of the rod.rp11) b) About which point on the dumbbell is its polar moment of inertia Izz a maximum and what is this maximum value? ∗ m problem 7.151. as depicted in the figure.4. b) Find the moment of inertia of the rod O Izz by considering it as a point mass located at its center of mass.108: (Filename:pfigure4.107 Think first. g y m x O problem 7.107: 7.3) . in two different 3 ways: a) by using the basic definition of polar O moment of inertia Izz = r 2 dm. ∗ b) Calculate the three moments of inertia to check your guess.3 m and y = 0. a) Guess (no calculations) which of the O O three moment of inertia terms I x x .rp6) (Filename:pfigure4. which is pinned at hinge O.4. A light rigid rod AB of length 3 has a point mass m at end A and a point mass 2m at end B. will your answer to part (a) change? d) Find the radius of gyration of the system for the polar moment of inertia. ∗ c) If the orientation of the system is changed. First.110 Locate the center of mass of the tapered rod shown in the figure and compute the polar cm moment of inertia Izz . ∗ 7. smaller. O is the smallest and which is the Izz biggest.5 kg is located at x = 0. [Hint: use the variable thickness of the rod to define a variable mass density per unit length. or C. Locate a point along the length of the bar about which the polar moment of inertia of the system is 20% more than that calculated about the mid point of the bar. Plot the percent error versus h/ .104 A small ball of mass 0. and O /3 M 2 /3 θ m problem 7.110: (Filename:pfigure4.2 kg is attached to a 1 m long inextensible string.4. answer the following questions without any calculations and then do calculations to verify your guesses.109: (Filename:pfigure4.111: (Filename:pfigure4.rp3) 7. shown in the figure. For what values of h/ is the percentage error less than 5%? 7.106 A dumbbell consists of a rigid massless bar of length and two identical point masses m and m.109 Show that the polar moment of inertia O Izz of the uniform bar of length and mass m.108 Do you understand the perpendicular axis theorem? Three identical particles of mass m are connected to three identical massless rods of length and welded together at point O as shown in the figure. 115: B x m C problem 7. and base b lies in the x y-plane. height h .120 A uniform thin circular disk of radius r = 100 mm and mass m = 2 kg has a rectangular slot of width w = 10 mm cut into it as shown in the figure. The mass of the plate is m = kg. c) Locate the center of mass of the plate cm and calculate Izz .116 A uniform square plate (2 m on edge) has a corner cut out.119: (Filename:pfigure4.blue.4.119 A uniform square plate of side = 250 mm has a circular cut-out of radius r = 1 50 mm. problem 7. ∗ 7. and is connected to a thin disk of mass M and radius R at B.rp7) 7.115 Perpendicular axis theorem and symmetry. the polar moment cm cm of inertia Izz = 0. 2 a) Find the polar moment of inertia of the plate. a) Find the polar moment of inertia ∗ cm Izz .118: b) Find a point P on the plate about which the system’s moment of inertia Izz is maximum? ∗ c) Find the radius of gyration of the system. Determine the natural frequency ωn of the system for small oscillations θ . ∗ m O b) Show that Izz = (h 2 + 3b2 ) by eval6 uating the integral in part (a). ∗ CONTENTS A m D m y A h m O y B b m x problem 7. [Hint: It may be easier to O set up and evaluate the integral for Izz and then use the parallel axis theorem cm to calculate Izz . Find I x x of the system. a) Set up the integral to find the polar moO ment of inertia Izz of the plate.117: (Filename:pfigure4. The spring is uncoiled when θ = 0. cm c) Find the limiting values of Izz for r = 0 and r = .rp9) K g A m θ R B (Filename:pfigure4.113: (Filename:pfigure.4. For the massless square plate with four point masses on the corners.6 kg · m2 . It spins about the origin at a constant rate of one revolution every π s. .rp2) 7.blue. A m D m 1m 2m x y r x m y B x m m C O problem 7.117 A uniform thin triangular plate of mass m.4. ∗ y 2m 1m ω 7.] y 7.151.54 7. O a) Find the polar moment of inertia Izz of the disk.116: (Filename:pfigure.4) 7.118 A uniform thin plate of mass m is cast in the shape of a semi-circular disk of radius R as shown in the figure. Izz . cm b) Plot Izz versus r/ .rp10) problem 7. a) What is the moment of inertia of the plate about point O? b) Where is the center of mass of the plate at the instant shown? b) Find the polar moment of inertia of the cm plate.87b.113 A thin rod of mass m and length is hinged with a torsional spring of stiffness K at A. and ∗ b) pinned frictionlessly to the rod. assuming that the disk is: a) welded to the rod.114: (Filename:pfigure4.1) problem 7.4.4. c) What are the velocity and acceleration of the center of mass at the instant shown? d) What is the angular momentum of the plate about the point O at the instant shown? e) What are the total force and moment required to maintain this motion when the plate is in the configuration shown? f) What is the total kinetic energy of the plate? m R O x (Filename:pfigure4.114 Do you understand the parallel axis theorem? A massless square plate ABC D has four identical point masses located at its corners.rp4) 7. a) Find the location of the center of mass of the plate M problem 7. The total mass of the remaining plate is 3 kg. what is H O .) f) If F1 were applied to m 2 instead of m 1 . The bar is hinged at end A. The object is hinged at O. Two uniform bars of length and mass m are welded at right angles.122: 7. and θ . a) What is the moment of inertia of the O object about point O (Izz )? ˙ ¨ b) Given θ. g. What is the period of small oscillations for this pendulum? Can you explain why it is longer or shorter than when it is is hung by its end? ˆ m ω x (Filename:pfigure. y w O x O r θ F1 1 = 1+ 2 m x m1 2 m2 (Filename:pfigure.3. A uniform bar of length and mass m is turned by a motor whose shaft is attached to the end of the bar at O.125: (Filename:pfigure. a) Draw a free body diagram of the bar. zero for some purposes’). the rate of change of angular momentum about point O? ˙ ¨ d) Given θ . h crit . the spring is unstretched.blue.153.123 A uniform rigid rod rotates at constant speed in the x y-plane about a peg at point O.122 An object consists of a massless bar with two attached masses m 1 and m 2 . h. what is T .123: (Filename:pfigure4.120: (Filename:pfigure4.4 m from one end. ¨ What is θ ? (Neglect gravity.126 The rod shown is uniform with total mass m and length . θ. θ or θ but you do know that F1 is applied to the rod. perpendicular to the rod at m 1 . a) Which point on the rod has the same acceleration as the ball.121 Motor turns a dumbbell.127: (Filename:pfigure. ˆ ˆ g. A linear spring with stiffness k is attached at the point A at height h above 0 and along the rod as shown. ı . θ . ¨ would θ be bigger or smaller? 55 7. b) If you have done the calculation above correctly there is a value of h for which the natural frequency is zero. a) What is the period of small oscillations for this pendulum? b) Suppose the rod is hung 0.127 A uniform stick of length and mass m is a hair away from vertically up position when it is released with no angular velocity (a ‘hair’ is a technical word that means ‘very small amount.s94h5p5) 7. what is H O .) 7.121: 7.82.6 Using and Izz in mechanics equations 7. Carefully define any coordinates. base vectors.1) problem 7. θ.124 A uniform one meter bar is hung from a hinge that is at the end. What is the force on the stick at point O when the stick is horizontal.128 Acceleration of a trap door. At the ends of the horizontal bar are two more masses m. ∗ b) What is the reaction force on the bar at end A just after release? ∗ problem 7.125 A motor turns a bar. The center of mass of the rod may not exceed a specified acceleration amax = 0. The bottom end of the vertical rod is attached to a hinge at O where a motor keeps the structure rotating at constant rate ω (counter-clockwise). ˆ ı O motor problem 7. What is the net force and moment that the motor and hinge cause on the structure at the instant shown? cm Izz O g θ uniform rod. d) What is the power produced by the motor at t = 1 s? ˆ ˆ ı motor O θ 7. immediately after release. It is allowed to swing freely. ˆ ˆ ı O problem 7.5 m/s2 . The angle that the bar makes (measured counter-clockwise) from the positive x axis is θ = 2π t 2 /s2 . g = 10 m/s2 .rp3) h x A O problem 7. The rod is pinned at point 0. and k.blue.s94q8p1) 7. Find the maximum angular velocity of the rod. Solve in terms of . or angles that you use. Call this value of h. Neglect gravity. b) Find the force acting on the bar from the motor and hinge at t = 1 s. a) Find the natural frequency of vibration (in radians per second) in terms of m. and θ . and θ .rp8) problem 7. Assume that θ is small for both parts of this problem. mass m k y = 0. A uniform bar AB of mass m and a ball of the same mass are released from rest from the same horizontal position. and . When θ = 0.2) m y m m 7.4. the angular momentum about point O? ˙ ˙ ¨ c) Given θ. ˙ ¨ e) Assume that you don’t know θ . It falls to the right. There is gravity.126: (Filename:pfigure.f93f1) . the total kinetic energy? c) Find the torque applied to the bar from the motor at t = 1 s.Problems for Chapter 7 b) Locate the center of mass of the disk cm and calculate Izz .5 m O problem 7. What is the behavior of the system when h < h crit ? (Desired is a phrase pointing out any qualitative change in the type of motion with some justification. m. and then solve. the period of the pendulum does not change much. ˙ as a function of m. no gravity. the pulley.128: (Filename:pfigure. you must first find the equations of motion. It is not only the fastest pendulum but also the “most accurate pendulum”. A bullet of mass m and horizontal velocity Vo strikes the end A of the bar and sticks to it (an inelastic collision).134: (Filename:pfigure. The claim is that even if d changes slightly over time due to wear at the support point. Assuming it swings near to the position where its center of mass G is below the hinge: a) What is the period of its swinging oscillations? b) If.16 m. c) How does the period of the pendulum vary with d? Show the variation by plotting the period against d . f) Find the reaction force on the rod at C.131 A slender uniform bar AB of mass M is hinged at point O. stick.45 m and after some wear d = 0. so it can rotate around O without friction. Calculate the angular velocity of the system — the bar with its embedded bullet.134 A thin hoop of radius R and mass M is hung from a point on its edge and swings in its plane.133 2-D problem. instead. What is α = ω = θ ˙ immediately after release? .78. linearize for small θ . Which pendulum shows the least error in its time period? Do you see any connection between this result and the plot obtained in (c)? problem 7.blue.29 m and after some wear d = 0.3) B 1/3 L O M 2/3 L 7. F Rp O massless pulley problem 7.blue.1) A frictionless pivot 7. How long does it take for the pulley to make one full revolution? ∗ stick. The pulley has negligible mass and radius R p . . A uniform stick with length and mass Mo is welded to a pulley hinged at the center O. d.145.133: String wraps around a pulley with a stick glued to it. Verify this claim by calculating the percent error in the time period of a pendulum of length = 1 m under the following three conditions: (i) initial d = 0. and string are at rest and a force F is suddenly applied to the string. and (iii) initial d = 0.30 m. mass = Mo 7. and θ. g) For the given rod.135 The uniform square shown is released ¨ from rest at t = 0. One end is attached to a hinge at O where a motor keeps the structure rotating at a constant rate ω (counterclockwise). b) Find the rate of change of angular momentum of the rod about C. The pendulum with the value of d obtained in (g) is called the Schuler’s pendulum. Two uniform bars of length and uniform mass m are welded at right angles.130 Given θ .15 m and after some wear d = 0.56 L A problem 7. and θ . A uniform bar of mass m and length hangs from a peg at point C and swings in the vertical plane about an axis passing through the peg. a) neglecting gravity ∗ b) including gravity. the hoop was set to swinging in and out of the plane would the period of oscillations be greater or less? Vo m problem 7. immediately after the impact.s94h8p3) CONTENTS y m B m C A C d θ G O m motor ω ˆ ˆ ı m x m 7. a) Find the angular momentum of the rod as about point C. ¨ e) Find θ when θ = π/6. (ii) initial d = 0. What is the net force and moment that the motor and hinge cause on the structure at the instant shown. θ.129? 7. See also problem 7. what should be the value of d (in terms of ) in order to have the fastest pendulum? ∗ h) Test of Schuler’s pendulum.] ∗ d) Find the total energy of the rod (using the height of point C as a datum for potential energy). Initially the bar is at rest in the vertical position as shown.130. A string is wrapped many times around the pulley. what is the total kinetic energy of the pegged compound pendulum in problem 7.132: A bent bar is rotated by a motor. At time t = 0.131: (Filename:pfigure. θ.3) ¨ ˙ 7. [Hint. ∗ O R θ G problem 7. (Filename:pg85.s94h8p6) problem 7.1) 7.132 Motor turns a bent bar. (Filename:p2.129: B (Filename:pfigure.46 m.129 A pegged compound pendulum. The distance d from the center of mass of the rod to the peg can be changed by putting the peg at some other point along the length of the rod. 138: (Filename:pfigure. As the drum rotates.5 kg · m2 . ∗ k R problem 7. VA=0 MA µ 1/2 RC MB h θ a) The hoop is hung from a point on its edge and swings in its plane. What is the period of oscillation of the disk if it is disturbed from its equilibrium configuration? [You may use the fact that. respectively. writing the equations of motion for each body. a) find the angular acceleration of the pulleys.7 kg·m2 . The moment of inertia of the disk about O is Io . mass of the reel. and solving the equations simultaneously.Problems for Chapter 7 y θ uniform square.139 Oscillating disk.1) 7. slope angle θ.42. Given that x A (0) = 0 and that x A (0) = 0. for ˙ ¨ˆ the disk shown. ∗ 7.137: (Filename:pfigure.42. m 1 = 40 kg and m 2 = 100 kg. and that there is no energy loss during the motion. a 5 kg mass m is pulled up by a string wrapped around the drum.1) 7. g 7. moment of inertia I about point O radius of gyration of the reel K C . the hoop was set to swinging in and out of the plane would the period of oscillations be greater or less? 7. where θ is 2 the angle of rotation of the disk. A point A is .KC O At t=0. IO problem 7. are released from rest in the configuration shown. Find the period of small oscillation. The rope supports a weight W .141 The compound pulley system shown in the figure consists of two pulleys rigidly connected to each other. b) What is the period of its swinging oscillations? V=2 At t=0. Initially the system is at rest.2) 57 marked on the string. Calculate the velocity for the block B after it has dropped a vertical distance h.135. VB=0 problem 7. Gear A is driven by gear C which has radius RC = 300 mm.135: (Filename:pfigure. Assume that the mass of the rope is negligible. H O = 1 m R 2 θ k . mass 2 of the block B M B . Just after release.]. Assume small rotations and the consequent geometrical simplifications.2) g m problem 7.141: A Spool has mass moment of inertia.142: (Filename:pfigure. mass of block A.136: (Filename:p. For reference.f96.136 A square plate with side and mass m is hinged at one corner in a gravitational field g. A uniform disk with mass m and radius R pivots around a frictionless hinge at its center.blue. Write the equation of conservation of energy for this system.139: (Filename:pg147. 7. Given: h. M A . as shown in the figure. and ∗ b) find the tension in each string.2 m and Ro = 0. The two masses. The disk is free to rotate about the axis through O normal to the page.137 A wheel of radius R and moment of inertia I about the axis of rotation has a rope wound around it. here are dimensions and values you should use in this problem: mass of hoop m hoop = 1 kg radius of hoop Rhoop = 3 m. The combined moment of inertia of the gear and the drum about the axis of rotation is Izz = 0. At the instant of interest. ∗ W problem 7. Assuming its swings near to the position where its center of mass is below the hinge.p3.136. The combined moment of intertia of the two pulleys about the axis of rotation is 2. what is x A (t)? ˙ up Ro Ri O F xA x (0) = 0 Inextensible xA (0) = 0 ˙ string wound A xA (t) = ? around spool m1 problem 7. The radii of the two pulleys are: Ri = 0. Assume the spring can carry compression. MC .142 Consider a system of two blocks A and B and the reel C mounted at the fixed point O. outer radius of the reel RC . coefficient of friction µ. RC MC.143 Gear A with radius R A = 400 mm is rigidly connected to a drum B with radius R B = 200 mm.1) m2 (Filename:summer95p2.2) 7.blue. Find the acceleration of the mass m. and differentiate to find the equation of motion in terms of acceleration. mass=M R x hinge problem 7.blue.4 m. The radius of gyration K cm = m K 2 .) ∗ 7. (Apply the workis defined by Izz energy principle.1) θ R c) If.138 A disk with radius R has a string wrapped around it which is pulled with a force F. and gravitational acceleration g= 10 m/s2 . Check the solution obtained by drawing separate free-body diagrams for the wheel and for the weight. It is attached to a massless spring which is horizontal and relaxed when the attachment point is directly above the center of the disk. instead. The angular speed and angular acceleration of the driving gear are 60 rpm and 12 rpm/s.140 This problem concerns a narrow rigid hoop. inner radius of the reel 1 RC .blue. h.129. g. (Note also that φ ≡ θ during such motions). set up a differential equation.blue. .147: Two racks connected by a gear.145 Frequently parents will build a tower of blocks for their children. θ P ω 6 in This? or This? uniform disk m = 10 lbm problem 7. . The spring is relaxed when the angle θ = 0. what is FB ? b) If the gear bearing had friction would that increase or decrease FB ? c) If the angular velocity of the gear is increasing at rate α does this increase or decrease FB at the given ω. would that increase or decrease FB to achieve the same constant rate? c) If the angular velocity of the gear is increasing at rate α. θ. A mass M falls vertically but is withheld by a string which is wrapped around an ideal massless pulley with radius a.149 A stick welded to massless gear that rolls against a massless rack which slides on frictionless bearings and is constrained by a linear spring.145: (Filename:pfigure. What is FB ? b) If the gear bearing had friction.150 A tipped hanging sign is represented by a point mass m. Assume you also know the input angular acceleration ω and the ˙ moments of inertia I A .1) massless racks FB = ? 7. use angular momentum balance. Assume the system is released from rest at θ = θ0 . You might want to draw FBD(s). In falling (even when they start to topple aligned). Neglect gravity. Assume that the machine starts from rest with a given output load.146 Massless pulley. (a) For what values of FB is the output power positive? (b) For what values of FB is the output work maximum if the machine starts from rest and runs for a fixed amount of time? MA problem 7.58 ωC. At the output the machinery (not shown) applies a force FB to the other rack.1) problem 7.aa) 7. d) If the output load FB is given then the motion of the machine can be found from the input load. a) Write the angular momentum of the mass about 0 when the rod has an an˙ gular velocity θ. dumbell and a hanging mass.144: Two gears.149: (Filename:p. a) What is the input power? b) What is the output power? c) What is the angular velocity ωoutput = ωC of the output shaft? A B RB RC C no slip y R A O a M θ m problem 7. (Filename:pg131. kids knock them down. and .148: A set of gears turning at variable rate. find the acceleration of the mass.148 2-D accelerating gear train. does this increase or decrease FB at the given ω. The tension T in the very stretchy cord is idealized as constant during small displacements. and use the result to find the quantities of interest. I B and IC of each of the disks about their centers . See also problem 7. θ . The sign sits at the end of a massless.3.16. solve it. [Hint: There are several steps of reasoning required. each of whose center is a distance from O. Assume you know the torque Minput = M A and angular velocity ωinput = ω A of the input shaft. In terms of some or ˙ ˆ all of m. a) Assume the gear is spinning at constant rate and is frictionless. R. ı . Point C is a distance h down from O. Just as frequently. no slip FB = ? RA RC C A FA = 100 lb B RB problem 7. rigid rod which is hinged at its point of attachment to the ground. At the output the machinery (not shown) applies a force of FB to the output gear. these towers invariably break in two (or more) pieces at some point along their length. See also problems 7. The pulley is welded to a dumbell made of a massless rod welded to uniform solid spheres at A and B of radius R.2) problem 7. k R ˆ ˆ ı m. Why does this breaking occur? What condition is satisfied at the point of the break? Will the stack bend towards or away from the floor after the break? 7.74. R. 100 lbf problem 7. a) Assume the gear is spinning at constant rate and is frictionless. (Filename:pg129. At the instant in question. So long as rack B moves in the opposite direction of the output force FB the output power is positive. θ0 . a. and . (Filename:pg131. At the input to a gear box a 100 lbf force is applied to gear A.αC RA the dumbell makes an angle θ with the positive ˙ x axis and is spinning at the rate θ . Consider all hinges to be frictionless and motions to take place in the plane of the paper. A 100 lbf force is applied to one rack. A taut massless elastic cord helps keep the rod vertical. M.1) 7.146: ωA RA no slip RC ωC RB A B C MC 7.144 Two gears accelerating. .147 Two racks connected by a gear.143: x (Filename:summer95f.f96.77 and 4. k. plug values into this solution.1) d) What is the output torque Moutput = MC ? 7.2) R B M g C m h ˆ ˆ ı (Filename:s97p3.p2. What is the acceleration of the point P at the end of the stick when θ = 0? Answer in terms of any or ˆ ˆ all of m. CONTENTS 7. Assume the bearings and contacts are frictionless. 59 L string φ m θ L rigid rod g O problem 7.blue.153. T = 2 mg mg 2 . and T < 2 .150: (Filename:pfigure.Problems for Chapter 7 b) Find the differential equation that governs the mass’s motion for small θ . c) Describe the motion for T > mg . Interpret the differences of these cases in physical terms.1) . 5 mı + 3.87 m .2 A circular disk of radius r = 100 mm rotates at constant speed ω about a fixed but unknown axis. The bearing at A distinct points on the axis and draw vecprevents translation of its end of the shaft in any tors r 1 and r 2 from these two points direction but causes no torques other than that to the particle of interest.83 rad/sk . r = 3 mı + 4 m .blue. A small sphere of mass 1 kg axis of rotation. shaft? a) For planar circular motion about the zaxis.7: x 300 mm (Filename:pfigure.5 The following questions are about the velocity and acceleration formulae for the nonconstant rate circular motion about a fixed axis: c) The acceleration a A of point A. 8. what is |A × (A×B )| in terms of |A| and |B |? (Hint: make a neat 3-D sketch. ˆ ˆ b) ω = 4.9. and v . b) The velocity v A/B of point A relative to point B. 8. that the direction). Showing ω ×(ω × r ) = (ω × ω) × r is almost trivial. Take two imposes this rate if needed. z ω O 30 o y A 8. In the configuration of interest along one of its edges. (why?) Take ω1 = ˆ ˆ ˆ ˆ ˆ ω1 k . c) Find the acceleration of points A and B. The center of mass of the plate goes around at a constant linear speed v = 2.rp10) problem 8. Find the following quantities: a) The vector ω. v a = = ω×r ˙ ω × r + ω × (ω × r ) B problem 8. r = ˆ ˆ 0.8 A rigid body rotates about a fixed axis ˆ and has an angular acceleration α = 3π 2 (ı + ˆ ˆ 1. the magnitude of the acceleration of the center of mass C is 50 m/s2 . r .2: 8.3 In the expression for normal acceleration a n = ω × (ω × r ). At the same instant. draw the circular orbit.5 m − ˆ 4. the angular velocity of each point on the diameter AB (which is parallel to the xˆ axis) is v = −2 m/sk . ˆ ˆ ˙ ˆ ω = α k . ˆ ı ˆ ˆ k 0. At a certain instant. b) Draw a schematic picture of a particle See also problem 8. The shaft is spinning counterclock8.10 Reconsider the particle on a shaft in problem 8.1 If A and B are perpendicular.1) A r C B ˆ ˆ k ˆ ı (Filename:pfigure4.1 m massless 0.6 A rectangular plate with width w = 1 m ˆ wise (when viewed from the positive k direcand length = 2 m is rigidly attached to a shaft tion) at 3 rad/s.5 + 1.5 m/s.1 m C problem 8. A shaft of negligible (or a point on a rigid body) going in cirmass spins at a constant rate. r = 3.rp1) motor x m = 1 kg D A 0.37.3.60 CONTENTS Problems for Chapter 8 Advanced topics in circular motion ˆ ˆ c) ω = 2 rad/sı + 2 rad/s + ˆ ˆ ˆ 2. The vector ω lies in the yz-plane. Show. used and thus conclude that the vector r can be taken from any point on the A bar of negligible mass is welded at right angles to the shaft. The motor at A cles about a fixed axis in 3-D.) 8.10.3. Find the magnitude v A ation formulae reduce to the formulae of the velocity of point A. Compute v in the three cases below.4 In circular motion the velocity is v = ω × r . take z 8.5 k ) rad/ s2 . no gravity. ˆ from elementary physics: v = ω R eθ ˆ ˆ and a = −ω2 R e R + Rα eθ . a) What is the position vector of the mass in the given coordinate system at the a) Find the angular velocity of the plate. . using ˆ of the motor (which causes a torque in the k the formulae for v and a above.95 mk . is attached to the free end of the bar.5 rad/s .9: (Filename:ch4. The shaft rotates about ˆ the rod BD is parallel to the ı direction. At the instant when the disk is in the x y-plane. The origin O of the coordinate frame is at the center of the cylinder and is at rest. instant of interest? b) At the instant shown. There is no gravity.33 rad/sı + 2. ω2 = ω2 ı + ω1 k and r = a ı − b and show that ω1 × (ω2 × r ) = (ω1 × ω2 ) × r . the x-axis. ω = ωk . respectively. 8.2 m B y z 8. c) What are the velocity and acceleration of the particle at the instant of interest? y x w problem 8. The frictionless bearing at C holds acceleration and velocity of the particle ˆ that end of the shaft from moving in the ı and are the same irrespective of which r is ˆ ˆ directions but allows slip in the k direction. and r = R e Rthe velocity of a point A of the body has xcomponent and y-component equal to 3 and and show that the velocity and acceler−4 m/s. find the velocity b) What is angular velocity vector of the of point P. the parenthesis around ω × r is important because vector product is nonassociative.1 3-D description of circular motion 8.7 The solid cylinder shown has an angular velocity ω of magnitude 40 rad/s. and show the vectors ω.9) 8.5 mı − 0. b) Find the location of the axis of rotation.9 Particle attached to a shaft. ˆ ˆ ˆ a) ω = 2 rad/sk . a) Find the angular velocity of the disk. where ω is the angular velocity and r is the position vector of the point of interest relative to a point on the axis of rotation.6: (Filename:pfigure4. crooked. a) In the coordinate system shown. c) Find the position vector r ≡ r P/O of point P.5 m φ P Q y x motor x m = 1 kg D A 0. the rod O P is in the the x z-plane. what is the position vector of the particle at the instant of interest? b) In the coordinate system shown. .13.2 m B 0. also of negligible mass. a) In the coordinate system shown. draw the basis vectors ˆ ˆ er and eθ and express them in terms of ˆ ˆ ˆ the basis vectors ı . what are angular velocity and angular acceleration vectors of the rod and shaft? c) In the coordinate system shown. k . Reconsider the crooked rod on a shaft in problem 8. See also problem 8. (Filename:ch4. What is the velocity and acceleration 8.particleshaft1.25 m 0.0 m 0. what is the position vector of the point P on the rod? b) In the coordinate system shown. At the instant of interest.a) y 0. The motor at A imposes this rate if needed.25 m 0.10: Particle attached to a shaft. The shaft is supported by bearings at its ends and spins at a constant rate ω.12. (Filename:ch5.11: (Filename:ch5. A one meter long uniform rod is welded at its center at an angle φ to a shaft as shown. the shaft is spinning counterclockwise (when viewed from ˆ the positive k direction) at ω = 3 rad/s and ω = 5 rad/s2 . The shaft is supported by bearings at its ends and spins at a variable rate. In the configuration of interest ˙ ˆ the rod BD is parallel with the ı direction.2. the ˙ crooked rod lies in the x y-plane.7) 8.2 m B y z ˆ ı ˆ ˆ k y z 0.a) 8. what is the velocity and acceleration of point P at the instant of interest? ∗ 1. what is the velocity and acceleration of point P? d) What is the velocity and acceleration of point P relative to point Q? motor x m = 1 kg D A 0.14: A crooked rod spins on a shaft. again. The shaft is welded orthogonally to a bar.5 m O y z weld A O x P 30o B ω ˆ ı ˆ ˆ k problem 8.5 m O weld problem 8. the crooked rod lies in the x y-plane.1 m massless 0. what is the position vector of the point P on the rod at the instant of interest? ∗ b) In the coordinate system shown.3) 8. Use this position vector to compute the velocity and acceleration of point P using the general formulae v = ω × r and a = ω × ω × r .rp1) .4) 1. A one meter long uniform rod is welded at its center at an angle φ to a shaft as shown.1 m C problem 8.13: A crooked rod spins on a shaft.12 Reconsider the particle on a shaft in problem 8.0 m 0.13 Crooked rod. The shaft rotates about its longitudinal axis at a constant rate ω = 120 rpm.25 m 0.copy) 0.1 m C ˆ ı ˆ ˆ k problem 8. The bearing at A prevents translation of its end of the shaft in any direction but causes no torques other than that ˆ of the motor (which causes a torque in the k direction).1 m C problem 8.1 m massless 0.3. At the instant of interest.11. Show that the answers obtained here are the same as those in (b).16: (Filename:pfigure4.16 A rigid rod O P of length L = 500 mm is welded to a shaft AB at an angle θ = 30◦ . 0.11 Particle attached to a shaft. what is the velocity and acceleration of the particle at the instant of interest? 8.12: (Filename:ch5. A shaft of negligible mass spins at variable rate.14 Crooked rod. At the instant of interest the shaft is spinning counterclockwise (when viewed from the positive z problem 8. b) Find the radius of the circular path of point P and calculate the velocity and acceleration of the point using planar ˙ˆ circular motion formulae v = θ eθ and ˙ ˆ a = −θ 2 R er .1 m massless 0.0 m 0.5 m φ P Q y x 0. There is no gravity. which is attached to a small ball of (non-negligible) mass of 1 kg. The frictionless bearing at C holds ˆ that end of the shaft from moving in the ı and ˆ ˆ directions but allows slip in the k direction.15: A crooked rod spins on a shaft.2 m B motor x m = 1 kg D A 0. At the instant of interest. a) Neatly draw the path of point P. What are the velocity and acceleration of point P relative to point Q? 1.5 m O weld problem 8. (Filename:pfigure4.15 Crooked rod. At the instant shown. no gravity (Filename:pfigure4. what are angular velocity and angular acceleration vectors of the shaft? c) In the coordinate system shown. no gravity. a) In the coordinate system shown.Problems for Chapter 8 a) What is the velocity of a point halfway down the rod B D relative to the velocity of the mass at D? b) What is the acceleration of a point halfway down the rod B D relative to the acceleration of the mass at D? of a point halfway down the rod B D relative the velocity and acceleration of the particle? 61 x direction) at angular speed ω with angular acceleration ω.5 m φ P Q y x 8. At the instant shown. 20 Let P(x. Find the angular velocity of the cone as a function of the cone angle β. In particular. x D 2L β A B r A y O C L ω z B problem 8. no gravity. The cone is attached to shaft AB 8. At the instant of interest. In the configuration of interest the rod BD is ˆ parallel with the ı direction.17 A uniform rectangular plate ABC D of width and length 2 rotates about its diagonal AC at a constant angular speed ω = 5 rad/s. Thus. a) Find the linear momentum L and its ˙ rate of change L for this system. [Moral: In ˙ general. Is this point unique? problem 8. which is attached to a small ball of(nonnegligible) mass of 1 kg. b) Find the angular momentum H O about ˙ point O and its rate of change H O for this system.18 A rectangular plate of width w = 100 mm and length = 200 mm rotates about one of its diagonals at a constant angular rate ω = 3 rad/s. y. axis of rotation normal to the plane).2. the plate is in the x z-plane. The shaft is but allows slip in the k welded orthogonally to a bar.rp5) (Filename:pfigure4. 8. 8. The motor at A imposes this rate if needed. Take m = 1 kg and L = 1 m. Assume that at the instant shown. The disk rotates at a constant angular speed ω = 120 rpm. a specific axis and a particular rotation rate.17: 8. Find the velocity of the corner point P as follows: a) Write the angular velocity ω of the plate in terms of its components in the two coordinate systems.rp4) C where R = + is the radius of the circular path of point P.22 The rotating two-mass system shown in the figure has a constant angular velocity ω = ˆ 12 rad/sı .rp11) 8. You may do the general problem or you may pick particular appropriate locations for the masses.20: ˙ˆ ω = θk (Filename:pfigure4.23 Particle attached to a shaft. Assume for the problems below that you have two mass points going in circles about a common axis at a fixed common angular rate.21 System of Particles going in a circle. b) Find the velocity and acceleration of point B when the plate is in the yzplane. a specific point C.3. rotates with the plate. the plate is in the x y-plane and the two axes z and z coincide.2 Dynamics of fixed axis rotation 8. by ˙ calculating the sum in HC that the result is the same when using the two particles or when using a single particle at the center of mass (with mass equal to the sum of the two particles). 0. z) be a point mass on an abstract massless body that rotates at a constant rate ω about the z-axis. c) Find the perpendicular distance d of point P from the axis of rotation. Using particular mass locations that you choose.rpb) problem 8. Assume that the cone rolls without slip on the disk. show that 2 particles give a dif˙ ferent value for HC than a single particle at the center of mass. CONTENTS a) 2-D (particles in the plane. and show that the linear speed of point P calculated from velocities found in part (b) is ωd in each case.19 A cone of angle β sits on its side on a horizontal disk of radius r = 100 mm as shown in the figure. At the instant shown.] 8. a) What is the tension in the rod BD? ∗ b) What are all the reaction forces at A and C? ∗ . A shaft of negligible mass spins at constant rate. There is a fixed coordinate system x yz with the origin at the center of the plate and the x-axis aligned with the axis of rotation. The shaft is spinning counterclockwise (when ˆ viewed from the positive k direction) at 3 rad/s. aligned with the principal axes of the plate. a) Show that the rate of change of angular momentum of mass P about the origin O is given by ˆ H O = −mω Rz eθ 2 m L A L y ˙ ˆ ˆ k ˆ ı z O L L m x (Filename:pfigure4. z) be the center of the circular path that P describes during its motion. a) Find the velocity and acceleration of point B at the instant shown.2. Another set of axes x y z .rp2) ω (Filename:pfigure4. the masses are in the x y-plane. that turns in the bearing. H C = r cm/C × m tot a cm . has the same origin but is attached to the plate and therefore.62 8.18: x problem 8. The bearing at A prevents translation of its end of the shaft in any direction but causes no torques other than that of the motor ˆ (which causes a torque in the k direction).22: y x' y' ω x axis P O y θ C z ˙ θ O d P problem 8. Let C(0. find the angular velocity of the cone for a) β = 30◦ . particular masses for the masses.19: (Filename:pfigure4. and b) β = 90◦ (or 90◦ minus a hair if that makes you more comfortable). b) Find the velocity of point P in both coordinate systems using the expressions for ω found in part (a). the disk drives the cone-and-shaft assembly causing it to turn in the bearing. b) 3-D (particles not in a common plane normal to the axis of rotation).2. There is no gravity. The frictionless bearing at C holds that end of the ˆ ˆ shaft from moving in the ı and directions ˆ direction. x2 y2 b) Is there any point on the axis of rotation ˙ (z-axis) about which H is zero. also of negligible mass. Show. . ∗ m α problem 8.26: (Filename:pfigure4.24 Particle attached to a shaft. . and k . a) What possible angle(s) could the pendulum make in a steady motion.29 Consider the configuration in problem 8. m ω O 8. X swings his daughter Y about the vertical axis at a constant rate ω = 1 rad/s. At the time of interest the rod BD is parallel to the +j direction.1 m massless 0. estimate the force on each of her shoulder joints. ω and ˆ ˆ ˆ the unit vectors ı . Assuming Y to be a point mass located at the center of mass. A perspective view and a side view are shown. Find the coefficient of friction between the block and the incline. 8. Repeat problems (b) and (c) doing it the way you did not do it the first time. ∗ c) Find the following quantities in any order that pleases you. At the instant of interest. Y is merely one year old and weighs 12 kg. weight and stop watch to check your calculation.25: φ φ g y z ˆ ı ˆ ˆ k (Filename:pfigure. bar and mass as one system and writing the momentum balance equations. At the edge of the 2 m radius turntable swings a pendulum which makes an angle φ with the negative z direction. The other is to use the result of (a) with action-reaction to do statics on the shaft BC.26 Mr.27 A block of mass m = 2 kg supported by a light inextensible string at point O sits on the surface of a rotating cone as shown in the figure. b) How long does it take to make one revolution? How does this compare with the period of oscillation of a simple pendulum. . The pendulum length is = 10 m. By experiments.2. 63 R ω SIDE VIEW z ω R 8. ∗ b) How many solutions are there? [hint. • The velocity of the rock ( v ).28: (Filename:pfigure. Consider the same situation as in problem 8.23 with the following changes: There is gravity (pointing in the −i direction).s94h7p1) 8. find φ for steady circular motion of the system. φ. • The rate of change of the angular momentum about the +x axis (d Hx /dt). your right thumb points up the z-axis. the limiting rate of rotation of the turntable before the block starts sliding up the inclined plane is found to be ω = 125 rpm. a) What is the force on D from the rod BD? (Note that the rod BD is not a ‘twoforce’ member. The cone rotates about its vertical axis of symmetry at constant rate ω.2) problem 8.Problems for Chapter 8 c) What is the torque that the motor at A causes on the shaft? ∗ d) There are two ways to do problems (b) and (c): one is by treating the shaft.27: (Filename:pfigure4. Why not?) b) What are all the reaction forces at A and C? c) What is the torque that the motor at A causes on the shaft? d) Do problems (b) and (c) again a different way. What is the relation between the angle (measured from the vertical) at which it hangs and the speed at which it moves? Do an experiment with a string. the block is at a distance L = 0. g. with gravity. The gravitational constant is g = 10 m/s2 pointing in the −z direction.28. ∗ motor x m = 1 kg D A 0. X L P d Y θ x z circle around z-axis y φ problem 8. All solutions should be in terms of m.s95q6) 0.14. a) Draw a free body diagram of the rock.2. sketch the functions involved in your final equation.rp11) 8.23: (Filename:pfigure. You are not asked to solve this equation but you should have the equation clearly defined.28 A small rock swings in circles at constant angular rate ω. It turns out that this quest leads to a transcendental equation. At the moment of interest the mass is passing through the yz-plane.] ∗ c) Sketch very approximately any equilibrium solution(s) not already sketched on the side view in the picture provided. Find the value of ω in rpm at which the block loses contact with the cone. It is hung by a string with length the other end of which is at the origin of a good coordinate system.rp12) 8.30 A small block of mass m = 5 kg sits on an inclined plane which is supported by a bearing at point O as shown in the figure. ω Mr. The rock has mass m and the gravitational constant is g.25 A turntable rotates at the constant rate of ω = 2 rad/s about the z axis.2 m B problem 8.1 m C problem 8. In other words. • The acceleration of the rock ( a ).blue.5 m from where the surface of the incline intersects the axis of rotation. The mass rotates so that if you follow it with the fingers of your right hand. 8. • The tension in the string (T ). The string makes an angle of φ with the negative z axis. The pendulum consists of a massless rod with a point mass m = 2 kg at the end. C and g.36 Flyball governors are used to control the flow of working fluid in steam engines. A spring-loaded flyball governor is shown schematically in the figure. diesel engines.81 m/s2 .2.3 what is the range of ω for which the block would stay on the ramp without slipping even if there were no rod? ∗ A ω y θ C Fixed A B F θ E m (Filename:pfigure4. Explain any oddities.1) 8. A block of mass m = 5 kg is held on an inclined platform by a rod BC as shown in the figure. in turn.35 A spring of relaxed length 0 (same as the diameter of the plate) is attached to the mid point of the rods supporting the two hanging masses.e. 8.30: k=2r 0=2r θ m problem 8. ˙ ˙ a) What are H O and L? ˙ b) Calculate H/O using the integral definition of angular momentum.37 Crooked rod. (a) Assuming ω = 5 rad/s and µ = 0 find the tension in the rod. Find the acceleration of mass B at the instant shown by a) calculating a = ω × ω × r where r is the position vector of point B with respect to point O. θ → 0 and θ → π/2 give sensible values of ω? CONTENTS m C θ ω r O θ θ /2 g ω r O (Filename:pfigure4. At a constant angular speed ω. when the governor rotates at 100 rpm. and that in this steady state. The rigid and massless links that connect the outer arms to the collar. and the length of each link is l = 0. find the required spring stiffness k. c) Assume that the entire system sits inside a cylinder of radius 20 cm and the axis of rotation is aligned with the longitudinal axis of the cylinder. problem 8. θ = 30o .31 Particle sliding in circles in a parabolic bowl.33. B. The maximum operating speed of the rotating shaft is 200 rpm (i.35: (Filename:pfigure4. (Your answer may include other given constants.64 ω L b) Find the steady state ω as a function of θ. As the governor rotates about the vertical axis. a) Find v in terms of any or all of R. The equation describing the bowl is z = C R 2 = C(x 2 + y 2 ).rp8) θ m 8.p2.36: m/4 (Filename:summer95p2. b) Now say you are given ω.rp9) g O problem 8. r = r B/O .34: platform ˆ k ˆ ı ˆ A problem 8. m problem 8. ∗ c) What are the six reaction components if there is no gravity? ∗ d) Repeat the calculations. Which reactions change and by how much? ∗ . A 2 long uniform rod of mass m is welded at its center to a shaft at an angle φ as shown. Find the amount the string is elongated. ωmax = 200 rpm).25 m.2) 8..f96. As if in a James Bond adventure in a big slippery radar bowl. your answer for θ reduces to that in problem 8.2. a) Find the steady angle θ as a function of the plate’s rotation rate ω and the spring constant k.33: m 8. The shaft is supported by bearings at its ends and spins at a constant rate ω.e. the two masses maintain a steady state angle θ from their vertical static equilibrium.s94h5p6) 8.rp7) B k D m θ C ω rod C m B O x problem 8. If the two masses are not to touch the walls of the cylinder. x. a particle-like human with mass m is sliding around in circles at speed v. The mass of each ball is m = 5 kg. (b) If µ = . try to lift the central collar (mass m/4) against the spring. E. etc. ∗ (Filename:pfigure4. and by ˙ ˆ b) calculating a = −R θ 2 e R where R is the radius of the circular path of mass B. including gravity g. Find v and R if you can. for k = 0. assume that θ = 60◦ . Two identical point masses hang from two points on the plate diametrically opposite to each other as shown in the figure. steam turbines. 8. ω ≡ θ = 5 rad/s.. i.32: θ L (Filename:pfigure. given that the spring constant k = 500 N/m.2.34 For the rod and mass system shown in the ˙ figure.33 A circular plate of radius r = 10 cm rotates in the horizontal plane about the vertical axis passing through its center O.31: (Filename:p.rp6) problem 8.) b) Verify that. The central collar can move along the vertical axis but the top of the governor remains fixed. and g = 9. The platform rotates at a constant angular rate ω. a) Find the linear speed and the magnitude of the centripetal acceleration of either of the two masses. g. C. F where the links are connected. There are frictionless hinges at points A. and C.32 Mass on a rotating inclined platform. the two masses tend to move outwards. D. and = 1 m. L = 1 m. The coefficient of friction between the block and the platform is µ. θ = 90o .2. Do the limiting values of θ. 4) 8. A massless. Ignore gravity. A bar with mass Mb and length is welded to the shaft at 45o . L L A θ B y ω z problem 8. One end of the rod is connected to the shaft by a ball-and-socket joint . b) Find the tension in the string DB. d. At the instant shown in the figure.) Use this calculation to find TD B and compare to method ii. including a bit of the strings. R and m. y motor B 45o z θ y problem 8. Which components in the matrix [I O ] will change if one mass is removed and the other one is doubled? . It is spun at constant rate ω.37: (Filename:h6.4) 8.1) problem 8.39: Rod on a shaft. find two positions along the rod where concentrating one-half the mass at each of these positions ˙ gives the same H O . rigid shaft is connected to frictionless hinges at A and B. b) The moment that the shaft applies to the rod. . (Filename:pg79. (ii) Write the equation of angular momentum balance about the axis AE. Find the tension in the string in terms of ω.39 Rod on a shaft. a) Draw a free body diagram of the rod AB. (That is. ω and Mmotor are known and the rod lies in the yz-plane. c) The reaction forces at the bearings.41 A uniform narrow shaft of length 2L and mass 2M is bent at an angle θ half way down its length. c) Find the reaction force (a vector) at A. Neglect gravity.40: ω z (Filename:pfigure. The rod has mass m and length and is attached at an angle γ at its midpoint to the midpoint of the shaft. The length of the shaft is 2d.s95f4) 8. (Filename:pg89. Two approaches: (i) Write the six equations of linear and angular momentum balance (about any point of your choosing) and solve them (perhaps using a computer) for the tension TD B . It is held by a hinge at A and a string at B.38 For the configuration in problem 8.37. find two positions along the bent portion of the shaft where concentrating one-half its mass at each ˙ ˙ of these positions gives the same H O and L. 8. The cross bar is welded to the main shaft. A motor applies a torque Mmotor along the axis of the shaft. It then falls under the action of gravity.25 kg and = 0.3 Moment of inertia matrices: [I cm ] and [I O ] D x y problem 8.45 Find the moment of inertia matrix [I O ] of the system shown in the figure. B R A 45 o string 8.40 Rod spins on a shaft. a) What is the magnitude of the force that must be applied at O to keep the shaft spinning? 65 m = 1 kg √ 1 = 2m B C string 45o 45 o A 45o O x weld b) (harder) What is the magnitude of the moment that must be applied at O to keep the shaft spinning? x E problem 8. assume m.2) A x Looking down y-axis x (Filename:p2.43: Uniform rod attached to a shaft. given that m = 0. A uniform 1 kg.41. 8.41: Spinning rod.Problems for Chapter 8 2d P y φ 8. ∗ problem 8.42 For the configuration in problem 8.44 3D. take the equation of angular momentum balance about point A and dot both sides with a vector in the direction of r E/A . √ 2 m rod is attached to a shaft which spins at a constant rate of 1 rad/s. At time t = 0 the assembly is tipped a small angle θ0 from the yz-plane but has no angular velocity. What is θ at t = t1 (assuming t1 is small enough so that θ1 is much less than 1)? z z x g 8. crooked bar on shaft falls down. The other end is held by two strings (or pin jointed rods) that are connected to a rigid cross bar. A uniform rod R with length R and mass m spins at constant rate ω about the z axis. No grav(Filename:f.43 Rod attached to a shaft.4) O ω D 1m 8. ity. A uniform rod is welded to a shaft which is connected to frictionless bearings.5 m.44: A crooked rod welded to a shaft falls down. a) Find the angular acceleration of the shaft. 48: (Filename:pfigure.m x y.46 A dumbbell (two masses connected by a massless rod of length ) lies in the x y plane. you are to calculate the moment of inertia matrix for the disk.50: (Filename:f92h7p1) m x φ x' z' φ z G uniform disk mass = Md. What are the principal axes of this body? y b c a homogeneous sphere homogeneous rectangular parallelapiped y point mass. The center of the jack is at the center of mass.blue. mark the approximate location of the center of mass and the principal axes about the center of mass 8.blue. m x O problem 8. 2b) relative to the origin O. radius = Ro b O problem 8.rp12) 8.m y r. Where should one put four masses on a massless disk to make it equivalent to a uniform and ‘massful’ disk in problem 8. a) Using a coordinate system that is lined up with the disk so that the z axis is normal to the disk and the origin of the coordinate system is at the center of mass (the center) of the disk.5 kg and each leg is = 0. O R d R dθ.50(a). b). It has three orthogonal bars with three pairs of point masses along the bars. and (−b. You may ignore the width of the rod. This problem concerns a uniform flat disk of radius R0 with mass Md = Mdisc . The y axis and the y axis are coincident.52: (Filename:pfigure4. do it by direct integration and. by using a change of basis. b. and the same moment of inertia matrix as a uniform disk? 8.y' problem 8.49 Find the moment of inertia matrix [I O ] for the ‘L’-shaped rod shown in the figure. the z axis makes an angle φ with the z axis which is normal to the plane of the (measured from the z axis towards the z axis). 0). calculate the moment of inertia matrix for the disk. for the purposes of dynamics.53 An ‘L’-shaped structure consists of two thin uniform rectangular plates each of width w = 200 mm.48 Three identical solid spheres of mass m and radius r are situated at (b.46: θ m b x L b massless rod holding two equal masses problem 8. 8. That is. 8. Find all nine components of [I O ] of the system of three spheres.rp16) 2 mr 2 5 0 0 0 2 mr 2 5 0 0 . every rigid body can be replaced by a massless rigid structure holding six masses. the same center of mass location. In the two parts of this problem. the disk intersects the x y plane along the y axis. m problem 8.4.66 y m /2 /2 b massless b bb O problem 8. z O x r y problem 8.4 m long. what set of four masses has the same total mass. In the first part. 0 2 mr 2 5 Use the parallel axis theorem.47 Equivalent dumbbell. The vertical plate has height . and z axes.4.52. For flat objects one only needs four point masses. It is a great fact that. y.49.52 Find the moment of inertia matrix [I cm ] for the uniform solid cylinder of length and radius r shown in the figure. Compute the moments and products of inertia of this body about the x. and use dm = and d A = (M/π R 2 ) d A. Hint: [I cm ] for each sphere is 8.2) (Filename:pfigure4.4.blue. capital "J" 8.45: x (Filename:pfigure4.2) CONTENTS r.38. Calculate the moment of inertia matrix for the disk.49: (Filename:pfigure.m z r. 0.rp13) 2 z point mass.50 Moment of Inertia Matrix. 0.] b) Use a coordinate system x y z aligned as follows: the origin of the coordinate system is at the center of the disk. The mass of the rod is m = 1.51: (Filename:pfigure. Alternate method: use the solution from (a) above with a change of basis.2) 8. in the second part. (0.51 For each of the shapes below. The bars are in the directions of the eigenvectors of the moment of inertia matrix of the body (the principal directions). [Hint: use polar coordinates. The structure looks like a child’s jack. 54 A solar panel consists of three rectangular plates each of mass m = 1. You should do them all three ways. find λ such that the angle between ω and H = [I G ] · ω is zero if 3 0 2 0 6 0 kg m2 . a) Assume that the configuration is as in problem 8. In each case ω = 2. At the moment of interest the hoop is in the x y-plane of a colbm ft2 . What are H O and H O ? b) Assume that the configuration is as in problem 8. The plates are welded at right angles along the side measuring 200 mm. each of mass m and length d.20 −2. if ω = (2ı − 3 ) rad/s and 5.50(a) and that the disk is spinning about the z axis at the constant ˙ rate of ωk. F P m problem 8.4. first using the way you are most comfortable with.8 8. a) What is [IG ] of the hoop? b) What is H O of the hoop? O x w y problem 8.89. These problems can be done at least three different ways: using the[I ]’s from problem 8.blue.60 Calculation of H O and H O . and z b) [I G ] = kg m2 . 0 0 10.63: 8. are welded perpendicularly to an axle of mass M and length .59 For ω = 5 rad/sλ.2) 8.1) 8. A force F = −F ı + G k acts at the point P as shown in the figure.57 Compute ω × ([I G ] · ω) for the following ˆ two cases.61: x (Filename:pfigure. if ω = 3 rad/sk and [I G ] = 20 0 0 0 20 0 0 0 40 kg· m2 .56 10. direct ˙ integration of the definition of H O .47. ) z problem 8.54: (Filename:pfigure4.62 A hoop is welded to a long rigid massless shaft which lies on a diameter of the hoop.1) d 8.4 0 0 0 5. you are concerned with calculating H O . [I G ] = 2 0 3 y O x 8. the other is in a coordinate system aligned nicely with the disk.20 0 0 0 14.50(b) and that the disk is spinning about the z axis at the constant ˙ rate of ωk.58 Find the z component of H = [I G ]ω for ˆ ω = 2 rad/sk and h [I G ] = 10.99.4 Mechanics using [I cm ] and [I O ] 8.61 What is the angular momentum H for: a) A uniform sphere rotating with angular velocity ω (radius R. and √ ˆ ˆ ω = ω(ı + )/ 2? 8.55 Find the angle between ω and H = ˆ [I G ]ω. The hoop spins about the x-axis at 1 rad/s. Evaluate all components of the moment of inertia matrix [I O ] of the solar panel.blue.rp14) ˆ ˆ 8. z w θ x O θ y problem 8.62: (Filename:pfigure. In the fixed x. For the most ˙ part.4 0 [I G ] = lbm ft2 .53: (Filename:pfigure4.63 Two thin rods.4. What are H O and H O ? (In this problem there are two ways to use [I]: one is in the x yz coordinate system. 8.5 rad/s . z coordinate system with origin at the center of mass (with which the plate is currently aligned) what are the vector components of ω? What are the components of H cm ? .56 Find the projection of H = [I G ]ω in the ˆ ˆ direction of ω.7 kg.64 The thin square plate of mass M and side c is mounted vertically on a vertical shaft which rotates with angular speed ω0 . and using the equivalent dumbbell (with the definition of ˙ H O ).blue. Every point on the disk moves in circles about the same axis at the same number of radians per second. mass m)? b) A rod shown in the figure rotating about its center of mass with these angular ˆ ˆ velocities: ω = ωı . There is no gravity. ∗ z problem 8. ordinate system which has its origin O at the center of the hoop. But.5 m The two side panels are inclined at 30◦ with the plane of the middle panel. ω = ω . The shaft is parallel to the x-axis. you should keep in mind that the purpose of the calculation might be to calculate reaction forces and moments.Problems for Chapter 8 h = 400 mm and the horizontal plate has depth d = 300 mm.50. Determine the initial angular acceleration α and initial reactions at point B. which is supported by a ball-and-socket joint at A and a journal bearing ˆ ˆ at B. You will consider a disc spinning about various axes through the center of the disc O. Find the moment of inertia matrix [I O ]. y.56 0 −2. 8 0 0 8 0 0 0 16 0 0 16 −4 0 0 8 0 −4 8 67 y mass = m L a) [I G ] = ∗ kg m2 . The total mass of the structure is m = 0. length = 1 and width w = 0.2 kg.00 8.rp15) ˙ 8. z B d /4 M /2 d m /4 x α A y (Filename:pfigure. Iz z is the moment of inertia of the object about the skewer axis. The rod is welded to a shaft that is along the y axis. Neglect gravity.g.) e) Find the reaction force at the ball and socket joint.67 From discrete to continuous. find the rotation rate ω in terms of φ. n. c) Now. L . a person who got rigor mortis and whose ankles went totally limp at the same time?. a simple pendulum?. Mtotal is the total mass of the body.3Dpend) A 8.68 c) Can you verify the formula (a simple pendulum and a swinging stick are special cases)? CONTENTS 8. ˙ a) Calculate H O and H O . a) Find the rate of rotation of the system as a function of g. calculating H .] ∗ 6 b) Set n = 1 in the expression derived in (a) and check the value with the rate of rotation of a conical pendulum of mass m and length . The shaft is spinning at a constant rate ω driven by a motor at A (Not shown).. d is the perpendicular distance from the axis to the center of mass of the object. and then calculating ˙ H from ω × H .66 A hanging rod going in circles. ˙ b) Using H O and angular momentum balance. Two point masses are connected by a rigid rod of negligible mass √ and length 2 2L. How long does it take to make one revolution? ∗ c) If another stick rotates twice as fast while making the same angle φ. ∗ c) What is the rate of change of angular momentum of the dumbbell at the instant shown? Calculate the rate of change of angular momentum two different ways: (i) by adding up the contribution of both of the masses to the rate of change of angular momentum ˙ r i × (m i a i )) and (i.65 If you take any rigid body (with weight ). g is the gravitational constant. Does this rate of rotation make sense to you? [Hint: How does it compare with the rate of rotation for the uniform stick of Problem 8.e. ∗ O ball & socket joint ˆ k ˆ ı ˆ φ m ω problem 8. φ is the amount of rotation about the axis measuring zero when the center of mass is directly above the z -axis.64: (Filename:pfigure. e. g. a box tipping on one edge?.blue.67: (Filename:pfigure. a broom upside down on your hand?. C= Mtotal gd sin γ /Iz z .65: hinge. H = (ii) by calculating I x y and I yz . Estimate the period with which it will swing back and forth. a) What is the moment of inertia matrix about point O for the dumbbell at the instant shown? ∗ b) What is the angular momentum of the dumbbell at the instant shown about point O. and m. the system is shown in the figure.x' Side View g y y' z' z problem 8. hold the axle with hinges that cause no moment about the axle. n equal beads of mass m/n each are glued to a rigid massless rod at equal distances l = L/n from each other. At the time of interest the dumbbell is in the yz-plane. The rod swings around the vertical axis maintaining angle φ. A uniform rod with length and mass m hangs from a well greased ball and socket joint at O. (a) (b) γ γ d 3-D pendulum on a (Filename:pfigure4. Is this force parallel to the stick? /4 m/4 φ m/4 ˆ k ˆ ı ˆ ω m/4 m/4 problem 8. what must be its length compared to ? d) Is there a simple way to describe what two uniform sticks of different length and different cone angle φ both have in common if their rate of rotation ω is the same? (If you accurately draw two such sticks. Use reasonable magnitudes of any quantities you need. and φ for constant rate circular motion.. but you are to consider the general problem with n masses. = −C sin θ. you should see the relation. For n = 4.) b) A door with frictionless hinges is misaligned by half an inch (the top hinge is half an inch away from the vertical line above the other hinge).66?] c M z c z' x y γ γ z y y' d ω0 problem 8. a door?. and hold it so that it cannot slide along the axle.2) x. [Hint: You may need these series sums: 1 + 2 + 3 + · · · + n = n(n+1) and 12 + 22 + 2 32 + · · · + n 2 = n(n+1)(2n+1) . . i.s94h6p4) 8. It swings in circles at constant rate ω = ωk while the rod makes an angle φ with the vertical (so it sweeps a cone). skewer it with an axle.66: (Filename:pfigure. put n → ∞ in the expression obtained in part (a). it will swing like a pendulum.. when the center of mass is in a plane containing the z -axis and the gravity vector.s94h6p3) . a bicycler stopped at a red light and stuck in her new toe clips?..35. That is. or = +C sin φ. γ is the angle of tip of the axis from the vertical. a) How many examples of such pendula can you think of and what would be the values of the constants? (e. it will obey the equations: ¨ θ ¨ φ where θ is the amount of rotation about the axis measuring zero when the center of mass is directly below the z -axis.68 Spinning dumbbell. θ 8. At the instant shown the plate and the y z axis attached to the plate are in the Y Z plane.68: Crooked spinning dumb(Filename:pg84.70 A uniform rectangular plate spins about a fixed axis. provoked by a teensy disturbance.72: (Filename:pfigure. Then use moment balance for this system.86.69: (Filename:pfigure.36. m.] ∗ O I yy O Izz cm I yy O Izy 2 1 (0.blue. at the instant shown. d. sin(β) = h/ (h 2 + b2 ) a) Compute the angular momentum of the plate about G (with respect to the body axis Gx yz shown). and width b spins about its diagonal at a constant angular speed ω. 2 mh 2 /6. . mb2 /2. B problem 8. 0+ (i. e y .3) z O x L h R y problem 8.Moving "Body" Frame ˆ ˆ ˆ (base vectors. M y . The plate lies in the Y Z plane at the instant shown. e z ) problem 8.s95q8) 8. (ii) By drawing free body diagrams of the masses alone and calculating the forces on the masses. (Filename:pfigure. and k (the base vectors aligned with the fixed X Y Z axes). The plate is initially vertical and then. ∗ d) In the case when h = b = 1 m what is the vertical reaction at C (still neglecting gravity).1) z m 45 weld O o ω y r cm O Ix x = = = = = = m x 8. 3 3 m 2 (b + h 2 /3). . b) Ignoring gravity determine the net torque (moment) that the supports impose on the body at the instant shown. as shown in the figure. Assume that the mass per unit length of the pipe is ρ. e x . To select properly the tubing dimensions it is necessary to know the forces and moments at the critical section AA. ı . The frame moves as shown. ı .71: (Filename:pfigure4. . height h.e.72 A thin triangular plate of base b. At t = 0 the plate is at rest. b) At t = 0+ .73 A welded structural tubing framework is proposed to support a heavy emergency searchlight. (Express your solution using base vectors and components associated with the inertial X Y Z frame. the frame has angular velocity ωo and angular acceleration αo .Problems for Chapter 8 d) What is the total force and moment (at the CM of the dumbbell) required to keep this motion going? Do the calculation in the following ways: (i) Using angular momentum balance (about any point of your choice) for the shaft dumbbell system. a) Determine the magnitude and direction of the dynamic moments (bending and twisting) and forces (axial and shear) at section AA of the frame if. what Looking down x-axis z 45o y a) what is the plate’s angular velocity as it passes through the downward vertical position? b) Compute the bearing reactions when the plate swings through the downward vertical position.69 A uniform rectangular plate of mass m. The plate is supported by frictionless bearings at O and C. A bell. Then use action and reaction and draw a free body diagram of the remaining (massless) part of the shaft-dumbbell system. The bearings are frictionless. The constant torque M y is applied about the Y axis.71 A uniform semi-circular disk of radius r = 0. Given. m. mh 2 /18.) c) Calculate the reactions at O or C for the configuration shown. and M y .blue. ˆ ı Y ω ω problem 8. when the framework is moving.1) 8. Neglect gravity. falls . The shaft is tipped very slightly from a vertical upright position.2 m and mass m = 1. d. b. what is the reaction at B? ∗ b g Z D c plate mass = m z' c A X x' problem 8. Or you could use this result from the change of coordinates formulae in lincm cm ear algebra: IY Y = (cos2 β)I y y + cm (sin2 β)Iz z to get an answer in terms of c. height h. The plate is held by bearings at the corners of the plate at A and B.Fixed Frame ˆ ˆ ˆ (base vectors. k) Gxyz . −mbh/4. and mass m is attached to a massless shaft which passes through the bearings L and R. just after the start). ∗ 69 Z b/2 G O X x β ˆ k z b/2 C h/2 h/2 y 8.rpi) OXYZ . (Note: cos(β) = b/ (h 2 + b2 ). cm and IY Y cm [You may use IY Y in your answer. a) At t = is the acceleration of point D? Give your answer in terms of any or all of ˆ ˆ ˆ c. h). Find the dynamic reactions on the bearings supporting the shaft when the disk is in the vertical plane (x z) and below the shaft.70: O β B d d y' My Y 8.. pivoting around the shaft.5 kg is attached to a shaft along its straight edge. The assembly is rotating about the z axis through ˆ O at the fixed rate ω = ωk which happens to be just right to make the plate horizontal and the shaft vertical. The upper end of the shaft is connected to a ball and socket joint at O.74 A uniform cube with side 0.73: (Filename:pfigure. The normal to the disk makes an angle β = 60o with the axle. A solid box B with dimensions a. The other end is held by two strings (or pin jointed rods) that are connected to a rigid cross bar. a) What is the x component of the angular momentum of the disk about point G the center of mass of the disk? b) What is the x component of the rate of change of angular momentum of the disk about G? c) At the given instant.79 A uniform disk of mass m = 2 kg and radius r = 0. what is the total force and moment (about G) required to keep the disk spinning at constant rate? x tˆ uniform disk of mass = m radius = r ˆ n z A G y A x E problem 8. At the instant shown in the figure. a) What is ωB ? ˙ b) What is ωB ? c) What is H B /cm ? ˙ d) What is H B /cm ? e) What is r B/A × F B (where F B is the reaction force on B at B)? A dimensional vector is desired.2 m is mounted rigidly on a massless axle.blue.75: (Filename:pfigure. as shown. b. the face ABC D is in the x y-plane.rp3) ˆ k ω y β ˆ ˆ ı B B problem 8.77: (Filename:s97f2) 1m A L A ω0 α0 ω y b z problem 8. a) What is ω in terms of m.75 A uniform square plate of mass m and side length b is welded to a narrow shaft at one corner.25 m is glued to a pipe along the diagonal of one of its faces ABC D. The cross bar is welded to the main shaft. ω z A round disk spins (Filename:pg92. the angular momentum vector changes. and k . ∗ c) As the disk rotates (with the axle). Find the reaction force at A.1) 8. or both? ∗ 8.77 Uniform rod attached to a shaft.s94q6p1) problem 8. a) Find the angular momentum H O of the disk about point O. A uni√ form 1 kg. and g? b) Is the reaction at O on the rod-plate assembly in the direction of the line from G to O? (Why?) a (Filename:pfigure.76: a 8.78 A round disk spins crookedly on a shaft. Neglect gravity. The shaft rotates at the constant rate ω. what would be ωB immediately afterwards? 8. f) If the hinges were suddenly cut off when B was in the configuration ˙ shown. The normal to the plane of the disk forms an angle β with the axle. a. ∗ b) Draw the angular momentum vector indicating its magnitude and direction. .78: crookedly on a shaft. It is supported with ball and socket joints at opposite diagonal corners at A and B. The gravitational constant is g. A thin homogeneous disk of mass m and radius r is welded to the horizontal axle AB. At the instant of interest. 2 m rod is attached to a shaft which spins at a constant rate of 1 rad/s.88. c) Suppose that the entire mass of the cube is concentrated at its center of mass.2) A 2a a = 0. b) Find the position of the center of mass of the cube and compute its acceleration a cm .76 A spinning box.74: D C B H G F (Filename:pfigure4.2. What is the angular momentum of the cube about point A? 8. The axle rotates at a constant rate ω = 60 rpm. The shaft is perpendicular to the plate and has length .s94f1p2) 8.70 L CONTENTS O ˆ k ˆ ˆ ı A C m = 1kg l= 2m B o 45 45 string o L b G 45o E x D problem 8. a) Find the velocity and acceleration of point F on the cube. and 2a (where a = . One end of the rod is connected to the shaft by a balland-socket joint. . . The pipe rotates about the y-axis at a constant rate ω = 30 rpm.1 m) and mass = 1 kg. Does it change in magnitude or direction. Ignore gravity.1m problem 8. It is spinning at constant rate of (50/π ) rev/sec and at the instant of interest is aligned with the ˆ ˆ ˆ fixed axis ı . A massless wire.25 m m=2 kg β B x ˆ n β ω z 1m 0. The uniform disk of mass m and radius r spins about the z axis at the conˆ stant rate ω. At the moment of interest the masses are all in the x y plane as shown.85: 8. 8. A rigid uniform disk of mass m and radius R is mounted to a shaft with a hinge. b) What is the reaction force at the hinge at O? 71 8.s94q7p1) problem 8. m. Is the system statically balanced? b) Find the rate of change of angu˙ ˙ lar momentum H /A .5 kg.5m B 1m ˆ ı ˆ z ω C 1m ˆ k problem 8. c) Find the angular momentum about point A. β and any base vectors you need.05m 0. At the instant shown the hinge axis is in the x direction. perpendicular to the shaft. a) Find the net reaction force (the total of all the forces) on the shaft. from the shaft at A to the edge of the disk at B keeps the normal to the plate (the z direction) at an angle φ with the shaft. Assume frictionless bearings and ignore gravity. A uniform thin circular disk with mass m = 2 kg and radius R = 0. Three equal masses (1 kg) are attached to the ends of the bars. Mass m 1 = 1 kg and mass m 2 = 1.82: O Front View (xz-plane) problem 8. only the direction is asked for. H /A .p3. using H /A = i r i/A × m i a i . φ and ω. An apparatus not shown keeps the shaft rotating around the z axis at a constant ω. The shaft spins at a constant speed ω = 3 rad/s.79: z E problem 8.1m 8. Where should what masses be added to make the system shown dynamically balanced? [The solution is not unique. ω.05m D m1 A (Filename:p.3) 8.3) 8.2m B m1 0. a) Find the rate of change of angular momentum of the disk about its center of mass.86 Dynamic balance of a system of particles.Problems for Chapter 8 d) Find the direction of the net torque on the system at the instant shown. the three masses are in the x z-plane.82 A disoriented disk rotating with a shaft. Use this intermediate result to find the rate of change of an˙ ˙ gular momentum H /A . r .81: (Filename:s97p3.81 Spinning crooked disk. (Filename:summer95p2.5 Dynamic balance ˆ ˆ 8. ∗ tˆ x uniform disk of mass = m radius = r ˆ ˆ n=k A O y 30o motor A D R=0. What is the total force and moment (relative to O) needed to keep this disk spinning at this rate? Answer in terms of m. At the instant shown. the disk is slanted at 30◦ with the x y-plane. [I G ] = 0 −3 10 0. At the instant shown. ∗ c) Find the total force and moment required at the supports to keep the motion going. d) What is the value of the resultant unbalanced torque on the shaft? Propose a design to balance the system by adding an appropriate mass in an appropriate location. ∗ b) Find the rate of change of angular momentum about point A. Three masses are mounted on a massless shaft AC with the help of massless rigid rods.s94h7p2) .86: (Filename:pfigure.80: motor (Filename:pfigure.85 Three equal length (1 m) massless bars are welded to the rigid shaft AB which spins ˆ at constant rate ω = ωı .83 Is ω = (2ı + 5k ) rad/s an eigenvector of 8 0 0 0 16 −4 kg m2 ? [I G ] = 0 −4 8 8.05m ˆ ˆ ı m2 ˆ k .05m C .84 Find the eigenvalues and eigenvectors of 5 0 0 0 5 −3 kg m2 .80 A uniform disk spins at constant angular velocity. Note. Ignore gravity. Its normal n is inclined from the z axis by the angle β and is in the x z plane at the moment of interest.f96. ∗ a) What is the tension in the wire? Answer in terms of some or all of R. The shaft rotates with a constant angular velocity ˆ ω = 5 rad/sk .25 m is mounted on a massless shaft as shown in the figure.1m problem 8. using H /A = ω × H /A and verify the result obtained in (b).s94h8p1) . You can answer this question based on parts (b) and (c).] tˆ x uniform disk of mass = m radius = r ˆ ˆ n=k z z' φ ω B A y' A O y ˆ β R φ y D A 120o 120o 15o C O 120o B ˆ ı B x ˆ n β ω z x O Front View (xz-plane) problem 8.3) Length of bars is 1 m Each mass is 1 kg (Filename:pfigure. . That is. at point B. the base of the triangle is parallel to the axis of rotation.0 m 0. b) Axis orthogonal to uniform rectangular plate and through its centroid. and d for which this block is dynamically balanced for rotation about an axis through a pair of corners? Why? 1. c and d.87 For what orientation of the rod φ is the system shown dynamically balanced? 8. and k .89: ω B 8. The total mass of all the objects is Mtot .] b) The General Result: Is M Az necessarily equal to zero? Prove your result or find a counter-example. ∗ CONTENTS d) Are there any values of b. b. But. .3) problem 8. it is moving as if it were a rigid body. all in a common plane with the axis. The center of mass G is on the ˆ ˆ ˆ axis of rotation.87: (Filename:ch4. This problem concerns a system rigidly attached to a massless rod spinning about a fixed axis at a fixed rate. c. The axis is coplanar with the triangle. d) Axis orthogonal to uniform but totally irregular shaped plate and through its centroid. Bx . The FBD of the system is shown in the figure. M Az . c.5 m O weld ˆ ı problem 8.s94h7p3) B φ Side View A φ problem 8. h) Axis with two equal masses attached as shown.f96. m. f) Axis through the center of a uniform sphere. The angle φ is arbitrary. Consider each of the objects shown in turn (so to speak). the rod is constrained from moving in the z-direction. each spinning about the z axis which is horizontal. In each case the center of mass is half way between the bearings (not shown). Gravity is ˆ in the − direction.72 8. Answer in terms of some or all of ω.89 An equilateral uniform triangular plate with sides of length plate spins at constant rate ω about an axis through its center of mass. whether or not the system is a rigid body. ˆ ˆ ˆ d. The position of a particle of the system relative to the axis is R . When φ = 0◦ . c) What is the sum of all the moments about point G of all the reaction forces at A and B? (Filename:Mikef91p3) . The ı . .c) ˆ k ˆ 8.92 Dynamic Balance. a) The general case: Imagine that you ˙ ˙ ˆ ˙ ˆ ˙ ˆ know H O = H0x ı + H0y + H0z k ˙ ˙ ˆ ˙ ˆ ˙ ˆ and also L = L x ı + L y + L z k . g) Axis through the diagonal of a uniform cube. For what angle φ is the plate dynamically balanced? Mz Ay x d Ax 1m y problem 8. What is the net force and moment required to maintain rotation at a constant rate? 30o ω side view 30o end view 8.25 m 0. with φ not equal to 30◦ . Referring to the free body diagram. Every point in the system is moving in circles about the same axis at the same number of radians per second.90: By ω c) Axis orthogonal to uniform rectangular plate and not through its centroid. The magnitude of the angular velocity is ω.87d.7. the body rotates about the axis shown. How would you find the six unknown reactions A x .p2.91 A uniform block spins at constant angular velocity. i) Axis with three equal masses attached as shown. A y . Point O is at the center of mass of the system and is at a distance d from the rotation axis and is on the x-axis at the moment of interest. If d = 0 is it true that all reaction forces are equal to zero? Prove your result or find a counter example. j) An arbitrary rigid body which has center of mass on the axis and two of its moment-of-inertia eigenvectors perpendicular to the axis. problem 8.5 m φ P Q y x c d H b 0. Assume there is no gravity. e) Axis along the diagonal of a uniform rhombus.blue. B y . In all cases. ∗ c) Static Balance:. a) Find v H . respectively. a) Axis along centerline of a uniform rectangular plate.91: (Filename:p.88 A uniform square plate with 1 m sides and 1 kg mass is mounted (as shown) with a shaft that is in the plane of the square but not parallel to the sides.88: (Filename:pfigure.1) 8.90 An abstract problem. (no picture) 1m Bz Bx A (Filename:pfigure. ı . Find the net force and moment required to maintain rotation of the plate at a constant rate for any φ. the rod is free to slide in the z-direction. b) Find a H . The position relative to some point C is r C . Give a brief explanation of your answer (enough to distinguish an informed answer from a guess). At the point A. Bz ? [Reduce the problem to one of solving six equations in six unknowns. and k directions are parallel to the sides with lengthes b. State whether it is possible to spin the object at constant rate with ˆ constant bearing reactions of mg /2 and no other applied torques (‘yes’) or might not be (‘no’). there are six scalar unknown reactions in this 3 dimensional problem. Where should you place what weights in order to dynamically balance the tire? Assume that a car tire can be modeled as a uniform disk of radius one foot and mass 20 lbm. They figure out where to put the weights by using a little machine that measures the wheel wobble. Assume the tire has been mounted crooked by one degree (the plane of the disk makes an 89◦ angle with the car axle).] . [one approach: guess a good location to put two masses and then figure out how big they have to be to make the tire dynamically balanced.92: 120o φ (Filename:s97p3. There are many approaches to solving this problem and there are also many correct solutions. But you don’t need such a machine for this problem because you have been told where all the mass is located.93: (Filename:pfigure. You should find any correct solution.2) 8.93 Dynamic tire balancing.Problems for Chapter 8 θ (a) ω (b) problem 8. This crooked wheel is statically balanced since its center of mass is on the axis of rotation. But it is dynamically unbalanced since it wobbles and will wobble the car.s94h7p5) 73 (c) (d) (e) (f) (g) m R (h) R m 120o (i) problem 8. Car shops put little weights on unbalanced wheels to balance them. Also show the wheel and the pebble at some time in this interval. Using a clear well labeled drawing (use a compass and ruler or a computer drawing program).blue. Also show the outline of the wheel and the pebble itself at some intermediate time of interest. and vo .7 A stone in a wheel. find the acceleration of the particle in terms of ˙ b. using a sketch and/or words.1 A particle moves along the two paths (1) and (2) as shown.1) (1) 5 y r θ1/2 = b 9. In particular. and θ . you are to try to get a clear answer by looking at computer generated plots. The pebble is directly below the center of the wheel at time t = 0. 9. (The foot is simultaneously sliding out on the floor). . Here are some steps to follow: a) Assuming the wheel has radius Rw and the pebble is a distance R p from the center (not necessarily equal to Rw ).8 A uniform disk of radius r rolls at a constant rate without slip. while end “A” is moved along the ˙ ground at a constant velocity vo . You should pick the parameters that make your case for an answer the strongest. θ .62. At time t = 0 it picks up a stone the road.74 CONTENTS Problems for Chapter 9 General planar motion of particles and rigid bodies Carefully define. [Use any software and computer that pleases you..1. ∗ b) Find the x and y coordinates of the path as functions of b and r or b and θ . The top of the ladder is sliding down the wall at one foot per second. 9. 9. Is φ positive or ¨ negative? Is φ positive or negative? B 0 x -5 0 5 10 15 h φ x A vo (2) 4 2 0 -2 -4 y problem 9. What is the force required to hold the disk in place when the mass is above the center of the disk? 9. write a program to make a plot of the path of the pebble as the wheel makes a little more than one revolution. θ . h.] c) Change whatever you need to change to make a good plot of the pebble’s path for a small amount of time as the pebble approaches and leaves the road. You want to know the direction of the rock’s motion just before and after it next hits the ground.5 The slender rod AB rests against the step of height h. ∗ v d θ x problem 9.6 A ten foot ladder is leaning between a floor and a wall.1 Kinematics of planar rigid-body motion 9.you list the conditions. The stone is stuck in the edge of the wheel. d. Although you could address this question analytically. and θ.. The wheel spins at constant clockwise rate ω.1: (Filename:polarplot0) 9. A round wheel rolls to the right.axes have the same scale so that the path of the pebble and the outline of the wheel will not be distorted.2 For the particle path (1) in problem 9. e) How does your computer output buttress your claim that the pebble approaches and leaves the ground at the angles you claim? f) Think of something about the pebble in the wheel that was not explicitly asked in this problem and explain it using the computer. The x-axis is on the ground and x(t = 0) = 0. ˙ and θ.1. When the ladder makes a 45 degree angle with the vertical what is the speed of the midpoint of the ladder? 9. Find φ and ˙ ¨ φ in terms of x. The wheel rolls without slipping. ˙ a) Calculate θ in terms of V. • The stone approaches the ground at angle x (you name it). You may make more than one plot. and reference frames you use.4: (Filename:pfigure. show that x(t) = ωt Rw − R p sin(ωt) y(t) = Rw − R p cos(ωt) b) Using this relation.3 For the particle path (2) in in problem 9. a) In each case. you are to plot the pebble’s path for a small interval of time near when the stone next touches the ground.) 9.4 A body moves with constant velocity V in a straight line parallel to and at a distance d from the x-axis.and y. ˆ b) Calculate the eθ component of acceleration.98.5: (Filename:pfigure. A small ball of mass m is attached to the outside edge of the disk. coordinate systems. But make sure your x. d) In this configuration the pebble moves a very small distance in a small time so your axes need to be scaled. any variables. θ.blue. • When the stone approaches the ground its motion is perpendicular to the ground. Express your answers using any convenient coordinate system (just make sure its orientation has been clearly defined). determine the velocity of ˙ the particle in terms of b.1) x θ = br -5 0 5 problem 9. and θ . Here are some candidate answers: • When the stone approaches the ground its motion is tangent to the ground. and/or hand calculation and/or a drawing. find the acceleration of the particle in terms of b. • The stone approaches the ground at various angles depending on the following conditions(. Disc D of radius r rotates about point C at constant rate ω2 measured with respect to the arm OC. 10.] Draw the paths on the computer. b) Velocity of points.blue. What are the absolute velocity and acceleration of point P on the disc. and a point that is initially half way between the former two points. You can find the velocity by differentiating the position vector or by using relative motion formulas appropriately. What is the velocity of point A in the middle of the wheels shown? concentric wheels L A 2 cm 2 cm problem 9.13: O A P b ω c 9.11 Questions (a) .50. Define appropriate dimensions for the problem.0 m ˆ ˆ ı θ = 45o B (Filename:pfigure4. a) Particle paths.10 The concentric wheels are welded to each other and roll without slip on the rack and stationary support.s94h11p2) 9.1) disk D r C P ω2 9.9: (Filename:pfigure. The point A on the ladder moves parallel to the wall. respectively.9 Rolling at constant rate. ˆ B problem 9. the COR changes with time. P is attached to the outside edge of the little cylinder. at a given instant. . More than one choice is possible. Note that the acceleration of the points is parallel to the line connecting the points to the center of the disk. b) Find the acceleration of the center of mass of the plate. of course). Note that the velocity of the points is perpendicular to the line connecting the points to the ground contact. Accurately plot the paths of three points: the center of the disk C. both have velocities that are consistent with the ladder rotating about some special point.10: (Filename:pfigure. 4 2 4 Draw the velocity vector (by hand) on your plot. a point on the outer edge that is initially on the ground.rp7) 9. The rack moves to the right at vr = 1 m/s. Draw the disk at its position after one quarter revolution. b) Pick another suitable moving frame and redo the problem. A = 1.3.1) A D G P C problem 9.118.28. Draw the direction accurately and draw the lengths of the vectors in proportion to their magnitude. Find the velocity of the points at a few instants in the motion: after 1 . a) What is the speed (magnitude of velocity) of point c? b) What is the speed of point P? c) What is the magnitude of the acceleration of point c? d) What is the magnitude of the acceleration of point P? e) What is the radius of curvature of the path of the particle P at the point of interest? f) In the special case of A = 2b what is the curve which particle P traces (for all time)? Sketch the path.12 See also problems 10.3. The point B moves parallel to the floor. Arm OC rotates with constant rate ω1 . b) As the ladder moves. First find a relation between ω and vC .39. c) Acceleration of points. It rolls 1 1 revolutions over the 4 time of interest. that other point C must be in the direction perpendicular to the motion of A.31.11: A little cylinder (with outer radius b and center at point c) rolls without slipping inside a bigger fixed cylinder (with inner radius A and center at point O). The absolute angular velocity of the little cylinder ω is constant. At the instant of interest. At the instant shown.rp6) problem 9.Problems for Chapter 9 9. What is the set of points on the plane that are the COR’s for the ladder as it falls from straight up to lying on the floor? no slip rack vr ω1 O θ stationary support problem 9.13 A uniform rigid rod AB of length = 1 m rotates at a constant angular speed ω about an unknown fixed point. P is on the line between O and c. a) Find the COR for the ladder when it is at some given position (and moving.) ∗ a) Pick a suitable moving frame and do the problem. a) Find the angular velocity of the rod. A round disk rolls on the ground at constant rate. Answer the questions in terms 9. if a point is A is ‘going in circles’ about another point C. make sure x and y scales are the same (In MATLAB.12: (Filename:Mikes93p1) 9. 1 . Yet. 3 .14: C ˆ ı (Filename:pfigure4. the velocities of the two corner points A and D are ˆ ˆ ˆ v A = −2 ft/s(ı + ) and v D = −(2 ft/s)ı . problem 9.35. (Filename:pfigure. Then note that the position of a point is the position of the center plus the position of the point relative to the center. and 11. b) Find the center of rotation of the rod. v P and a P ? (To do this problem will require defining a moving frame of reference. [Hint: Write a parametric equation for the position of the points. the center of rotation (COR). use the axis(’equal’) command for the same scaling). Question (f) is closely related. 75 the velocities of the two ends of the rod are ˆ ˆ v A = −1 m/sı and v B = 1 m/s . Do the same as above but for acceleration. At the instant shown.14 A square plate ABC D rotates at a constant angular speed about an unknown point in its plane. a) Find the center of rotation of the plate. Hint. See also problem 9.(e) refer to the cylinders in the configuration shown figure.blue. and 1 revolution. of the given quantities (and any other quantities you define if needed).15 Consider the motion of a rigid ladder which can slide on a wall and on the floor as shown in the figure. Make sure the answers are the same. 2 Unconstrained Dynamics of 2-D rigid-body planar motion 9. A constant force of F = 1000 Ni is applied to the peg for . Its center of ˙ˆ ˙ ˆ mass translates with velocity v = x ı + y in the x y-plane.25 A uniform thin flat disc is floating in space. This disk has radius of 1 m and mass of 10 kg. Treat this problem as two-dimensional. and the radius of curvature ρ. A T a) What is the velocity of the center of mass of the center of the disk after the force is applied? b) Assuming that the idealizations named in the problem statement are exact is your answer to (a) exact or approximate? c) What is the angular velocity of the disk after the force is applied? d) Assuming that the idealizations named in the problem statement are exact is your answer to (c) exact or approximate? 9.22: (Filename:pfigure.24 A uniform disk.17 Find expressions for et .16 A car driver on a very boring highway is carefully monitoring her speed.1) a 30o b B 9. the center of mass? ∗ b) What is the angular acceleration of the stick? ∗ c) What is the acceleration of the point A? ∗ 9.82. and f) problem 5.1) problem 9. The force F = F ı with F > 0 is suddenly applied at point A. a) What is the acceleration of the center of the disc? ∗ b) What is the angular acceleration of the disk? ∗ d) (relatively harder) What additional force would have to be applied to point B to make point B’s acceleration zero? ∗ .96.20 The uniform rectangle of width a = 1 m.1) 9. an . No gravity. For various positions of the particle on the path. At the moment in question.84.) [Hint: You have been given some useless information. at .20: (Filename:pfigure. 9. c) problem 10. A tension T is suddenly applied at A.] CONTENTS A ˆ F = Fı wall ladder y no applied forces x C C ˆ ˆ ı B floor problem 9. At what points on the path are er and ˆ ˆ ˆ en parallel(or eθ and et parallel)? x 2 ).blue.81.19 A uniform disc of mass m and radius r ˆ rotates with angular velocity ωk . a) Find the path of the center of mass.95.22 Force on a stick in space.21: (Filename:pfigure. e) problem 5.83. A force F is applied to it a distance d from the center in the y direction. with one end resting on a frictionless surface and the other held by someone’s hand. A massless peg is attached to a point on its perimeter.blue. en . at rest on a frictionless surface.85. a) What is the acceleration of point C.5. B 9. A uniform thin stick with length and mass m is. The questions below concern the instant after the force F is applied. is sitting motionless on the frictionless x y plane.21 The vertical pole AB of mass m and length is initially. displaced slightly from the vertical. parallel to the y axis and has no velocity and no angular ˆ velocity. point C is at xC = 3 m and yC = 2 m. Find the acceleration of any point on the body that you choose.23 A uniform thin rod of length and mass m stands vertically. at any position (or time) on the given particle paths for a) problem 5. It has radius R and mass m. ∗ b) problem 5. with mass center labeled as point G. b) Find the force of the floor on the end of the rod just before the rod is horizontal.15: (Filename:pfigure.18 A particle travels at non-constant speed on an elliptical path given by y 2 = b2 (1 − Carefully sketch the ellipse for particular values of a and b. a2 problem 9. sketch the position vector ˆ r (t). and the path coordinate basis vectors en ˆ ˆ and et . the polar coordinate basis vectors er and ˆ ˆ eθ . (Mark it.0001 s (one ten-thousandth of a second). and its speed increases uniformly from 50 mph to 60 mph.101. What is ¨ ¨ xcm ? What is θ AB ? What is x B ? Gravity may ¨ be ignored. No forces are applied during the release. What is the acceleration of the of the car half-way through this one hour period. There is gravity. at the instant of interest.s94h10p2) 9. and mass m = 1 kg in the figure is sliding on the x − y plane with no friction. d) problem 5. m y x problem 9. Over a one hour period.blue. What is the total kinetic energy of the disk? 9. 2-D . the car travels on a curve with constant radius of curvature.76 A ˆ is H cm = 5k ( kg·m/s2 ). The linear ˆ ˆ momentum is L = 4ı + 3 ( kg·m/s) and the angular momentum about the center of mass 9. ρ = 25 mi. length b = 2 m. The rod is released from rest. in path coordinates? ˆ ˆ 9. b) the velocity of sphere A.33: 9.2W θ (Filename:pfig2. and b) the output torque of the system assuming there is no power loss in the system. determine at that instant a) the angular acceleration of the bar.34: Positive power out (Filename:pfigure. ∗ b) Is Fout greater or less than the Fin ? (Assume Fin > 0.1) 2 ft B.26: (Filename:pfig2. problem 9. If spring 2 breaks. find: a) the input power to the system.blue.0 ft/s which sends the assembly into free fall. W = F S.28 Two small spheres A and B are connected by a rigid rod of length = 1. If the driven gear rotates at a constant speed ωout . problem 9.1) 9.33 The driving gear in a compound gear train rotates at constant speed ω0 .3. and by a moment M. Determine the angular momentum of the assembly about its mass center at point G immediately after A is hit.rp6) 9.4 Mechanics of contacting bodies: rolling and sliding 9.Problems for Chapter 9 y R d d) the acceleration of sphere A.25: (Filename:pfigure.W x F W = 5 lb v = v0 problem 9. The assembly is hung from a hook. driven gear radius R (Filename:pfig2.31 A point mass ‘m’ is pulled straight up by two strings.28: (Filename:pfigure.rp2) 9.138. and evaluate b) Repeat (a) immediately after the left string is cut.27: 2 B (Filename:pfigure. W = M θ.1) (Filename:pfig2.26 A uniform rectangular metal beam of mass m hangs symmetrically by two strings as shown in the figure. and m y x problem 9.2. are dimensionally correct if S and θ are linear and angular displacements respectively. The two strings pull the mass symmetrically about the vertical axis with constant and equal force t. The driving torque is Min . Draw a free body di˙ ˆ agram of the mass at the instant of interest and evaluate the left hand side of the energy balance equation (P = E˙K ).34 An elaborate frictionless gear box has an input and output roller with Vin = const.102. After the center of mass has fallen two feet.3 Special topics in planar kinematics 9. i. b) the acceleration of point A. determine: a) the angle θ through which the rod has rotated.rp7) 1 A problem 9. 77 ω r g G F = 50 N problem 9.27 A uniform slender bar AB of mass m is suspended from two springs (each of spring constant K ) as shown.31: h Positive power in problem 9. power and energy m problem 9.0 ft and negligible mass.32 In a rack and pinion system.3. Assuming that Vout = 7Vin and the force between the left belt and roller is Fin = 3 lb: a) What is Fout (draw a picture defining the signs of Fin and Fout )?. as shown. At an instant in time t. and c) the acceleration of point B.50. suddenly breaking its contact with the hook and giving it a horizontal velocity v0 = 3.29 Verify that the expressions for work done by a force F. Sphere A is struck. ωin M ωout driving gear radius r 9.32: A. respectively.105.blue.blue. the ˆ position and the velocity of the mass are y(t) and y (t) .30 Calculate the energy stored in a spring using the expression E P = 1 kδ 2 if the spring is 2 compressed by 100 mm and the spring constant is 100 N/m.blue. c) the total kinetic energy of the assembly of spheres A and B and the rod.) Why? ∗ gear box black box (frictionless) T L T Vin Vout 9.2) L 9. a) Draw a free body diagram of the beam F. 9. A 3 2 B The next several problems concern Work.3.. the rack is acted upon by a constant force F = 50 N and has speed v = 2 m/s in the direction of the force. power in = power out. .e. Find the input power to the system.rp8) 9. The disk skids for a while and then is eventually rolling.44 A block of mass M is supported by two rollers (uniform cylinders) each of mass m and radius r .a) 9. They roll without slip on the block and the ground. R0 . a) What is the angular acceleration of the ¨ˆ disk. no slip problem 9. I = 0. You can make a reasonable physical model of this situation with an empty soda can and a piece of paper on a flat table.39: 9.44: (Filename:pfigure. The ring starts at the right end (x = d). 2-D .38 A uniform disk with radius R and mass m has a string wrapped around it.2) 9. a force F is applied vertically to the cord wound around the wheel. g. radius r ) rolls without slip as a mischievous child pulls the tablecloth (mass M) out with acceleration A. Dimensions are as marked. energy is lost to friction.9.s94h11p2.45 Dropped spinning disk.2) (Filename:p.41: ˆ Ro C Ri ˆ ı F g F M problem 9. An inextensible string is wrapped around the hoop and attached to the ceiling. a) What is the acceleration of the center of the spool? ∗ b) What is the horizontal force of the ground on the spool? ∗ c) Clearly describe the subsequent motion of the ring. R. m. 9. find: a) the acceleration of the block. Assume that the film and reel (together) have mass distributed the same as a uniform disk of radius Ri . A uniform disk of radius R and mass m is gently dropped onto a surface and doesn’t bounce. θ0 . c) the reaction at the wheel bases.4) uniform disk.42. what is the maximum speed of the pebble during the motion? When is the force on the pebble from the wheel maximum? Draw a good FBD including the force due to gravity.43.blue. d) the force of the right wheel on the block. b) In the transition from slipping to rolling. ∗ 9. r .37 Falling hoop. m. α Disk = −θ k ? Make any reasonable assumptions you need that are consistent with the figure information and the laws of mechanics. Given F. Which way does it end up rolling at what speed? d) Would your answer to the previous question be different if the ring slipped on the cloth as the cloth was being pulled out? 9. and t. How does the amount lost depend on the coefficient of friction (and other parameters)? How does this loss make or not make sense in the limit as µ → 0 and the dissipation rate → zero? ∗ .42 Again. What is the total kinetic energy of the disc when the center of mass is directly above the center of the disc? problem 9. State your assumptions. ı . and F are the accelerations of points C and B at the instant shown (the start of motion)? B frictionless contact (no friction) problem 9. . and m)? ∗ 9. The hoop is released from rest at the position shown at t = 0.39 If a pebble is stuck to the edge of the wheel in problem 9.p3. When it is ˙ released it is spinning clockwise at the rate θ0 . what is the acceleration of the center of the spool? Is it possible to pull the cord at some angle between horizontal and vertical so that the angular acceleration of the spool or the acceleration of the center of mass is zero? If so.36 A non-uniform disc of mass m and radius r rolls without slip at constant rate.40: (Filename:pfigure. or hub) is idealized as a hoop with mass m and radius R.1) 9. The thin ring (of mass m. The string is pulled with a force F.blue. b) the acceleration of the center of mass of this block/roller system. g.108. Coordinate directions are as marked. tire. Reconsider the spool from problem 9.R problem 9.51.s94h11p5) C Ri ˆ ı F yG m. The lightweight spool is nearly empty but a lead ball with mass m has been placed at its center. Ro .38: F = 1N A 2m x (Filename:pfigure.35 A uniform disc of mass m and radius r rolls without slip at constant rate.37: (Filename:p. A force F is applied in the horizontal direction to the right. find the angle in terms of Ri .1) ˆ Ro roll without slip problem 9. b) Does G move sideways as the hoop falls and unrolls? m x d problem 9. What. What is the total kinetic energy of the disk? CONTENTS P C a) What is the ring’s acceleration as the tablecloth is being withdrawn? b) How far has the tablecloth moved to the right from its starting point x = 0 when the ring rolls off its left-hand end? (Filename:pfigure.43: napkin ring r tablecloth A (Filename:pfigure. and f) the angular acceleration of the wheels. 9.78 9. and F.f96.s96. a) Find yG at a later time t in terms of any or all of m. G is at the center of the hoop. A bicycle rim (no spokes.40 Spool Rolling without Slip and Pulled by a Cord. µ.blue.43 A napkin ring lies on a thick velvet tablecloth. as shown in the figure.40. in terms of ˆ ˆ Ri . A force F is applied in the horizontal direction to the cord wound around the wheel. This time. m. R. The center of mass is located at a distance r from the center 2 of the disc.f.41 A film spool is placed on a very slippery table. e) the acceleration of the wheel centers. The disk rolls without slipping. and M. ∗ b) Find the acceleration of the point A in the figure. a) What is the speed of the center of the disk when the disk is eventually rolling ˙ (answer in terms of g. In this case.5 kg m2 M = 1 kg G Rolling contact. tube. See also problem 9. Spool Rolling without Slip and Pulled by a Cord. 9. It does not slip. Determine: (a) the angular acceleration α of the disk.Problems for Chapter 9 y R µ problem 9. the other a rolling hoop (no slip).47 A rigid hoop with radius R and mass m is rolling without slip so that its center has translational speed vo .48: (Filename:pfigure.52 A uniform cylinder of mass m and radius r rolls down an incline without slip.blue. however. I G massless pulley m 9. I. It does hinge freely about the bar.1) m = 200 kg r = 0.50 Two objects are released on two identical ramps. as shown below. m.3) hoop of mass m g R vo m φ L g L rigid bar R/2 R m φ problem 9.51 The hoop is rolled down an incline that is 30◦ from horizontal. It is let go from rest at t = 0. at least in part. modeled as massless. The gravitational constant is g.46 Disk on a conveyor belt.s94h11p3) (Filename:s97f4) g r 9. B.blue. How long does it take the hoop? [Useful formula: “s = 1 at 2 ”] 2 30o problem 9.1m no rotation y rolling disk Vo x g R r no slip α problem 9.) problem 9. When the hoop hits the bar suddenly it sticks and doesn’t slide. How fast is it moving (i. or are there ties? First try to construct any plausible reasoning.49.s95q10) 79 ˆ g R P d) After the hoop has descended 2 vertical meters (and traveled an appropriate distance down the incline) what is the acceleration of the point on the hoop that is (at that instant) furthest from the incline? ˙ θ ˆ ı g x problem 9. The disk is delivered to a flat hard platform where it slides for a while and ends up rolling. The wheel makes contact with coefficient of friction µ. ∗ a) Block A has the same mass and has center of mass a distance R0 from the ramp. It rolls on massless wheels with frictionless bearings.s94f1p1) 30o 9. It is raced against four other objects (A. Good answers will be based. A uniform metal cylinder with mass of 200 kg is carried on a conveyer belt which moves at V0 = 3 m/s. r. It then hits a narrow bar with height R/2. on careful use of equations of mechanics.45: (Filename:pfigure.51. How big is vo if the hoop just barely rolls over the bar? ∗ 9. (b) the minimum value of the coefficient of (static) friction µ that will insure no slip.49 Spool and mass. A reel of mass M and cm moment of inertia Izz = I rolls without slipping upwards on an incline with slope-angle α.51: (Filename:pfigure.87. no slip belt problem 9. R. The wheel is released from rest at the position shown.e. c) Hollow pipe C has the same mass (MC = M0 ) and the same radius (RC = R0 ). It does not fall over sideways. Can you find a round object which will roll as fast as the block slides? How about a massless cylinder with a point mass in its center? Can you find an object which will go slower than the slowest or faster than the fastest of these objects? What would they be and why? (This problem is harder.2) 9.47: (Filename:pfigure. has a point mass (mass = m) at its perimeter.5) 9. Who wins the races. one at a time. C and D). A wheel.46: (Filename:pfigure. a) At t = 0+ what is the acceleration of the hoop center of mass? b) At t = 0+ what is the acceleration of the point on the hoop that is on the incline? c) At t = 0+ what is the acceleration of the point on the hoop that is furthest from the incline? . A uniform disk with mass M0 and radius R0 is allowed to roll down the frictionless but quite slip-resistant (µ = 1) 30◦ ramp shown. d) Uniform disk D has the same radius (R D = R0 ) but twice the mass (M D = 2M0 ). α. Find the acceleration of point G in terms of some or all of M. by the way.blue. The disk is not rotating when on the belt. b) Uniform disk B has the same mass (M B = M0 ) but twice the radius (R B = 2R0 ). It takes the sliding mass 1 s to reach the bottom of the ramp. problem 9.52: (Filename:pfigure. g and any base vectors you clearly define.50: (Filename:pfigure. are in the same gravity field and have the same distance to travel. It is pulled up by a string attached to mass m as shown. Both have the same mass. One is a sliding block (no friction).48 2-D rolling of an unbalanced wheel.blue.53 Race of rollers.47. what is the speed of the center of mass) when it eventually rolls? ∗ 9.49: M. that the friction be high: µ = ∞) what is the velocity of the point P just before it touches the ground? 9. a) What is the acceleration of the point P just after the wheel is released if µ = 0? b) What is the acceleration of the point P just after the wheel is released if µ = 2? c) Assuming the wheel rolls without slip (no-slip requires. m. 46. Perfect slipping (no friction)..) C problem 9.80 k A x k RC CONTENTS R0 θ C R problem 9. Use the point on the cylinder which touches the disk for the angular momentum balance equation.blue. (You may or may not need to use this hint. The disk rolling in the cylinder is a one-degree-of-freedom system. velocities and accelerations of all points. The sketch should show all variables.53: D (Filename:pfigure.] e) Equation of motion. 9. Write the equations of linear and angular momentum balance for the disk.56: A disk rolls without slip inside a bigger cylinder. At what angle θ does the hoop leave the track and what is its angular velocity ω and the linear velocity v of its center of mass at that instant? If the hoop slides down the track without friction. Here we are interested in the free body diagrams only during collision.141. The springs. d) Kinematics. and 2.2) G K problem 9. 9.3) A B 9. will it leave at a smaller or larger angle θ than if it rolls without slip (as above)? Give a qualitative argument to justify your answer. [Hint: you’ll need to think about the rolling contact in order to do this part. State in words why the forces you choose to show should not be ignored during the collision.1) L R0 problem 9. Write the angular momentum balance equation as a single second order differential equation. b) Calculate the natural frequency of the system for small oscillations. (Filename:h12.55 A uniform cylinder of mass m and radius R rolls back and forth without slipping through small amplitudes (i. Leave as unknown in these equations variables which you do not know.5 Collisions 9. f) Simple pendulum? Does this equation reduce to the equation for a pendulum with a point mass and length equal to the radius of the cylinder. Therefore.152. Draw a free body diagram of the disk. so that it does not rotate. Draw a neat sketch of the disk in the cylinder. HINT: Here is a geometric relationship between angle φ hoop turns through and angle θ subtended by its center when no slipping occurs: φ = [(R1 + R2 )/R2 ]θ. . M. a) What is the period of oscillation in the first case? b) What is the period of oscillation in the second case? c) Momentum balance. are unextended when A is directly over C.57: (Filename:pfigure. Consider two kinds of friction between the roller and the surface it moves on: 1. Assume that no slipping occurs. ignore all forces that are much smaller than the impulsive forces. or why not? g) How many? How many parts can one simple question be divided into? A O R1 ω φ O A θ c R2 (Hint) ˆ n tˆ roller.s94h11p4) a) Sketch.57 A uniform hoop of radius R1 and mass m rolls from rest down a semi-circular track of radius R2 as shown. IG hinge friction problem 9. which act both in compression and tension. b) FBD. For all of the problems below.56 A disk rolls in a cylinder.1) 9. 9.58 The two blocks shown in the figure are identical except that one rests on two springs while the other one sits on two massless wheels. the springs attached at point A on the rim act linearly and the vertical change in the height of point A is negligible). coordinates and dimension used in the problem. Perfect rolling (infinite friction). the disk rolls without slip and s back and forth due to gravity. That is. The angle that the line from the center of the cylinder to the center of the disk makes from the vertical can be used as such a variable. the values of only one coordinate and its derivatives are enough to determine the positions. Find all of the velocities and accelerations needed in the momentum balance equation in terms of this variable and it’s derivative.e. when the disk radius gets arbitrarily small? Why.55: RD (Filename:pfigure.blue.54 A roller of mass M and polar moment of inertia about the center of mass IG is connected to a spring of stiffness K by a frictionless hinge as shown in the figure.blue. a) Determine the differential equation of motion for the cylinder’s center. Draw free body diagrams of each mass as each is struck by a hammer.54: (Filename:pfigure. Problems for Chapter 9 9.62 Two equal masses each of mass m are joined by a massless rigid rod of length . The assembly strikes the edge of a table as shown in the figure, when the center of mass is moving downward with a linear velocity v and the sys˙ tem is rotating with angular velocity φ in the counter-clockwise sense. The impact is ’elastic’. Find the immediate subsequent motion of the system, assuming that no energy is lost during the impact and that there is no gravity. Show that there is an interchange of translational and rotational kinetic energy. 81 9.64 A gymnast of mass m and extended height L is performing on the uneven parallel bars. About the x, y, z axes which pass through her center of mass, her radii of gyration are k x , k y , and k z , respectively. Just before she grasps the top bar, her fully extended body is horizontal and rotating with angular rate ω; her center of mass is then stationary. Neglect any friction between the bar and her hands and assume that she remains rigid throughout the entire stunt. a) What is the gymnast’s rotation rate just after she grasps the bar? State clearly any approximations/assumptions that you make. b) Calculate the linear speed with which her hips (CM) strike the lower bar. State all assumptions/approximations. c) Describe (in words, no equations please) her motion immediately after her hips strike the lower bar if she releases her hands just prior to this impact. m k k problem 9.58: m (Filename:pfig2.1.rp9) 9.59 These problems concerns two colliding masses. In the first case In (a) the smaller mass hits the hanging mass from above at an angle 45◦ with the vertical. In (b) second case the smaller mass hits the hanging mass from below at the same angle. Assuming perfectly elastic impact between the balls, find the velocity of the hanging mass just after the collision. [Note, these problems are not well posed and can only be solved if you make additional modeling assumptions.] ˙ φ v problem 9.62: (Filename:pfigure.blue.137.2) V 45 o L m M 45 (a) o M m V (b) problem 9.59: (Filename:pfig2.2.rp7) 9.63 In the absence of gravity, a thin rod of mass m and length is initially tumbling with constant angular speed ωo , in the counterclockwise direction, while its mass center has constant speed vo , directed as shown below. The end A then makes a perfectly plastic collision with a rigid peg O (via a hook). The velocity of the mass center happens to be perpendicular to the rod just before impact. a) What is the angular speed ω f immediately after impact? b) What is the angular speed 10 seconds after impact? Why? c) What is the loss in energy in the plastic collision? L/2 z ω y cm upper bar L/2 θ lower bar Just as top bar is grasped 9.60 A narrow pole is in the middle of a pond with a 10 m rope tied to it. A frictionless ice skater of mass 50 kg and speed 3 m/s grabs the rope. The rope slowly wraps around the pole. What is the speed of the skater when the rope is 5 m long? (A tricky question.) problem 9.64: (Filename:pfigure.blue.130.2) Before impact O hook A vo 10 m 5m rope pole problem 9.60: (Filename:pfigure.blue.49.1) m = 50 kg v0 = 3m/s G d ωo 9.65 An acrobat modelled as a rigid body. An acrobat is modeled as a uniform rigid mass m of length l. The acrobat falls without rotation in the position shown from height h where she was stationary. She then grabs a bar with a firm but slippery grip. What is h so that after the subsequent motion the acrobat ends up in a stationary handstand? [ Hints: Note what quantities are preserved in what parts of the motion.] ∗ L 9.61 The masses m and 3m are joined by a light-weight bar of length 4 . If point A in the center of the bar strikes fixed point B vertically with velocity V0 , and is not permitted to re˙ bound, find θ of the system immediately after impact. DURING h After impact O Perfectly plastic collision ωf problem 9.63: (Filename:pfigure.blue.78.2) bar BEFORE problem 9.65: (Filename:pfigure.s94h10p4) 4 2 3m A B Neglect gravity problem 9.61: Neglect gravity! (Filename:pfigure.blue.81.1) AFTER m V0 G 9.66 A crude see-saw is built with two supports separated by distance d about which a rigid plank (mass m, length L) can pivot smoothly. The plank is placed symmetrically, so that its 82 center of mass is midway between the supports when the plan is at rest. a) While the left end is resting on the left support, the right end of the plank is lifted to an angle θ and released. At what angular velocity ω1 will the plank strike the right hand support? b) Following the impact, the left end of the plank can pivot purely about the right end if d/L is properly chosen and the right end does not bounce. Find ω2 under these circumstances. • the hands on the bat). Call the magnitude of the impulse of the ball on the bat (and visa versa) Fdt. Use various momentum equations to find the angular velocity of the bat and velocity of the ball just after the collision in terms Fdt and other quantities of above. Use these to find the energy of the system just after collion. Solve for the value of Fdt that conserves energy. As a check you should see if this also predicts that the relative separation speed of the ball and bat (at the impact point) is the same as the relative approach speed (it should be). You now know can calculte vhit in terms of m b , m s , Ms , L , , and vb . Find the maximum of the above expression by varying m s and . Pick numbers for the fixed quantities if you like. CONTENTS • L c.m. θ • • d Just Before Collision ω1 b) Can you explain in words what is wrong with a bat that is too light or too heavy? Just After Collision ω2 problem 9.66: (Filename:pfigure.blue.81.2) c) Which aspects of the model above do you think lead to the biggest errors in predicting what a real ball player should pick for a bat and place on the bat to hit the ball? d) Describe as clearly as possible a different model of a baseball swing that you think would give a more accurate prediction. (You need not do the calculation). 9.67 Baseball bat. In order to convey the ideas without making the calculation to complicated, some of the simplifying assumptions here are highly approximate. Assume that a bat is a uniform rigid stick with length L and mass m s . The motion of the bat is a pivoting about one end held firmly in place with hands that rotate but do not move. The swinging of the bat occurs by the application of a constant torque Ms at the hands over an angle of θ = π/2 until the point of impact with the ball. The ball has mass m b and arrives perpendicular to the bat at an absolute speed vb at a point a distance from the hands. The collision between the bat and the ball is completely elastic. a) To maximize the speed vhit of the hit ball How heavy should a baseball bat be? Where should the ball hit the bat? Here are some hints for one way to approach the problem. • Find the angular velocity of the bat just before collision by drawing a FBD of the bat etc. • Find the total energy of the ball and bat system just before the collision. • Draw a FBD of the ball and of the bat during the collision (with this model there is an impulse at Problems for Chapter 10 83 Problems for Chapter 10 Kinematics using time-varying base vectors Problems for all three sections of this chapter are combined. 10.3 Find the components of (a) v = ˆ ˆ ˆ 3 m/sı + 2 m/s + 4 m/sk in the rotated baˆ ˆ ˆ ˆ sis (ı , , k ) and (b) a = −0.5 m/s2 + x y-plane. At time t = 0, the engraved line is aligned with the x-axis, the bug is at the origin and headed towards the positive x direction. a) What is the x component of the bug’s velocity at t = π s? b) What is the y component of the bug’s velocity at t = π seconds? c) What is the y component of the bug’s acceleration at t = π? 10.1 Polar coordinates and path coordinates 10.2 Rotating frames reference ˆ ˆ ˆ ˆ 3.0 m/s2 k in the standard basis (ı , , k ). (y and y are in the same direction and x and z are in the x z plane.) z z' θ d) What is the radius of curvature of the bug’s path at t = 0? 10.8 A bug walks on a turntable. In polar coordinates, the position of the bug is given by ˆ R = R er , where the origin of this coordinate system is at the center of the turntable. The ˆ ˆ (er , eθ ) coordinate system is attached to the turntable and, hence, rotates with the turntable. The kinematical quantities describing the bugs ˙ ˙ ¨ ˙ motion are R = − R0 , R = 0, θ = ω0 , and ¨ θ = 0. A fixed coordinate system O x y has origin O at the center of the turntable. As the bug walks through the center of the turntable: a) What is its speed? b) What is its acceleration? c) What is the radius of the osculating circle (i.e., what is the radius of curvature of the bug’s path?). 10.9 Actual path of bug trying to walk a straight line. A straight line is inscribed on a horizontal turntable. The line goes through the center. Let φ be angle of rotation of the ˙ turntable which spins at constant rate φ0 . A bug starts on the outside edge of the turntable of radius R and walks towards the center, passes through it, and continues to the opposite edge of the turntable. The bug walks at a constant speed v A , as measured by how far her feet move per step, on the line inscribed on the table. Ignore gravity. a) Picture. Make an accurate drawing of the bug’s path as seen in the room (which is not rotating with the turntable). In order to make this plot, you will need to assume values of v A ˙ and φ0 and initial values of R and φ. You will need to write a parametric equation for the path in terms of variables that you can plot (probably x and y coordinates). You will also need to pick a range of times. Your plot should include the instant at which the bug walks through the origin. Make sure your x− and y− axes are drawn to the same scale. A computer plot would be nice. b) Calculate the radius of curvature of the bug’s path as it goes through the origin. 10.3 General expressions for velocity and acceleration ˆ ˆ 10.1 Express the basis vectors (ı , ) associated with axes x and y in terms of the standard ˆ ˆ basis (ı , ) for θ = 30o . y, y' θ x x' problem 10.3: (Filename:efig1.2.25) 10.4 For Q given in problem 12.22 and M = ˆ ˆ ˆ 25 N·mE 1 − 32 N·mE 2 , find the e2 component of M without calculating [Q]{M }. 10.5 A particle travels in a straight line in the x y-plane parallel to the x-axis at a distance y = in the positive x direction. The position of the particle is denoted by r (t). The angle of r measured positive counter-clockwise from the x axis is decreasing at a constant rate with magnitude ω. If the particle starts on the y axis at θ = π/2, what is r (t) in cartesian coordinates? ˆ 10.6 Given that r (t) = ct 2 ı and that θ(t) = d sin(λt) find v (t) ˆ ˆ (a) in terms of ı and , ˆ ˆ (b) in terms of ı and . y' y x' θ x problem 10.1: (Filename:efig1.2.23) ˆ ' 10.2 Body frames are frames of reference attached to a body in motion. The orientation of a coordinate system attached to a body frame B is shown in the figure at some moment of interest. For θ = 60o , express the basis vectors ˆ ˆ ˆ ˆ (b1 , b2 ) in terms of the standard basis (ı , ). ˆ ˆ ı' θ ı ˆ r(t) y problem 10.6: (Filename:pfig3.1.DH1) B ˆ b2 θ problem 10.2: ˆ b1 x (Filename:efig1.2.24) 10.7 A bug walks on a straight line engraved on a rotating turntable (the bug’s path in the room is not a straight line). The line passes through the center of the turntable. The bug’s speed on this line is 1 in/s (the bug’s absolute speed is not 1 in/s). The turntable rotates at a constant rate of 2 revolutions every π s in a positive sense about the z-axis. It’s surface is always in the ∗ ∗ CONTENTS e) What is the total force acting on the train? f) Sketch the path of the train for one revolution of the turntable (surprise)? ˆ North turntable. Evaluate the absolute velocity and acceleration of the bug in the following situations (in cartesian coordinates): xT (a) (b) (c) (d) (e) (f) (g) (h) (i) 4m ” ” ” ” ” ” ” ” xT ˙ 3 m/s ” ” ” ” ” ” ” ” xT ¨ 0 m/s2 2 m/s2 0 m/s2 ” ” ” 2 m/s2 ” 4 m/s2 10. k ).. exactly as in problem 10.blue.13: (Filename:pfigure. The track itself is on a turntable B that is rotating counter.f93f4) C D a E B y ω problem 10. is a distance = 6 cm from the center of the turntable.1 kg toy train engine is going clockwise at a constant rate (relative to the track) of 2 m/s on a circular track of radius 1 m.clockwise at a constant rate of 1 rad/s.14 A giant bug walks on a horizontal disk. e) Force. The dimensions are as shown.10: (Filename:pfigure. The bug is walking. d) Force. a b.15. The scratch. the angular velocity of the disk (i. At the instant of interest. The bug has a mass m B = 1 gram.12: vB/T (Filename:Mikes93p2) bug x 10. and ˙ m = 1 kg = mass of the bug. At the instant shown. At the instant of interest the train is pointing due south (−j) and is at the center KinematicalQuantities turntable. ωD ˙ problem 10. the line D E is currently aligned with the z-axis. point C. relative to the turntable. The turntable is turning clockwise at a constant angular rate ω = 2 rad/s. u = du/dt.11 A bug of mass m = 1 kg walks on a straight line marked through the center of a turntable on the back of a pick-up truck. Find the absolute acceleration of the bug. the turntable is at its maximum amplitude x = A.84 c) Accurately draw (say.17. at a constant rate v B/T = 12 cm/s. In each of the cases shown in the figure. O of the rb rb ˙ rb ¨ θ ω ω ˙ a) What is the velocity of the train relative 2 o 2 1 m 0 m/s 0 m/s to the37 0 v /B turntable rad/s? 0 rad/s ” ” ” ” ” ” ” ” ” b) What is rad absolute velocity of the train π/2 the 1 rad/s ” ” ” 1 m/s2 v ? 0 rad 0 rad/s ” 0 m 1 m/s 0 m/s2 ” 1 rad/s ” c) What is the acceleration of the train rel1 m 0 m/s ” ative to rad turntable a 3 rad/s2 π/2 the 0 rad/s /B ? ” 5 m/s −2 m/s2 0 rad ” 0 rad/s2 ” 0 m/s 0 m/s2 What israd absolute acceleration of the π/2 the ” 2 rad/s2 d) a? ” ” ” train0 rad 2 rad/s 0 rad/s2 r = the position of the center of the disk. a = the acceleration of the center of the disk. In each case. 10. An x − y − z frame is attached to the disk. What is the force on the bugs feet from the turntable when she starts her trip? Draw this force as an arrow on your picture of the bug’s path. In this problem: problem 10. ˙ = the angular acceleration of the disk. straight along the scratch in the y-direction. The disk is rotating about the z axis (out of the paper) and simultaneously translating with respect to an inertial frame X − Y − Z plane. on the computer) the osculating circle when the bug is at the origin on the picture you drew for (a) above. A small bug walks along line D E with ˙ velocity vb relative to the turntable. At the instants shown. What is the force on the bugs feet when she is in the middle of the turntable? Draw this force as an arrow on your picture of the bug’s path. the dotted line is scratched on the disk and is the path the bug follows walking in the direction of the arrow. determine the total force acting on the bug. A small .e.13 See also problem 10. the x − y − z coordinate system shown is aligned with the inertial X − Y − Z frame (which is not shown in the pictures).10 A small bug is crawling on a straight line scratched on an old record.12 A turntable oscillates with displacement ˆ x C (t) = A sin(st)ı . v = the velocity of the center of the disk. at its closest. everything is aligned as shown in figure. a) What is the bug’s velocity? b) What is the bug’s acceleration? c) What is the sum of all forces acting on the bug? d) Sketch the path of the bug in the neighborhood of its location at the time of interest (indicate the direction the bug is moving on this path). = . B track 1m O East ˆ ı y z x ωD . Point B is a distance a from the center of the turntable. The disc of the turntable rotates with angular speed and acceleration ω D and ω D . u = the speed of the bug traversing the dotted line (arc length on the disk per unit time). 10. and the bug is passing through point B on line D E.55.2) 10. The location of the bug is marked with a dot. 19: (Filename:f.67.13 going counter-clockwise at constant rate (relative to the track) of 2 m/s on the circular track. . and k components. Consider a turntable on the back of a pick-up truck.1) 10.117. θ. the radius of the record is 7 inches.blue. Vectors should be expressed in terms of ˆ ˆ ˆ ı . θ.16: (Filename:pfigure. the bug as the particle of interest.75. Clearly define all variables you are going to use. The crank mechanism parts move on the x y plane with the x direction being along the piston. The line may or may not be drawn through the center of the turntable.20 The crank AB with length L AB = 1 inch in the crank mechanism shown rotates at a con˙ stant rate θ AB = ω AB = 2π rad/s counterclockwise. The honeybee is at the outer edge of the phonograph record in the position shown in the figure.17: Picking apart the five term acceleration formula.) ∗ 10.18 Slider crank kinematics (No FBD re˙ ¨ quired!). The truck may or may not be going at constant rate. The bug may be anywhere e) What is the angular velocity of the connecting rod AB? [Geometry fact: ˆ ˆ r AB = 2 − R 2 sin2 θ ı − R sin θ ] ∗ f) For what values of θ is the angular velocity of the connecting rod AB equal ˙ to zero (assume θ = 0)? (you need not answer part (e) correctly to answer this question correctly. Given θ . 2-D . crank R O motor θ A L rod y xT x O' turntable B M piston problem 10. (harder) How many situations can you find where a pair of terms is not zero but all other terms are zero? (Don’t try to do all 10 cases unless you really think this infamous formula is fun. The initial angle of rotation is θ = 0 at t = 0. a) What is the angular velocity of the crank OA? ∗ b) What is the angular acceleration of the crank OA? ∗ c) What is the velocity of point A? ∗ c) What is the angular velocity of the connecting rod BC when θ = π/2 rad? d) What is the angular velocity of the connecting rod BC as a function of time? LBC C B θ LAB A ωAB = 2π rad/s 12' turntable problem 10. ˆ ˆ ı Blow-up of turntable: bug rb O' Truck problem 10. The carousel has angular velocity of 5 rpm. alights on a phonograph turntable that is being carried by a carnival goer who is riding on a carousel.Problems for Chapter 10 1m y 85 on the line and may or may not be walking at a constant rate.16 A honeybee.) ∗ 10. The connecting rod BC has a length of L BC = 2 in.22. For each term in the five term acceleration formula. which is increasing (accelerating) at 10 rev/min2 .17 See also problem 10. Assume R. ˙ ¨ θ . ∗ 45o u = 0 ˙ C x y 45 o a) 1m u = 0 r = 0ˆ + 0 ˆ ı ˆ v = 0ˆ + 0 ı =0 ˙ = 0 a = 3 m/s2 ı + 0 ˆ ˆ u = 1 m/s u=0 ˙ =0 ˙ =0 ˆ r = 0ˆ + 0 ı ˆ v = 0ˆ + 0 ı ˆ a = 0ˆ + 0 ı crank R O motor θ A rod B M piston b) ˆ ˆ ı C y x c) u=0 ˆ r = 0ˆ + 0 ı u=0 ˙ ˆ v = 0ˆ + 0 ı 1m =0 C x ˙ 1m ˆ ı = 1 rad/s2 a = 0ˆ + 0 y problem 10.34. A bug walks on a line on the turntable. it may or may not go at a constant rate. find a situation where all but one term is zero. The situation is sketched below. if it spins. . θ.2) d) What is the acceleration of point A? ∗ problem 10. At the instant shown. See also problem 11. Consider a slider-crank mechanism.blue.14: 10.20: (Filename:pfigure. can you find the velocity and acceleration of B? There are many ways to do this problem. (Filename:pfigure.1) 10.11. rod AB is rotating clockwise with angular speed ω = 3 rad/s and angular acceleration α = 2 rad/s2 .1) c) Two terms at a time. What is the moving frame.blue. and what is the reference point on that frame? b) One term at a time. L. a) What is the velocity of point B at the end of the crank when θ = π/2 rad? b) What is the velocity of point C at the end of the connecting rod when θ = π/2 rad? 30 carousel o bee 10. The turntable may or may not be spinning.19 Slider-Crank. rotate in the vertical plane. Find the angular velocity of rod DE. Try at least one or two.21 See also problem 10.15 Repeat questions (a)-(f) for the toy train in problem 10. and R.s95q12) d) u=0 ˆ r = 0ˆ + 0 ı u=0 ˙ 1m ˆ v = 0ˆ + 0 ı C 1m x ˙ = 1 rad/s ˆ a = 0ˆ + 0 ı =0 45o y e) C u = 1 m/s r = 0ˆ + 0 ˆ ı u=0 ˙ ˆ v = 0ˆ + 0 ı = 1 rad/s x ˙ =0 ˆ a = 0ˆ + 0 ı (Filename:pfigure. the phonograph rotates at a constant 33 1/3 rpm. and. connected together through a collar C. Calculate the magnitude of the acceleration of the honeybee. ∗ .5) ω θ P problem 10. a) Draw a picture of the situation. The two rods AB and DE.blue. 10. sensing that it can get a cheap thrill.116. The collar C is pinned to the rod AB but is free to slide on the frictionless rod DE. Use the turntable as the moving frame. θ are given.18: (Filename:pfigure. 26: (Filename:pfigure.21: 10. The speed and acceleration of slider B relative to the frame L are ˆ ˆ v B/L = −2 ft/s and a B/L = −2 ft/s2 . slider A is x A = 4 ft from point O. α 45 o E (Filename:summer95f.36. a) Find the velocity of point B.4.blue. and . ω O r β A problem 10. b) Find P’s acceleration a P and the angular acceleration α of the rod. and b) The absolute angular velocity of bar AB.23 Collar A is constrained to slide on a horizontal rod to the right at constant speed v A . The link AB is supported by a wheel at D and its end A is constrained so that it only has horizontal velocity. A thin rod of mass m and length L is attached by a frictionless pin P to the cylinder’s rim and its right end is dragged at a constant speed v A along the (frictionless) horizontal ground. v A/F . No slipping occurs between the wheel and the link. Alternatively you can think of a small bead that slides on a rod. rotate in the vertical plane. Knowing that rod AD is 250 mm long and distance d = 150 mm. The wheel has an angular velocity ω and radius r . B D L yB P y x C problem 10. ∗ .1) B problem 10.blue.4) problem 10.28: D C (Filename:pfigure.25: (Filename:pfigure.22: 10. a) For the position shown (where P is directly above the contact point). rod AB is rotating clockwise with angular speed ω = 3 rad/s and angular acceleration α = 2 rad/s2 .28 The rod of radius r = 50 mm shown has a constant angular velocity of ω = 30 rad/s counterclockwise. Assume the mass is a little bug that can walk as it pleases on the rod (or in the slot) and you control how the turntable/rod rotates. See also problems 10.blue. The two rods AB and DE. . ωD /F . Name two situations in which one of the terms is zero but the other is not in the two term polar coordinate formula for velocity. A solid cylinder of radius R and mass M rolls without slip along the ground.120. respectively. ˙ b) Find the angular speed θ of the rod? ˆ ˆ Answer in terms of v A . The collar C is pinned to the rod AB but is free to slide on the frictionless rod DE.21. (β = sin−1 r ).25 See also problem 11.30 and 11. θ. Find the angular acceleration of rod DE.115. The pin P is attached to ED at fixed radius d and engages the slot on CB as shown.5 rad/sk .65. ωL/F = 0.blue. ˙ˆ ˙ˆ R e R + R θ eθ . The mass always stays in the slot (or on the rod). A second collar B connected by a pin joint to the other end of the rigid bar AB slides on a vertical rod.23: B C (Filename:s93q5) A 10.a) problem 10. ı and .24 A bar of length = 5 ft. connects sliders A and B on an L-shaped frame. body D .blue.86 B C 10.27 The slotted link CB is driven in an oscillatory motion by the link ED which rotates about D with constant angular velocity ˙ θ = ω D . At the instant shown. Determine: a) The absolute velocity of the slider A.27: E d B D B C A xA problem 10. slider B is y B = 3 ft from O.22 Reconsider problem 10.5 m problem 10.2) A ω. body L is aligned with the x − y axes.29 Picking apart the polar coordinate formula for velocity. The distance O A = . body L. body D .121.52.5 m A ω.26 See also problem 11. CONTENTS D P R L m M θ vA 0.1) (Filename:pfigure. r ω θ A d B D 10. Determine the angular velocity of the link AB and velocity of the point A. determine the acceleration of collar D when θ = 90◦ .24: O ωL/F φ D θ (Filename:pfigure. Given: ω.35. find P’s velocity v P and the rod’s angular velocity ω. θ.3) 0. ˙ Find the angular velocity and acceleration θ ¨ and θ of CB when θ = π/2. which itself is rotating at constant speed about an axle perpendicular to the plane of the figure through the point O and relative to a fixed frame F . 10.116. ∗ 10. Answer ˆ ˆ in terms of v A . r . This problem concerns a small mass m that sits in a slot in a turntable. ı and . At the instant shown. connected together through a collar C. α 45 o E (Filename:summer95f. . It is connected by a pin joint to one end of a rigid bar AB with length which makes an angle θ with the horizontal at the instant of interest. θ problem 10. You should thus gain some insight into the meaning of each of the two terms in that formula.1) ˆ n ˆ λ y x B 10. φ1 .83. and φ2 . each of length ˙ and mass m. en . φ1 .36 Double pendulum. φ2 .37: 10. and φ2 . the polar coordinate basis vectors er and ˆ ˆ eθ .35 For the configuration in problem 9. For the instant shown. sketch the position vector ˆ r (t). answer the following questions in terms of . φ2 .34 A particle travels at non-constant speed on an elliptical path given by y 2 = b2 (1 − Carefully sketch the ellipse for particular values of a and b. a2 Top view y ω O φ1 x A φ2 disk D L problem 10. You should thus gain some insight into the meaning of each of the four terms in that formula.31 For the configuration in problem 9. and the radius of curvature ρ. a) What is the absolute velocity of point A? b) What is the velocity of point B relative to point A? c) What is the absolute velocity of point B? O (Filename:doublerot1) problem 10. φ1 and φ2 are also known at the instant shown.Problems for Chapter 10 Top view y ω b) problem 5.36. a = ( R − R θ 2 )e R + ˙˙ ¨ ˆ (2 R θ + R θ)eθ . are zero. a) What is the absolute acceleration of point A? b) What is the acceleration of point B relative to point A? c) What is the absolute acceleration of point B? x 10. ∗ 10. φ1 . answer the following questions in ˙ ¨ ˙ ¨ terms of . ρ = 25 mi. At what points on the path are er and ˆ ˆ ˆ en parallel(or eθ and et parallel)? 10. d) problem 5. D? What is the absolute angular acceleration of the disk if ω1 and ω2 are not constant? x 2 ).37. name four situations in which all of the terms. At the instant shown.37 Double pendulum.81. and the path coordinate basis vectors en ˆ ˆ and et .s94h9p4b) r C P ω2 problem 10.30 Picking apart the polar coordinate formula for acceleration.33 Find expressions for et .a) problem 10. For the dou¨ ¨ ble pendulum in problem 10. an .84.8) . D? ∗ ω1 O θ B (Filename:ch7. Again. ˙ φ2 . For various positions of the particle on the path. and φ2 are known.31: 10.12 what is the absolute angular velocity of the disk. Reconsider the configurations in problem 10.accel) disk D L ω1 O θ r C P ω2 10.8. ∗ φ1 A φ2 problem 10.82. For the instant shown. taking ω2 to be the angular velocity of disk D relative to the rod.36: B (Filename:ch7.85. This time. and f) problem 5. The double pendulum shown is made up of two uniform bars.5. what is the absolute angular acceleration of the disk. at any position (or time) on the given particle paths for a) problem 5.29: (Filename:pfigure. at . in the four term polar coordinate for¨ ˙ ˆ mula for acceleration. φ1 .12.32 A car driver on a very boring highway is carefully monitoring her speed.35: (Filename:doublerot. φ2 . in path coordinates? ˆ ˆ 10. What is the acceleration of the of the car half-way through this one hour period. ˙ ˙ φ1 . φ1 . Over a one hour period. e) problem 5. 87 10. See also problem 10. c) problem 10.29. the car travels on a curve with constant radius of curvature. Each situation should pick out a different term. φ1 .ang. but one. and its speed increases uniformly from 50 mph to 60 mph. gravitational acceleration g = 10 m/s2 (or zero).25.3: M x (Filename:pfigure. a) Draw free body diagrams of the blocks together and separately.blue. radius of hoop Rhoop = 3 m. such that the crate comes to rest at point C.1 Several of the problems below concern a narrow rigid hoop and/or a small bead. There are no external forces applied. calculate the maximum speed attained by the crate at point B. mass of hoop m hoop = 1 kg.2 A warehouse operator wants to move a crate of weight W = 100 lb from a 6 ft high platform to ground level by means of a roller conveyer as shown. i) At t = 0+ what is the linear momentum of the bead and hoop system? j) At t = what is the rate of change of the linear momentum of the bead and hoop system? 0+ m St h ee p St . A bead slides on a frictionless circular hoop . which is of height h and slope θ. upon reaching the bottom of the street. causing it to move along the track. d) If the crate slides a distance x. m 1 = m 2 = m. The hoop is on the x y plane.3 On a wintry evening. a) What is her initial kinetic and potential energy? b) What is the energy lost to friction as she slides down the hill? c) What is her velocity on reaching the bottom of the hill? Ignore air resistance and all cross streets (i.e. what is the energy lost to sliding in terms of µ.blue.2) . as shown below. The hoop is kept from moving by little angels who arrange to let the bead slide by unimpeded. b) Write the equations of linear momentum balance for each block.4 Reconsider the system of blocks in problem 3. on the horizontal conveyor. now. both blocks are frictional and sitting on a frictional surface. e) Assuming that friction still acts on the level flats on the bottom. and it is on the x-axis. The center of the circular hoop is the origin O of a fixed (Newtonian) coordinate system O x yz. a) Assuming that the track is perfectly frictionless from point A to C and that the car never leaves the track.5: ˆ ı (Filename:pfigure. assume that the horizontal track C − D is frictional and determine the coefficient of friction required to bring the car to a stop a D.2: (Filename:pfigure. how much time will it take before they come to rest? 11. it is traveling counterclockwise (looking down the z-axis). this time with equal mass. say. the speed of the bead is 4 m/s. what is their instantaneous mutual velocity just after the embrace? B 20 m y A x 10 m C ˆ µ=? g D 30 m problem 11. For reference here are dimensions and values you should use in working the problems: mass of bead m bead = 2 grams. The coefficient of friction between the lower block and the floor is µ2 . At the top of the slope she starts to slide (coefficient of dynamic friction µ). point B. Assume that both blocks are sliding to the right with the top block moving faster. b) In the ensuing motion. assume the hill is of constant slope). 11. she collides with another student of mass M and they embrace. W . Call this time t = 0. For the frictional blocks. c) Find the acceleration of each block for the case of frictionless blocks.1) A 6 ft C 8 ft 11. Also. µ.21. Gravity is in the negative z direction. Assume the rollers are massless. the rollers on the horizontal conveyer are frictional. The coefficient of friction between the two blocks is µ1 .30. d) If.88 CONTENTS Problems for Chapter 11 Mechanics of planar mechanisms k) At t = 0+ What is the value of the acceleration of the bead? (Hint:abead = ahoop center + abead/hoop center ) 11. Steep Street. point A? b) What are the kinetic and potential energies of the cart at the end of the inclined plane just before it moves onto the horizontal conveyor. a) At t = 0 what is the bead’s kinetic energy ? b) At t = 0 what is the bead’s linear momentum ? c) At t = 0 what is the bead’s angular momentum about the origin? d) At t = 0 what is the bead’s acceleration? e) At t = 0 what is the radius of the osculating circle of the bead’s path ρ? f) At t = 0 what is the force of the hoop on the bead? g) At t = 0 what is the net force of the angel’s hands on the hoop? h) At t = 27s what is the x-component of the bead’s linear momentum? Just when the bead above has traveled around the wire exactly 12 times the angels suddenly (but gently) let go of the hoop which is now free to slide on the frictionless x y plane. assuming it starts moving from rest at point A. thus providing an effective friction coefficient µ. θ problem 11. a student of mass m starts down the steepest street in town. It is known that µ1 > 2µ2 .1 Dynamics of particles in the context of 2D mechanisms 11. At t = 0. a) What are the kinetic and potential energies(pick a suitable datum) of the crate at the elevated platform.30. The rollers on the inclined plane are well lubricated and thus assumed frictionless. At t = 0+ we can assume the angels have fully let go.blue. find the acceleration of each block. and x? e) Calculate the value of the coefficient of friction.2) 11.5 An initially motionless roller-coaster car of mass 20 kg is given a horizontal impulse F dt = P i at position A. What happens if µ1 >> µ2 ? 100 lb frictionless frictional B 24 ft problem 11. c) Using conservation of energy. determine the magnitude of the impulse I so that the car “just makes it” over the hill at B. Answer the following questions in terms of φ. and its rate of change. if for a range of values of t.8 A scotch yoke. Assume ω = ω0 is a constant for the rod in the figure. Initially.6 Masses m 1 = 1 kg and m 2 kg move on the frictionless varied terrain shown. d) What condition must be met so that the bug does not bounce the platform? e) What is the maximum spin rate ω that can be used if the bug is not to bounce off the platform? y R θ ω 11. ∗ l ω b) Turn this equation into a differential equation for R(θ ). the rod is aligned with the x-axis and the bead is one foot from the origin and has no radial velocity (d R/dt = 0). m.34.55. The motion of the platform is y = A sin(ωt) with constant ω. b) Write the equation F = ma for a particle in polar coordinates and think of a force that would be relevant to this problem.10 and 11. a) Draw a free body diagram of the bug.) A newer kind of gun. What. the angle the stick makes ˙ with the x axis. Assume the mass is free to slide. The friction coefficient is µ. This solution could be checked by plugging back into the differential equations — you need not do this (tediousl) substitution.s94h9p5) O ˙ φ. (See also problem 11.1) 11. and v are arbitrary constants.s94f1p5) (v(t − t0 )/d) where θ0 . v A . Assume there is no friction between the bead (mass m) and the wire. ¨ a) What is φ? b) What is the force exerted by the rod on the support? x c) What is the acceleration of the bug? y massless rod A bug. the values of R and θ were calculated and then plotted using polar coordinates. That is. is used to make a horizontal platform go up and down. Ignore gravity. b) What is the acceleration of the platform and. t0 .9. and ω0 .1) problem 11.9: (Filename:pfigure. The bead has mass m b . the bug? c) Write the equation of linear momentum balance for the bug.blue. The angle θ is zero and ω is a constant. slides on the stick.Problems for Chapter 11 11. φ. There is gravity. ¨ a) What is R at this instant? Give your ˙ answer in terms of any or all of R0 . m.9 A new kind of gun. rotates at a constant rate ω = ω0 . Initially the mass is at a distance R0 from the hinge point on the stick ˙ and has no radial velocity ( R = 0). Find l(t) in terms of 0 .10 The new gun gets old and rusty. φ. Her feet have no glue on them. ˆ ˆ ω. friction cannot be neglected. m. hence.8: g v1 m1 m2 h1 h2 Peg fixed to disc (Filename:pfigure. a person adds a length 0 to the shaft on which the bead slides. m 1 has speed v1 = 12 m/s and m 2 is at rest. and v A . . a device for converting rotary motion into linear motion. A long stick with mass m s rotates at a constant rate and a bead.5. a) Find a differential equation for R(t).7 The two differential equations: ¨ ˙ R − Rθ 2 ˙ θ + Rθ ˙ ¨ 2R have the general solution R θ = = d 2 + (v(t − t0 ))2 θ0 + tan −1 = = 0 0 11. See also problems 11. ∗ b) After a very long time it is observed that the angle φ between the path of the bead and the rod/trough is nearly constant.9. What can you say about the shape of this curve? [Hints: a) Actually make a plot using some random values of the constants and see what the plot looks like. m problem 11. This time.13 Slippery bead on straight rotating stick. At the instant of interest the bead with mass m has ˙ radius R0 with rate of change R0 . The coefficient of restitution in the collision is e = 0. The rigid rod. v0 . ı .11 (See also problem 11.10. .11: (Filename:pfigure. in terms of µ. A dead bug with mass m is standing on the platform.blue. Neglect gravity. Assume the bead starts at l = 0 with l˙ = v0 . R0 .6: (Filename:Danef94s2q7) 11. φ x 11. As an attempt to make an improvement on the ‘new gun’ demonstrated in problem 11. The stick rotates at ˆ constant rate ω = ωk .12: (Filename:pfigure. Assume that the distance the bug is from the origin . A ω problem 11.11. There is no gravity. and . measured counterclockwise from the positive x axis.) Reconsider the bead on a rod in problem 11. e R . At t = 0. on which the bead slides. are known at the instant of interest. µ. a curve would be drawn. modeled as a point mass. The solution describes a curve in the plane.11. is this φ? ∗ frictionless hinge problem 11. Find the speed of m 2 at the top of the first hill of elevation h 1 = 1 m.12 A bug of mass m walks straight-forwards ˙ with speed v A and rate of change of speed v A on a straight light (assumed to be massless) stick. ˙ ˙ φ. . ∗ c) How far will the bead have moved after one revolution of the rod? How far after two? ∗ d) What is the speed of the bead after one revolution of the rod (use ω0 = 2π rad/s = 1 rev/sec)? ∗ e) How much kinetic energy does the bead have after one revolution and where did it come from? ∗ problem 11. The two masses collide on the flat section. Does m 2 make it over the second hill of elevation h 2 = 4 m? 89 bug platform y g 11. eθ . The initial angle of the stick is θ0 . c) The answer is something simple. ] 11. The stick is hinged at the origin so that it is always horizontal but is free to rotate about the z axis. d. you may assume that K > mω2 . Assume the bead is released pulled to a distance d from the center of the rod and ˙ then released with an initial R = 0. a) Derive the equation of motion for the position of the bead R(t). ∗ b) Calculate the angular velocity of the tube just after the ball leaves the tube. Assume that the rod/turntable in the figure is massless and also free to rotate. and of mass m t = 1.17 Bead on springy leash in a slot on a turntable.14 Mass on a lightly greased slotted turntable or spinning uniform rod. d.16: (Filename:pfigure. The bead in the figure is held by a spring that is relaxed when the bead is at the origin.2 m.] e) A finite rigid body.125) revolutions.90 a) What is the torque. the angular velocity of the rod/turntable is 1 rad/s. The turntable spins a constant rate ω. If needed. v0 .5 kg. (Hint: Treat the tube as a slender rod. [Note. How does the speed of the bead over one revolution depend on µ. The bead is also attached to a spring with constant K . K. the coefficient of friction between the bead and the wire? Make a plot of | v one revolution | versus µ. but contact at various places on the stick? (Do the frictionless case only. Answer in terms of m. The turntable speed is controlled by a strong stiff motor.5 m/s.18 A small bead with mass m slides without friction on a rigid rod which rotates about the z axis with constant ω (maintained by a stiff motor not shown in the figure). Make the unreasonable assumption that the slot is long enough to contain the bead for this motion. calculate the angular velocity of the tube just before the ball leaves the tube. Where is the bead at t = 5 sec? ˙ R = 0 at t = 0.f93q11) Top view y ω 11.18: (Filename:pfigure.14: (Filename:pfigure. determine the transverse and radial components of velocity of the ball as it leaves the tube. you have been working with µ = 0 in the problems above.) a) If the angular velocity of the tube is ω = 8 rad/s as the ball passes through C with speed relative to the tube.1a) c) If the speed of the ball as it passes Cis v = 1. as a function of the net angle the stick has rotated. What are possible motions of the bead? b) Assume ω = ω0 is a constant. What is going on here? problem 11. θ − θ0 .16 A bead of mass m = 1 gm bead is constrained to slide in a straight frictionless slot in a disk which is spinning counterclockwise at constant rate ω = 3 rad/s per second. The slot is straight and goes through the center of the turntable. CONTENTS other end of which is attached to the rod. is zero. the slot is parallel to the x-axis and the bead is in the center of the disk moving out ˙ (in the plus x direction) at a rate Ro = . required in order to keep the stick rotating at a constant rate ? b) What is the path of the bead in the x yplane? (Draw an accurate picture showing about one half of one revolution. Give an example of a situation in which this acceleration is important and explain why it arises.s94h9p4a) (Filename:pg57. d R/dt. ω. Assume that at t = 0. The constant of the spring is k. d) Write an expression for the Coriolis acceleration. a) Assume ω = 0 for all time. the . What would change if instead of a point mass you modeled the bead as a finite rigid body (2-D).) c) How long should the stick be if the bead is to fly in the negative x-direction when it gets to the end of the stick? d) Add friction.) 11. y ω y x ω = constant O x problem 11. ∗ d) After the ball leaves the tube. problem 11. At time t = 0. After a net rotation θ of one and one eighth (1. b) How is the motion affected by large versus small values of K ? c) What is the magnitude of the force of the rod on the bead as the bead passes through the center position? Neglect gravity. 11. and that the radial velocity of the bead. that the radius of the bead is one meter. ∗ problem 11. what is the force F of the disk on the bead? Express this answer in terms ˆ ˆ of ı and . what are the components of the acceleration vector in the directions normal and tangential to the path of the bead after one revolution? Neglect gravity. The spring is relaxed when the bead is at the center position. The bead is free to slide on the rod.s94q9p1) x y Top view y ω O x x ω 11. If the bead is at radius Ro with 11.17: Mass in a slot.19 A small ball of mass m b = 500 grams may slide in a slender tube of length l = 1. with center of mass on the stick. What are possible motions of the bead? Notice there are two cases depending on the value of ω0 .15 A bead slides in a frictionless slot in a turntable. The tube rotates freely about a vertical axis passing through its center C. what constant torque must be applied to the tube (about its axis of rotation) to bring it to rest in 10 s? ∗ 11.8 m/s. find the force of the barrel on the shell. ∗ . The 0. 11. b) Determine the circular frequency of small oscillations of the system in part (a).5θ m rad θ A x problem 11.21 A shell moves down a barrel. Also assume that your hand is accelerating both vertically and horizontally with ˆ ˆ a hand = ahx ı + ahy .24: Support(2-D).) ∗ problem 11. θ . . a) Draw a Free Body Diagram of the rod.5m ˆ O ˆ ı ω= 2 rad/s turntable bead hyperbolic spiral track r r =0. ω ˆ er C m C B problem 11.21: A shell moves along a canon barrel. Coordinates and directions are as marked in the figure. At time t = 0. it passes due north of point O.f96.1 kg toy train’s speedometer reads a constant 1m/s when.) ˆ eθ y g m C ˆ er 0. ˙ m(L θ)2 −mgL(1−cos θ)+K (aθ)2 = Constant a) Obtain the differential equation of motion of this system by differentiating this energy equation with respect to t.s94h12p1) 11. a) Determine the radial and transverse components of the acceleration of the bead after 2 s have elapsed from the start of its motion.24 A rod is on the palm of your hand at point A. θ . the string AB is cut. m.5θ( m/ rad). The ball B has mass m B . ı . HINT: (Let sin ∼ θ ∼ and cos θ ∼ = = = 1 − θ 2 /2).Problems for Chapter 11 A y b) the absolute acceleration of the mass at the instant of cutting.2) θ O problem 11. the barrel is tipped up an angle θ and the mass m is moving at speed v along the frictionless barrel. θ . What is the force of the turntable on the train? (Don’t worry about the z-component of the force. . 11. θ.25: ˆ ˆ ı problem 11. θ. heading west.blue. ¨ c) If the hand is not stationary but θ has been determined somehow. Assume the hand is accelerating to the right with acceleration ˆ a = a ı .1) C A 2/3L 1/3L θ ˆ ˆ ı 11. At the end of a stick connected ˙ to this motor is a frictionless hinge attached to a second massless stick. g. (Filename:p.27 A motor at O turns at rate ωo whose rate of change is ωo . At the instant of interest the barrel is being raised at a constant angular speed ω1 . a. At the end of the second stick is a mass O θ z x problem 11. The arm starts from rest at θ = 0o . ∗ 91 ˆ eθ y g L B 2L A x problem 11. Its mass m is assumed to be concentrated at its end at C. and ˙ θ. 11. The arm rotates about its pivot point at O with ¨ constant angular acceleration θ = 1 rad/s2 driven by a motor (not shown). ∗ c) the tension in string BC at the instant of cutting.2) 11.blue.25 Balancing a broom. the cart moves to the right with constant acceleration ax . ¨ Solve for θ in terms of g. m. The train is on a level straight track which is mounted on a spinning turntable whose center is ‘pinned’ to the ground.23 The forked arm mechanism pushes the bead of mass 1 kg along a frictionless hyperbolic spiral track given by r = 0. Its length is .20: (Filename:pfigure.1) y turret shell vo 11. ahx . (Filename:pg122.f.22 Due to forces not shown. .23: (Filename:pfigure. The turntable spins at the constant rate of 2 rad/s.26 A conservative vibratory system has the following equation of conservation of energy. . find the vertical force of the hand on the rod in ¨ ˙ terms of θ. As˙ sume that you know θ and θ at the instant of interest.53. b) Assume that the hand is stationary. Both sticks have length L. What is the force of the hand on the ˙ ˆ ˆ broom in terms of m. Neglecting gravity. b) Determine the magnitude of the net force on the mass at the same instant in time. Find a) the tension in string BC before cutting.19: (Filename:pfigure. and ˆ ˆ g? (You may not have any e R or eθ in your answer. and ahy .1) toy train one way North 11.22: (Filename:Danef94s1q6) Rod on a Moving (Filename:p3.91.20 Toy train car on a turn-around. At the instant of interest. Simultaneously the gun turret (the structure which holds the barrel) rotates about the vertical axis at a constant rate ω2 .2 Dynamics of rigid bodies in one-degreeof-freedom mechanisms 11. 35. The system is released from rest when the rod makes an angle of 45◦ . ∗ problem 11.30: (Filename:pfigure. θ x K problem 11. a G . ∗ e) Find the acceleration of point B.1) problem 11. ω O r β A problem 11. a) Draw a free body diagram of the rod. P problem 11. How do these results compare with those from part(b)? CONTENTS B 2.159.33: B grease disk D L ω1 O θ r C P ω2 (Filename:pfigure. and a load P applied at the top end. find the acceleration of point A.blue. Write the equation of motion of the rod. Gravity pulls it down.29 Robot arm.force) .28 (See also problems 10. B D 11.tension) A ωAB . d) Do the reaction forces at A and B add up to the weight of the bar? Why or why not? (You do not need to solve for the reaction forces in order to answer this part.s94h12p5) 11.mom) A L G θ problem 11.ang.blue. They are attached with a rigid massless rod of length that is pinned at both ends.98.30 A crude model for a column.34: (Filename:slidercrank.1) A L θ rod R y 1m 2m o B ωBC/AB 30 m C P B motor M piston ˆ ˆ ı 11.31 A thin uniform rod of mass m rests against a frictionless wall and on a frictionless floor. 2-D . What is the acceleration of the block B immediately after the system is released? grease A mA = m B = m massless rigid rod of length π/4 m d) the rate of change of angular momentum of the disk about point C. ∗ 11. crank B O (Filename:ch7.) In problem 9.28: (Filename:doublerot. m.33 Two blocks. find the net force acting on the object P which has mass m = 1 kg. The robot arm AB is rotating about point A with ω AB = 5 rad/s and ω AB = 2 rad/s2 . Gravity cannot be neglected.119. a linear torsional spring of stiffness K attached to the center hinge (the spring is relaxed when θ = 0). 2-D . Refer to the figure in problem 10. ωAB ˙ problem 11. and the acceleration of the center of mass. ωo . ω AB . b) The rod is released from rest at θ = θ0 = 0. b) Kinematics: find the velocity and acceleration of point B in terms of the velocity and acceleration of point A c) Using equations of motion. a) Obtain the exact nonlinear equation of motion.35 In problem 10.25.32 Bar leans on a crooked wall. ∗ c) Using the equation of motion.31 and 10. that the piston has mass M and the crank has radius R. L. consists of two identical rods of mass m and length with hinge connections.31: 11.19.29: x (Filename:pfigure.12. A uniform 3 lbm bar leans on a wall and floor and is let go from rest.blue. each with mass m. find the ˙ initial angular acceleration. that the connecting rod AB is massless.92 ¨ m. At the instant shown. and θ. What is the tension in the massless rod AB (with length L) when the slider crank is in the position with θ = 0 (piston is at maximum extent)? Assume the crankshaft has constant angular velocity ω. Meanwhile the forearm ˙ BC is rotating at a constant angular speed with respect to AB of ω BC/AB = 3 rad/s. There is gravity. ∗ 11.27: (Filename:pfigure. m problem 11.32: m. ∗ d) Find the reactions on the rod at points A and B.2) 0 f) When θ = 2 . Ignore gravity. ∗ b) the rate of change of angular momentum of the disk about point O. No gravity. of the rod. find ω AB and the acceleration of point A.34 Slider Crank.5 ft A 30o 45o (Filename:pfigure. shown in the figure. find the force on the wheel at point D due to the rod. θ ∗ 11. ∗ Assume the rod is massless and the disk has mass m. For the configuration shown. that the cylinder walls are frictionless.2) 11.) problem 11. ωO ˙ m π/2 at time of interest y m.s94h10p3) L θ L ωO . find a) the angular momentum about point O. a) Draw a Free Body Diagram of the bar. b) Obtain the squared natural frequency for small motions θ . c) Check the stability of the straight equilibrium state θ = 0 for all P ≥ 0 via minimum potential energy. slide without friction on the wall and floor shown. that there is no gas pressure in the cylinder. what is θ? ˙ ˙ Answer in terms of ωo . ∗ c) the angular momentum about point C.35: (Filename:roddisc. θ . 38: Free Body Diagram Exercise (Filename:pfigure. Do not ignore gravity. find the force on the rod at point P. determine the compression of the spring required to stop the car. bar CD is driven by the motor which spins at constant rate. CD and the ground DA.26. The string length is such that the spring is relaxed when the mass is on top of the hole in the plane. state the circumstances in which the statement is true (assuming the particle stays on the plane). ∗ b) While the car is in contact with the spring but still moving forward. 11.s94h2p7) 11. ∗ c) Repeat part (a) assuming now that the ground is perfectly frictionless from points A to B shown in the figure.42: (Filename:pfigure. Initially in motion with speed v0 .42 Which way does the bike accelerate? A bicycle with all frictionless bearings is standing still on level ground.37: (Filename:pfigure. A horizontal force F is applied on one of the pedals as shown. and C.2) Lemond's stomach C A 60o 60o M D P N 11. You tie a string to the vertically down right pedal and pull back with a force of 70 N. It is held by a string which is in turn connected to a linear elastic spring with constant k. do this problem a few times. c) Repeat problem (b) but this time assume that the mass of bar BC is negligible. Mc R Vo R g Mw k F A B (Filename:pfigure.2) No slip problem 11. The bars are connected by well lubricated hinges at A. find the vertical reactions (in the y direction) at A . A particle with mass m slides on a rigid horizontal frictionless plane.s94h11p1) B 11.37 Slider-Crank mechanism.135.38 Consider the 4-bar linkage AB. you can regard his and the bike’s combined mass (all 70 kg)) as concentrated at a point in his stomach somewhere.40: C B 1m .41 While walking down a flat stretch of road you see a dejected Greg Lemond riding by. BC and CD. BC. Neglecting gravity. what is the force on P at the instant shown? ∗ problem 11.Problems for Chapter 11 11. backward.3 Dynamics of multidegree-of-freedom mechanisms 11. Justify your answer as clearly as you can. Use the law of action and reaction in these diagrams.s94h14p2) problem 11.41: (Filename:pfigure. relative to the ground the right foot is only going 3/4 as fast as the bike (since it is going backwards relative to the bike). Further. At D. clearly enough to convince a person similar to yourself but who has not seen the experiment performed. Arm AB and BC are each 0. as shown in the figure.] 11. Greg’s left foot has just fallen off the pedal so he is only pedaling with his right foot. The position of the particle ˆ ˆ is r = x ı + y .36: 11.40 An idealized model for a car comprises a rigid chassis of mass Mc and four identical rigid disks (wheels) of mass Mw and radius R. You note that. whose corners follow the guide. in which direction is the tangential force on any one of the wheels (due to contact with the ground) pointing? Why? (Illustrate with a sketch). For each of the statements below. assuming ¨ ¨ y ˙ = 0 initially.2) problem 11. B enters the curved guide with velocity V . Assume no frictional losses. You would like to know Greg’s acceleration. and B. a) Assuming no wheel slip. What acceleration do you measure? b) What is Greg’s actual acceleration? [Hint: Greg’s massless leg is pushing forward on his body with a force of 70 N] P R L m M θ vA (Filename:rodcyl. [Hint. The bicycle is gently balanced from falling over sideways. (Hint: for the rigid planar body ABC. The slider crank mechanism shown is used to push a 2 lbm block P. Greg. each with a different vector notation.blue. Does the bicycle accelerate forward.39: (Filename:pfigure. There is no slip between the wheels and the ground.blue.85. ¨ find y A and y B in terms of a cm and θ . the car is momentarily brought to rest by compressing an initially uncompressed spring of stiffness k. His riding “tuck” is so good that you realize that you can totally neglect air resistance when you think about him and his bike. If you can. which has connections at the two points B and C.36 In problem 10.blue. which at the moment in question is at its lowest point in the motion. D motor turns bar CD problem 11.34. Justify your answer with convincing explanation and/or calculation. though you can’t make out all the radii of his frictionless gears and rigid round wheels. this implies that the laws of statics apply to bar BC. It is heavy enough so that both wheels stay on the ground.43 Particle on a springy leash.75m A π/4 α 11. B. ∗ problem 11. tells you he is pushing back on the pedal with a force of 70 N.5 ft long.39 The large masses m at C and A were supported by the light triangular plate ABC. Given that arm AB rotates counterclockwise at a constant 2 revolutions per second. or not at all? Make any reasonable assumptions about the dimensions and mass.) θ 93 a) In your first misconceived experiment you set up a 70 kg bicycle in your laboratory and balance it with strings that cause no fore or aft forces.force) A m a y m C x b B V R problem 11. ever in touch with his body. a) Draw a Free Body Diagram of the linkage ABCD (with ‘cuts’ at A and D but not at B and C.) b) Draw Free Body Diagrams of each of the bars AB. 45 A particle with mass m is held by two long springs each with stiffness k so that the springs are relaxed when the mass is at the origin. m. ∗ 11. (Filename:pg69. its time derivatives. is “de-spun” by the following mechanism. a) What is the net displacement of the cart xC at t = tvert ? ∗ b) What is the velocity of the cart vC at t = tvert ? ∗ c) What is the tension in the string at t = tvert ? ∗ d) (Optional) What is t = tvert ? [You will either have to leave your answer in the form of an integral you cannot evaluate analytically or you will have to get part of your solution from a computer. Assume this dumbell is released from . ∗ c) Express the conservation of angular momentum in cartesian coordinates.] C y x mC problem 11. tube. At time t = 0 all masses are stationary and the string BC is cleanly and quietly cut.p3. A sattelite.blue. At the start of de-spinning the two masses are released from their position wrapped against the satelite. The length of string AB is r = 1 m.49: (Filename:p. ∗ g) Express the conservation of energy in polar coordinates. ∗ h) Show that (a) implies (f) and (b) implies (g) even if you didn’t note them a-priori. The hoop is released from rest at the position shown at t = 0. It is a nonlinear equation and you are not being asked to solve it numerically or otherwise.73.s96.47 A dumbell slides on a floor. k) The trajectory is not a closed curve. sketches or words. and (b) the strings are wrapped in the opposite direction as the initial spin.R problem 11. Before string BC is cut string AB is horizontal. ∗ e) Show that (a) implies (c) and (b) implies (d) even if you didn’t note them a-priori. and .1) 11.48 After a winning goal one second before the clock ran out a psychologically stunned hockey player (modeled as a uniform rod) stands nearly vertical. and t. ∗ i) Find the general motion by solving the equations in (a). b) Find the velocity of point A just before it hits the floor e) r = constant ˙ f) θ =constant ˙ g) r 2 θ = constant. At the end of the strings are placed 2 equal masses. The players perfectly slippery skates start to pop out from under her as she falls.94 a) The force of the plane on the particle is ˆ mg k .44 “Yo-yo” mechanism of satelite despinning. k b) x + m x = 0 ¨ k c) y + m y = 0 ¨ k d) r + m r = 0.2) 11. Before launch two equal length long strings are attached to the sattelite at diametrically opposite points and then wound around the sattelite with the same sense of rotation. Describe all possible paths of the mass. After some unknown time tvert the string AB is vertical. a) Find yG at a later time t in terms of any or all of m. Assume that the particle displacement is much smaller than the lengths of the springs.49 Falling hoop. or hub) is idealized as a hoop with mass m and radius R.47: ˆ ˆ ı B (Filename:s97f5) y x ˆ k y v0 ˆ ˆ ı O θ x m k problem 11. Assume the motion is planar. and the gravitational constant is g. Find the motion of the sattelite as a function of time for the two cases: (a) the strings are wrapped in the same direction as the initial spin. R. tire. ∗ d) Express the conservation of angular momentum in polar coordinates.) 11. a millisecond before she sticks our her hands and brakes her fall (first assume her skates remain in contact the whole time and then check the assumption)? c) Find a differential equation that only involves θ. 11.s94h12p3) strings A r 11. j) The trajectory is a circle. A bicycle rim (no spokes. modeled as a uniform disk. her mass m. g.46 Cart and pendulum A mass m B = 6 kg hangs by two strings from a cart with mass m C = 12 kg. h) m(x 2 ˙ + y 2 ) + kr 2 ˙ i) The trajectory is a straight line segment. g. G is at the center of the hoop. where r = | r | ¨ CONTENTS f) Express the conservation of energy in cartesian coordinates.43: Particle on a springy leash. An inextensible string is wrapped around the hoop and attached to the ceiling. ∗ = constant j) Can the mass move back and forth on a line which is not the x or y axis? ∗ rest in the configuration shown. a) What is the path of her center of mass as she falls? (Show clearly with equations. a) Write the equations of motion in cartesian components.46: (Filename:pfigure. b) Does G move sideways as the hoop falls and unrolls? B mB yG m. [Hint: What is the acceleration of A relative to B?] a) Find the acceleration of point B just after the dumbell is released.45: problem 11. Mass B slides on a frictionless floor so that it only moves horizontally.1) (Filename:pfigure. ∗ b) Write the equations of motion in polar coordinates. A g θ problem 11. Her height is . Two point masses m at A and B are connected by a massless rigid rod with length .) b) What is her angular velocity just before she hits the ice. her tip from the vertical θ . (This equation could be solved to find θ as a function of time. stationary and rigid. (Below.53 Numerically simulate the coupled system in problem 11.5 x theta 2 1.A Ruina 4/27/95 z0 = [0.5. [’tfinal = ’ . ∗ G M function zdot=newgunfun(t. thetamax=max(z(:.50 A model for a yo-yo comprises a thin disk of mass M and radius R and a light drum of radius r. a) Draw separate free body diagrams of the bead and rod.52: Coupled motion of bead and rod and turntable. andys plot f) As t goes to infinity does the bead’s distance go to infinity? Its speed? The angular velocity of the turntable? The net angle of twist of the turntable? ∗ (Filename:pfigure.a) y A x problem 11.5. zdot = [ omega.1) 11.5 1 0. and ω1 = 1 rad/s b) Consider the special case m R = 0.1)). determine the acceleration aG of the center of mass. title(’andys plot’). mass mR collar. There is no gravity. I=2. -2*m*omega*R*v/(I + m*Rˆ2). rigidly attached to the disk.54.s94h9p4b. = 3 m. omega.num2str(tfinal.9)] ) text(-4. a) Write the equations of motion for the system from part (b) in problem 11.9)] ) text(-4.2.51 A uniform rod with mass m R pivots without friction about point A in the x y-plane.2.1)). Sketch (approximately) the path of the motion of the collar from the time the string is cut until some time after it leaves the end of the rod. that the radius of the bead is one meter and that the radial velocity of the bead.1. the rod has angular velocity ω1 . is zero.5 . (Filename:pfigure.29 has polar moment of inertia I ozz.2. ylabel(’theta’).3. plot(log10(t). d R/dt. axis([-5. omega = z(2).2) Top view 11. ∗ c) Use the equations of motion to show that angular momentum is conserved. m C = 3 kg.5. Also find the cart’s speed at the same instant. a) What is the speed of the collar after it flies off the end of the rod? Use the following values for the constants and initial conditions: m R = 1 kg. around which a light inextensible cable is wound. c) If two balls are dropped simultaneously through the tube. we show a solution using MATLAB. b) Compute the velocity of the cannon ball relative to the cart. v = z(4).) 11.52 (See also problem 11. ∗ b) Numerically integrate the equations of motion.m quiz 13 soln .25. v. tol = 1e-7 . a = 1 m. theta = z(1). v] tfinal = 10000 . Assume that at t = 0 the angular velocity of the rod is 1 rad/s.50: g d) Find one equation of motion for the system using: (1) the equations of motion for the bead and rod and (2) conservation of angular momentum. what speed does the cart have when the balls reach the bottom? Is the same final speed also achieved if one ball is allowed to depart the system entirely before the second ball is released? Why/why not? string a ω uniform rod. compute the horizontal speed (relative to the ground) that the cannon ball has as it leaves the bottom of the tube.51: (Filename:pfigure.blue.[’theta max= ’ . Assume it is free to rotate.z) m=.0]. say. 0. . A collar with mass m C slides without friction on the rod after the string connecting it to point A is cut. E 0 . 0 3]) text(-4.5 0 -5 theta max= 2. Assuming that the cable unravels without slipping on the drum. [t. Use the simulation to show that the net angle of the turntable or rod is finite. ∗ 95 %newgunscript.blue. ∗ e) Write an expression for conservation of energy. ) Assume the rod in the figure for problem 10.z] = ode45(’newgunfun’. tfinal. y ω 3 2.9)] ) ˆ ı ˆ r R problem 11. log10(tfinal). % [Theta. The cannon ball is dropped through a frictionless tube shaped like a quarter circle of radius R.53: (Filename:q13soln) x 11.[’tol = ’ . xlabel(’log10 (time)’).54 A primitive gun rides on a cart (mass M) and carries a cannon ball (of mass m) on a platform at a height H above the ground.52. Let the initial total energy of the system be. mass mC problem 11.z(:. omegaˆ2* R ]. R = z(3).25. Before the string is cut. The bead is free to slide on the rod. R.53.52 as a set of first order differential equations.Problems for Chapter 11 11. b) Write equations of motion for the system. a) If the system starts from rest.num2str(tol.num2str(thetamax.41.52859593 tol = 1e-07 tfinal = 10000 -4 -3 -2 Top view y ω -1 0 1 log10 (time) 2 3 4 problem 11. z0. tol). ] ∗ b) For what values of d0 . The mass of the pendulum bob ¨ M. so that the upper prism slides down along the lower one until it just touches the horizontal plane.58: (Filename:pfigure. it is momentarily horizontal but on the left side of the cart. and .115. The pendulum is massless except for a point mass of mass m p at the end. The cart is held from tipping over.f93q12) A C B b φ W nW φ a problem 11. There is no gravity.96 m R M L problem 11. in the direction)? ∗ M y F A problem 11. . ı .60: (Filename:pfigure.e.blue. t and a0 is the acceleration of the mass exactly vertical ˆ (i. The answer is a C = (g/3)i in the special case when m p = m c and θ = π/4. M. The rod makes an angle θ with the vertical. mass m pin connection 30 g x 30o problem 11. a) What is the force of the cart on the mass? [in terms of ˆ ˆ g. 2D.115. t. and a0 are given constants.59: (Filename:pfigure. The center of mass of a triangle is located at one-third of its height from the base. The angle of the pendulum θ with ˙ the x axis and its rate of change θ are assumed to be known. .blue. as shown in the figure.2a) released from rest problem 11. v0 . What is θ ? (Answer ˙ in terms of a A .61 Pumping a Swing Can a swing be pumped by moving the support point up and down? For simplicity. Compute the velocities of the two prisms at the moment just before the upper one reaches the bottom. The length of the massless pendulum rod is .. ∗ ˆ ı not to scale x C B θ P mP problem 11.60 Due to the application of some unknown force F.14.58 As shown in the figure.blue.2) θ x a) What is the kinetic energy of the system? b) What is the linear momentum of the system (momentum is a vector)? ˆ ˆ ı d µ=0 motor driven wheels o g y θ A m uniform rod. ] H ˆ mC C 11. m.28. g. There is a frictionless hinge at A.59 A pendulum of length hangs from a cart. Attached to the block is a uniform rod of length which pivots about one end which is at the center of mass of the block.s95q11) frictionless. Initially the mass is at rest with regard to the cart.56 Mass slides on an accelerating cart. A cart is driven by a powerful motor to move along the 30◦ sloped ramp according to the formula: d = d0 + vo t + a0 t 2 /2 where d0 . What is the acceleration of the cart just after the mass is released? [Hints: a P = a C + a P /C. v0 .blue. Let the rock be mass m attached to a string of fixed length . The cart rolls without friction and has mass m c . a) Determine the angular speed of the rod as it passes through the vertical position (at some later time).57 A thin rod AB of mass W AB = 10 lbm and length L AB = 2 ft is pinned to a cart C of mass WC = 10 lbm. neglect gravity and consider the problem of swinging a rock in circles on a string.18. the base of the pendulum A is accelerating with a A = a A i . θ and θ .blue. The system is released from rest with the rod in the horizontal position.55: (Filename:pfigure. The cart itself has a 30◦ sloped upper surface on which rests a mass (given mass m). Can you speed it up by moving your hand up and down? How? Can you make a quantitative prediction? Let x S be a function of time that you can specify to try to make the mass swing progressively faster. ∗ b) Determine the displacement x of the cart at the same instant.54: (Filename:pfigure. v0 .1) 11.) 11.57: some later time A (Filename:pfigure.55 Two frictionless prisms of similar right triangular sections are placed on a frictionless horizontal plane. ∗ c) After the rod passes through vertical. massless rollers problem 11. a0 . The cart is initially stationary and the pendulum is released from rest at an angle θ. d0 . the latter of which is free to move along a frictionless horizontal surface.56: (Filename:pfigure. How far has the cart moved when this configuration is reached? 11. The prisms are held in the initial position shown and then released. The rod and block have equal mass. The surface on which the mass rests is frictionless. The top prism weighs W and the lower one nW .1) CONTENTS 11. Use the numbers below for the values of the constants and variables at the time of interest: m θ dθ/dt d 2 θ/dt 2 x d x/dt d 2 x/dt 2 = = = = = = = = 1m 2 kg π/2 1 rad/s 2 rad/s2 1m 2 m/s 3 m/s2 11. a block of mass m rolls without friction on a rigid surface and is at position x (measured from a fixed point).3) 11. g = 9.100. y and their timederivatives.4 Advanced problems in 2D motion 11. h max .64: (Filename:pfigure.2) o C problem 11.99 and problem 8.1) 11. Define any variables you use in your solution.62 Using free body diagrams and appropriate momentum balance equations. Your equations should be in terms of θ.63: (Filename:pfigure. Try to balance a broom stick by moving your hand horizontally. and the rate of tip of the broom. See also problem 7.137. and you should probably just wonder at it rather than try to understand it in detail.1. θ2 and all mass and length quantities. θ1 . Imagine that your hand connection to the broom is a hinge. The masses of the carts and pendulum bob are m A = m B = m C = 10 kg.68. (b) What is the tension after 5 s? 11.rp11) (Filename:pfig2. find differential equations that govern the angle θ and the vertical deflection y of the system shown.59. θ2 . then find an expression for d 2 θ/dt 2 in terms of θ and r (t): 11. c) assuming the pendulum is a uniform rod of mass m and length . b) as above but with gravity. is stationary prior to impact.67 A pendulum is hanging from a moving support in the x y-plane. and g. You can model the broom as a uniform stick or as a point mass at the end of a stick — your choice.1.blue. after impact. as well as M.62: (Filename:pfig2. find the angular acceleration of the broom. as in problem 7.blue.3) 11.65 See also problem 3.Problems for Chapter 11 After Collision xs θ m problem 11.1) 97 Before collision (dotted lines) Frictionless Vo B hmax 11. ˙ ˙ θ1 . the current tip. m. For the following cases.18. as in problem 11. you will to try to balance the broom differently. The support moves ˆ in a known way given by r (t) = r (t)ı . There is no friction and the cable is initially taut. 11. You can balance a broom by holding it at the bottom and applying appropriate torques.99 or by moving your hand back and forth in an appropriate manner. Be clear about your datum for y. together with bob C.65: massless m θ m problem 11.s94h10p5) 2kg 2m 30 horizontal line 2kg problem 11.blue. and bob C is connected to cart B by a massless. In the language of robotics.blue. is an ‘open loop’ control. The first rod is massless.29. Model your hand contact with the broom as a hinge. Write two equations from ¨ ¨ and θ2 given which one could find θ1 .68 Robotics problem: balancing a broom stick by sideways motion.61: (Filename:pfigure. the system looks like the picture in the figure. Hint: ˙ M /A = H /A whole system ˙ M /B = H /B for bar BC 30 rigid surface (Newtonian frame) o y g B x θ2 A θ1 α problem 11. Find equations of motion for the second rod. each of length . 11.69 Balancing the broom again: vertical shaking works too. The lesson to be learned here is more subtle. which is simultaneously more simple minded and more subtle.68: x (Filename:pfigure. (a) What is the tension in the cable immediately after release? (Use any reasonable value for the gravitational constant). In this problem. Given the acceleration of your hand (horizontal only). ∗ b) Control? Can you find a hand acceleration in terms of the tip and the tip rate that will make the broom balance upright? ∗ a) with no gravity and assuming the pendulum rod is massless with a point mass of mass m at the end. confined to motion in the x − y plane.81 m/sec L B A C A problem 11. Find the maximum vertical position of bob C.64 Carts A and B are free to move along a frictionless horizontal surface. Now balance the broom by moving it in a way that ignores what the broom is doing. The new strategy.rp10) ˆ eθ y g φ L A m C ˆ er 11. In the previous balancing schemes. k M m m problem 11.66 A model of walking involves two straight legs. find a differential equation whose solution would determine θ(t). what you have been doing before is ‘closed-loop feedback’ control.63 The two blocks shown are released from rest at t = 0. measured clockwise from vertical. . as shown in the figure. During the part of the motion when one foot is on the ground. . inextensible cord of length . A double pendulum is made of two uniform rigid rods. you used knowledge of the state of the broom (θ ˙ and θ) to determine what corrective action to apply. Cart A moves to the left at a constant speed v0 = 1 m/s and makes a perfectly plastic collision (e = 0) with cart B which. a) Equation of motion.66: (Filename:pfigure. The wheel has k spokes (pick k = 4 if you have trouble with abstraction). (Filename:pfigure.71 A rocker. g) Stability? Can you find an amplitude and frequency of shaking so that the broom stays upright if started from a near upright position? You probably cannot find linear equations to solve that will give you a control strategy. Taking the hand motion as given. you cannot do this experiment with a broom.98 a) Picture and model. b) FBD. The broom is instantaneously at some angle from vertical. The body just swings around the contact point until the next spoke hits the ground. It can be answered at a variety of levels.70: B (Filename:pfigure. Assume that just before collision number n. The pendulum is released from rest at φ1 = 0 and φ2 = π/2. Assume that the mass of the wheel is m. the angular velocity of the wheel is ωn− . . What is the relation between ωn+ and ωn+1 ? d) Assume rolling down hill at slope θ . What is the relation between ωn+ and ωn+1 ? 11. You improve the stability a little by including a little friction in the hinge. says someone ∗ θ A L x ˆ ˆ ı yhand (h) Table Shake knife back-&-forth in this direction φ L G Press finger here problem 11.5 Dynamics of rigid bodies in multi-degree-offreedom 2D mechanisms 11.vert. R.] answer below is questionable.72 Consider a rigid spoked wheel with no rim. Just after collision n the angular velocity of the wheel is ωn+ . CONTENTS ˆ eθ y g m C ˆ er ˙ a) Given the tip angle φ. [You may assume φ and ˙ φ are small. ¨ Just after release what are the values of φ1 and ¨ φ2 ? Answer in terms of other quantities.] ∗ b) Using the result of (a) or any other clear reasoning find the conditions on the parameters (m. d) Kinematics. This problem is not easy. The Kinetic Energy is TN + . Because of the large accelerations required. The knife is different from the broom in some important ways: there is no gravity trying to ‘unalign’ it and there is friction between the knife and table that is much more significant than the broom interaction with the air. the kinetic energy is Tn− .] e) Equation of motion.shake) R problem 11. simulate on the computer the system you have found. g) find φ. Nonetheless. the tip rate φ and the values of the various parameters (m. Assume your hand oscillates sinusoidally up and down with some frequency and some amplitude. Write the equation of angular momentum balance about the point instantaneously coinciding with the hinge. the gravitational constant. Assume that when it rolls a spoke hits the ground and doesn’t bounce. [Stable means that if the person is slightly perturbed from vertically up that their resulting motion will be such that they remain nearly vertically up for all future time. this problem might best be solved by guessing on the computer. The feet do not slip on the floor.69: The sketch of the knife on the table goes with part (h). Successful strategies require the hand acceleration to be quite a bit bigger than g. you can see that this experiment might work. each of length and mass m. L. h) Dinner table experiment for nerdy eaters.71: (Filename:pfigure. ¨ R.s94q11p1) 11. Using the angular momentum balance equation. keeping the knife from sliding out from your finger but with the knife sliding rapidly on the table. The stability obtained is like the stability of an undamped uninverted pendulum — oscillations persist. So. L. The double pendulum shown is made up of two uniform bars. Rapidly shake your hand back and forth. The deeper you get into it the more you will learn. the experiment should convince you of the plausibility of the balancing mechanism. write a governing differential equation for the tip angle. Note that the knife aligns with the direction of shaking (use scratchresistant surface). the potential energy (you must clearly define your datum) is Un+ . ∗ 11. ∗ f) Simulation. a) What is the relation between ωn− and Tn− ? O φ1 A φ2 problem 11. the potential energy (you must clearly define your datum) is Un −.6 Advanced dynamics of planar motion 11. g) that make vertical passive dynamic standing stable. If you put a table knife on a table and put your finger down on the tip of the blade. Draw a FBD of the broom. The uniform spokes have length R. Use any mass distribution that you like.70 Double pendulum.s94h12p2) b) What is the relation between ωn− and ωn+ ? c) Assume ‘rolling’ on level ground. [Hint: There are many approaches to this problem. c) Momentum balance. and that the polar moment of inertia about its center is I (use I = m R 2 /2 if you want to get a better sense of the solution). Use any geometry and kinematics that you need to evaluate the terms in the angular momentum balance equation in terms of the tip angle and its time derivatives and other known quantities (take the vertical motion of your hand as ‘given’). A standing dummy is modeled as having massless rigid circular feet of radius R rigidly attached to their uniform rigid body of length L and mass m. Draw a picture which defines all the variables you will use. blue.1) . in what senses does this wheel become like an ordinary wheel? 99 k evenly spaced spokes 2R problem 11.109.72: (Filename:pfigure.Problems for Chapter 11 e) Can it be true that ωn+ = ω(n+1)+ ? About how fast is the wheel going in this situation? f) As the number of spokes m goes to infinity. z P m x R 30o massless disk problem 12. m v . does not change with time)? 12. The initial speed of the bead is v0 . with positions r C .13.2 A particle moves on a helix. There is no gravity. what is the direction of the force from the bead on the wire? i) For a helical wire. and 12. v = v (x ). Right next to the tire is the car fender (i. ˆ ˆ e) Write a in terms of et and en . The wire is twisted and curved in complicated ways. . and c) any point anywhere? 12. (a) how is linear momentum defined? (b) How is rate of change of linear momentum defined? 12. determine the force exerted on the car by the track when z = 5 m.6 For what points C. 2 constant? 12.8 A 500 kg roller coaster car travels along a track defined by the conical spiral r = z. and z = c2 t. where at a given instant in time velocity depends on position. 12.38. f) What is the radius of the osculating circle? Check your answer to see if it makes sense in the special case when c2 = 0. The only force on the bead comes from the wire.33. The radius of the tire is ro .1 Find a rotation from a picture. 12. a) Is it true that F = m a (where F is the total force on the bead and a its acceleration?)? b) Is the momentum of the bead. v E/G ? b) What is the absolute velocity of the potato eye. ˆ c) Find et at general time t.7 Bead on a stationary 3-D wire. θ = c1 t. that is.e. A disk with negligible mass has two point masses glued to it. Give examples of motions of a rigid body (you may also use a point mass) in which a) only one term at a time in the above for˙ mula for H O survives (three examples) and b) only two terms at a time in the formula survive (three examples).. Being a part-time dynamicist. 12. constant? c) Is the kinetic energy of the bead. at a particular instant in time. (The high quality potato came from the market with the ˆ ˆ center of mass already marked inside it. what is the acceleration of the point P? Is it up or down? ∗ 12. are the balance of angular momentum equations ˙ MC = H C . ˆ are basis vectors in the kitchen frame. A bead with mass m slides on a frictionless wire that is fixed in space. but it is held back by a balland-socket joint at one point on its perimeter.3 Instantaneous dynamics and “inverse dynamics” in 3-D 12. h) For a circular wire. b) Find a at general time t. and k a) What is the velocity of the potato eye relative to G.11 Disk suspended at corner by a ball and socket joint. is r E/G = 1 inı . See also problem 12. ) ı . Just after the disk is released. . ˙ θ = −2z. ˙ 12. At 5k the same instant in time. In fact the fender is rubbing on the tire just a little. ∗ 12.s94h13p2) 12. Add any extra assumptions if you feel they are required. but not quite enough to disturb any insects that might be in the neighborhood). If θ = 2 rad/s is constant.2 Velocity and acceleration of points on a rigid body: ω and α 12. point G.3 A curious chef tosses a potato in the air. she has just the right instruments in her kitchen and measures the absolute angular velocity of the potato to ˆ ˆ ˆ be ω = 1ı + 2 + 3k ( rad/s) and the absolute ˆ ˆ center of mass velocity to be v G = 3ı + 4 + ˆ ( in/s). 1 mv 2 .12 A car driving in circles counterclockwise at constant speed so that its left rear tire has speed vo (the middle of the tire. a) Find v at general time t.1 Orientation in 3-D 12.27.10 (See also problems 12. Assume that the gravitational force acts in the negative z direction. she notes the position of an eye on the potato relative to its center of ˆ mass. or the location of the ground contact point. b) the center of mass.9 When do you use which term in H O formula for a rigid body? To do the angular momentum balance about an arbitrary point O ˙ MO = H O where the most genwe write ˙ eral formula for H O for one rigid body is ˙ ˙ H O = r cm/o ×m a cm +[Icm ]·ω+ω×([Icm ]·ω). It is stationary and horizontal when it is allowed to fall. what is the direction of the force from the bead on the wire? ∗ m y 12. Remember to express the force in vector form. Say you know that a particle moves according to the equation ˆ ˆ ˆ r = R cos θ ı + R sin θ + z k where R is a constant. (say.5 For a continuous system.11: (Filename:pfigure. the fender overlaps the outer face of the tire. write an expression for the force F on the bead from the wire. v E ? d) What is the speed of the bead at t = 3 s? e) Is the acceleration always such that a · ˆ ˆ et = 0 (where et is the unit tangent to the wire)? f) Is the acceleration always such that a · ˆ ˆ en = 0 (where en is the unit normal to the wire)? g) Using any notation you like to describe the wire shape.4 Under what circumstances is the linear momentum of a system conserved (that is.35.) Find the reaction supplied by the environment to the system of bodies ABD so that each body spins at a constant rate in problem 12.) The radius of the circle that the tire travels on is Ro . ˆ d) Find en at general time t. r = ˆ ˆ ˆ r0 cos θ ı + r0 sin θ + Cθ k ). Justify your answers to the following questions with text and/or equations. correct: a) the origin.100 CONTENTS Problems for Chapter 12 General motion in 3-D 12. d. F.s94h13p3) 12. if you are really cocky.14 A square plate in space.18 A rigid body flies through the air and tumbles.37.) Find the reaction supplied by the environment to the system of bodies ABD so that each body spins at a constant rate in problem 12.13 The moment of inertia matrix. b) Find the angular acceleration of the system when released from rest in the configuration shown. What is his absolute velocity and acceleration? What is the total force acting on him? (Both bugs are at the same place at the instant of interest and walking on lines that are.3) i) A bug is climbing straight up on the car fender named at speed vg . ı . parallel. and φ is the acceleration of point B? b) For what angle φ is the acceleration of point B in the direction of the force? c) Find another direction for F so that the force at B and the acceleration at B ˆ ˆ are parallel (not necessarily in the -k plane).31. where e = {ı . At the instant of interest she is right next to the center of the wheel. Reconsider the system of problem 12.f93f5) 12. in the direction shown in the yz plane. A uniform rectangular plate of mass m = 10 lbm is floating in space.(j) of this problem are a special case of question (k). 12. F problem 12. Before the force is applied the plate is at rest in space with no other forces applied to it. you should skip the first questions and only do (k). but this time the car speed is increasing with acceleration ao . ∗ c) Some of the plate accelerates in the direction of the force and some accelerates back towards the tail of the force vector.Problems for Chapter 12 a) What is the angular velocity of the car? ∗ 101 12.4 Time evolving dynamics of an unconstrained rigid body in 3-dimensions 12. A constant force with magnitude F is applied. find the components ˆ ˆ ˆ of v = 2 m/sı − 3 m/s + 1 m/sk in the rotated coordinate system. Find the equations of motion and solve them for your choice of parameters and initial conditions. 12. Or. k . You may use your answer from problem 12. questions (a) -(j). .] d) Find yet another direction for F so that the force at B and the acceleration at B ˆ ˆ are parallel (not necessarily in the -k plane).16: (Filename:p. k }. He is in the middle of the tire at the instant of interest. Then.14: (Filename:pfigure.15 Force on a rectangular plate in space. . You should check that your answers from (k) reduce to the answers in questions (a) . a) Find the acceleration of the center of ˆ mass of the plate. 12. .)? ∗ g) What is the rate of change of angular momentum of the tire (relative to its center of mass)? ∗ h) What is the total torque of all the forces applied to the tire from the road and the car as calculated at the center of the tire? ∗ a) What is the acceleration of the center of the plate a G ? ∗ b) What is the angular acceleration of the ˙ plate ω ∗ c) What is the acceleration of point B a B ? ∗ ˆ k ˆ ı ˆ ˆ ˆ k φ ˆ ı y x z d m B d F G B 2L 2L problem 12.17 (See also problems 12. What part of the plate accelerates in the opposite direction of the applied force? 12. What is her velocity and acceleration? What is the total force acting on her? ∗ j) Another bug is climbing on a straight line marked on the tire at constant speed vb . the rotating engine is also massless) relative to the car center of mass? ∗ e) What is the total torque on the car relative to the center of the circle around which it is traveling? ∗ f) What is the angular momentum of the tire relative to its center of mass (use any sensible model of the tire.32.19 Find the rotation matrix [R] for θ = 30o ˆ ˆ ˆ ˆ ˆ ˆ such that {e } = [R]{e}.15: (Filename:pfigure. a) Find the moment of inertia matrix [I O ] of the system.(j). ∗ a = −16 ft/s2 k b) Find the angular acceleration of the plate.s96. F z m x 1 ft 2 ft y problem 12.5 Kinematics of motion relative to a moving and rotating reference frame in 3D 12.34. and 12.16 Force on a square plate. A uniform square plate with mass m and sides with length 2L is floating stationary in space. 12. Questions (a) . b) What is the angular velocity of the tire relative to the car? ∗ c) What is the angular velocity of the tire? ∗ d) What is the total torque applied to the car (not including the tires as part of the car. A force of magnitude F is suddenly applied to it at point A.p3.11 — a disk with negligible mass that has two point masses glued to it. Consider the time immediately after the time of the force application. in terms of m. Using the coordinates shown the moment of inertia matrix for this plate is: [I/cm ] = m L2 3 1 0 0 0 1 0 0 0 2 ˆ ˆ ˆ a) What. to the corner B of a uniform square plate with side length d and mass m. display your answers for the special case ao = 0. Using the rotation matrix.) k) Redo the problem.11. A force F = 5 lbf is suddenly applied at a corner that is perpendicular to the plate. at the moment. [Physical reasoning will probably work more easily than mathematical reasoning. say a rigid disk. Find the components of a = ˆ . ∗ y' b) What is the angular acceleration of the person’s arm? c) What is the linear velocity of the beer glass relative to inertial space? d) What is the linear acceleration of the beer glass relative to inertial space? e) Describe how you would point an absolutely full glass such that the beer will not spill out. 12.118. Two identical thin disks. c) Calculate the force F acting on the particle.2) − 3 2 1 2 . Determine: (a) The total angular velocity of each disk. and the particle is on the Y axis. Z mass.m F x y 12. a uniform disc of mass m.21 A rigid body B rotates in space with ˆ ˆ ˆ angular velocity ω = (2ı + 3 − k ) rad/s.25 A person sits on a bar stool which spins at angular speed .2. bodies C and D . x' problem 12. He notices that a drop of water on the wiper blade is moving along the blade at approximately 3 in/s.26: (Filename:pfigure. A set of local coordinate axes x y z is fixed to the body at its center of mass. The axle rotates about the fixed Z axis at a constant angular speed . find the absolute angular velocity of body D . Find the velocity ( r P ) of ˙ point P using the Q formula.1) C O R problem 12. as shown in the figure. rotates with respect to body A at a constant angular rate ω1 about axis P P . and 12.blue.102 z' z θ θ y x. ˆ ˆ ˆ ˆ (−2e1 +3e3 ) m/s2 in the (E E 2 .28: (Filename:pfigure. Find the rate ˆ ˆ ˆ of change basis vectors ı . a distance R from the origin. and rotates with. ωD ? ∗ b) What is the absolute angular acceleration of the disk α D ? ∗ c) What is the absolute velocity of point B. define it explicitly. the system is in the configuration ˆ shown with r C B = (D/2)k . Attached to this rod.28) CONTENTS a) What is the angular velocity of the person’s arm? of negligible mass. (s)he hoists a beer glass of mass m up at angular speed ω relative to the bar stool with a straight. as shown: a) Find the particle’s velocity relative to XY Z. m ω X O Y P R y R y' ω D B x. Find the velocity of the water-drop in George’s frame of reference. The rigid rod R rotates at constant rate about the z axis. a B ? ∗ 12. a) What is the absolute angular velocity of the disk.38. If you choose to use another coordinate system.25: (Filename:pfigure. e3 ) ˆ ˆ ˆ ˆ and (E 1 .24 George is driving his car on the highway on a rainy day. His windshield wipers are on. Body B . specify clearly his/her orientation and motion. rotates with respect to body B .s95f3a) B D 12. 12. Bodies A and B are 12. 12. at a constant angular rate ω2 about axis CC .53.26 Kinematics of a disk spinning on a spinning rod. 12.blue. X Y Z is an inertial coordinate system. b) Calculate the particle’s acceleration relative to X Y Z . as depicted below. The shaft O B rotates with angular speed rad/s. is the center C of a disk C with diameter D. ωD /F .35. Simultaneously. E 2 .19: (Filename:efig1. At the moment of interest. v B ? ∗ d) What is the absolute acceleration of point B.20 A circular plate rotates at a constant anˆ gular velocity ω = 5 rad/sk . e2 .27 See also problems 12. its angular speed is π/2 rad/s. If you use moving axes to do this problem. The position vector r P of a point P is given in local coordinates: ˙ ˆ ˆ r P = (5ı + ) m. and .123. θ measures the the angle of her/his arm with respect to the X Y plane. A set of coordinate axes x y z is glued to the plate at its center and thus rotates with ω.33. and k using the ˙ Q formula. 12. E 3 ) basis 1 1 0 0 √ if Q = 0 0 1 √2 3 2 z L ω θ m C' ω1 O' C ω2 E r P' y x problem 12. the rod. 12. and the disks roll without slipping. z O ω0 P G A L B D. of radius r are connected (perpendicularly) to opposite ends of a thin axle. Body D .23 Let B and F be two moving rigid bodB F ˙ ˙ ies. Each of the bodies is turned at constant rate by motors(not shown.29 See also problem 12.22 The rotation matrix between two coorˆ ˆ ˆ dinate systems with basis vectors (e1 . A point B is on the outer edge of the disk. Y Z lies in the plane of the paper. Body A rotates with respect to a Newtonian frame F at a constant angular rate ω0 about axis O O . Although the wipers rotate at variable angular speed. ωC and ωD . Show that ωB /F = ωB /F . rigid arm of length L. a ‘fork’.27: (Filename:Danef94s1q2) problem 12. he approximates that when the wiper is parallel to his nose. .x' xyz: fixed frame problem 12. The disk spins about its center line with constant angular speed ω rad/s relative to the shaft O B. In George’s coordinate system (attached to his frame of reference and moving with him).) At the instant shown. The disk is rotating at constant rate ω about an axis y that is attached to. E 3 ) is Q such that {e} = [Q]{E }. 12. For the instant when the disk lies in the Y Z plane.28 A particle P of mass m is attached to the edge of the disk of radius R. the wiper is rotating in the ˆ positive direction and the water-drop is movˆ ing in the positive k direction at the moment of interest. See also problem 8. a uniform disc of mass m.32: (Filename:bikefork.) For the system in problem 12. rotates about the AA axis. h. at a constant angular rate ω2 about axis CC . find ω1 in terms of ω2 .2) problem 12.s95q14) D.33 (See also problems 12.alt) x ω1 y' 12. 12.10.33: (Filename:bikefork.) For the configuration in problem 12.blue.alt) r D x ˆ k ˆ ıO ˆ C' D. The main arm is rotating at = 10 rpm and accelerating at ˙ = 5 rev/min2 . find the absolute velocity and acceleration of point Q on the disk.27.32. Point O is fixed in space. and 12. r and h. The seat for the pilot may rotate inside the capsule about an axis perpendicular to the page and going through the point B. that the pilot’s head is subjected to by choosing a moving coordinates system x yz centered at B and fixed to the capsule. particularly his ears.30 Kinematics of a rolling cone.34: 12.17.29: (Filename:pfigure. the capsule is rotating at a constant speed about CC at ωc = 10 rpm. find the velocity and acceleration of point Q on the disk using the alternative method.accel) D.27. and 12. ω2 . rotates with respect to body B .37.) At the instant shown. k }. and k ) ∗ b) At the instant shown what is the angular acceleration of the cone? (in terms of ˆ ˆ ˆ some or all of ω1 . Bodies A and B are of negligible mass.17. a) Determine the acceleration.35 (See also problems 12. find the velocity and acceleration of point the point on the edge of the disk directly above its center point E using the alternative method.33.37. rotates with respect to body A at a constant angular rate ω1 about axis P P . F ˙ ωD . ∗ d) If the disk rolls with no slip and point B has constant speed v find the magnitude of the angular acceleration of the cone in terms of v. r . and k ) ∗ 12. a ‘fork’.) 103 z O ω0 G A r C P ω 1 y F z O y G ω0 A C' D.10. at the instant shown. engineers have developed the centrifuge.m C' ω1 O' E r ω2 P' r C problem 12. . ı . z O ω0 G A P ω 1 F z O ω0 x y L P' B x y P G A L B Q E O' C ω2 (Filename:bikefork1.35. ω2 . . b) Repeat the calculations of (A) by letting x yz be fixed to the seat. The pilot sits in the capsule which may rotate about axis CC.m ω1 O' C ω2 E r C' 12.) For the system in problem 12.m P ω 1 F z L P' B x L P' B x y R R Q E O' C ω2 (Filename:Danef94s3q2) Q E O' C ω2 (Filename:bikefork1.38. and the seat rotates at a speed of ωs = 5 rpm inside the capsule.38.35: c) If the disk is rolls with no slip. A solid cone with circular-base radius r and height h moves as follows. r .ang.31. 12. at the instant shown. malfunctions of the control system. his head. 12.34 (See also problems 12.m F C' D.30: Cone rolling on a plane (Filename:pfigure.19. etc.Problems for Chapter 12 (b) The total angular acceleration of each disk. 12.119. A main truss arm of length = 15 m.31: problem 12. ∗ problem 12. Body B .31.37. When a pilot sits in the capsule. . a) At the instant shown what is the velocity of point B? (in terms of some or all of ˆ ˆ ˆ ω1 .m r r problem 12. ∗ 12. 12. shown diagrammatically in the figure.34. ˆ The cone spins about this moving axis with constant angular rate ω2 . 12. r and h. The axis of the cone O B rotates about the verˆ tical axis with constant angular velocity −ω1 ı .17. h.accel) 12. The cone surface always contacts the plane but may roll and/or slide. in terms of how many g’s. Body A rotates with respect to a Newtonian frame F at a constant angular rate ω0 about axis O O . Body D . and 12. find the absolute angular acceleration of frame D . 12. and 12.31 See also problems 12. These rotations are controlled by a computer that is set to simulate certain maneuvers corresponding to the entry and exit from the earth’s atmosphere. a distance r = 1 m from point B. P' problem 12. F ˙ ωD .36 To simulate the flight conditions of a space vehicle.32 (See also problems 12. αC and αD .34. find the absolute angular acceleration of frame D .32. (Express your answers relative to the roˆ ˆ ˆ tating basis { ı . and 12. ∗ ω2 y B h r A z O O ω0 z P G A F L B x y problem 12.ang. Each of the bodies is turned at constant rate by motors(not shown. ı .) For the configuration in problem 12. has the position shown in the enlarged figure. . blue.27. and 12. d) A pilot has different tolerances for acceleration components. ˆ ˆ − sin θ e1 + cos θ e1 .125. The shaft rotates at a constant angular speed = 120 rpm about the positive x-axis. and 12.blue. and the angular velocity of the rod. ω? b) Obtain the equations of motion of the rod in terms of the speed of point A and the angle φ. ˆ∗ ˆ ˆ∗ The unit vectors e1 and e3 lie in the (e3 − ∗ ˆ ˆ ˆ er ) plane and e2 is normal to the (e3 − er ) plane.5.34.33.6 Mechanisms in 3-D 12. Bodies A and B are of negligible mass and body D is a uniform disc of mass m. i. (Here vision becomes greatly distorted.e.ang. 12. (Here the pilot experiences blackout or may pass out completely.36: problem 12. Find the dynamic reactions on the bearings at A and B at the instant shown.104 c) Choose another moving coordinates system to solve the same problem.5 g’s. ˆ ˆ sin φ e3 + cos φ er . Rod CO is welded to the shaft and has negligible mass.39: ˆ k ˆ ˆ ı D m = 2 kg r = 0.41 A thin rod of mass m and length L moves while making contact with the frictionless surˆ ˆ face (the e1 − e2 plane) at its tip A.39 Reconsider problem 12. and y C B truss capsule bearings Y B D. θ is assumed to be constant and ˆ ˆ the rod always remains in the e3 .38 (See also problems 12. 12. Bodies A and B are of negligible mass and body D is a uniform disc of mass m.ang.2) .31. L P' B x y Q E O' C ω2 (Filename:bikefork1.32.35. θ and ω and base vectors you define (a) the velocity of the bead (a vector) and (b) the acceleration of the bead (a vector).41: (Filename:summer95f. in terms of θ. v G .37 (See also problems 12. A thin uniform disk D of mass m = 2 kg and radius r = 0.mom) 12.4 m A O 1m problem 12.40 A bead P rides on a frictionless thin circular hoop of radius R.2 m B A θ ˆ er ˆ1 e* problem 12.L ˆ e3 ˆ e1 ˆ e2 φ G ˆ2 e* ˆ3 e* ω C 0. er plane.37: 12.mom) C' D.half their tolerance levels. Find the angular momentum and rate of change of angular momentum of the system of bodies ABD about: a) point G ∗ b) point E.m F O C' ω1 O' C ω2 E r P' bearings A (Filename:pfigure.10. toe to head – 5 g’s.) Reconsider the system in problem 12. Make reasonable assumptions about moments of inertia.a) (Filename:pfigure.) Arrange a test program on the centrifuge so that each acceleration is reached separately while the other accelerations are kept below one. The starred basis vectors are written in terms of the the standard basis as follows: ˆ er ˆ1 e∗ ˆ3 e∗ ˆ2 e∗ = = = = ˆ ˆ cos θ e1 + sin θ e2 .128.. ∗ Make reasonable assumptions about moments of inertia.38: (Filename:bikefork.) (b) Front to rear – 1. 12.17.m ω r R problem 12. Find. The disk spins about its own axis at a constant rate ω = 60 rpm with respect to the shaft (or bar CO). They are roughly: (a) In a vertical direction.) Reconsider the system in problem 12.2 m is mounted on a massless shaft AB as shown in the figure. ∗ g m.2. a) What is the kinematical relation between the velocity of the center of mass.48. CONTENTS z O ω0 G A P ω 1 F 12.1) problem 12.rp10) ωc r y B seat & pilot bearing capsule x . ˙ A Z z O ω0 P G L A C x y 12.40: (Filename:pfigure4. x ωs θ P problem 12. ˆ ˆ cos φ e3 + sin φ er . The hoop rotates about its diameter at a constant angular speed ω ˙ in the vertical plane. Find the angular momentum and rate of change of angular momentum of the system of bodies ABD about: a) point G b) point E. as shown in the figure. 43 You are an astronaut in free space and your jet pack is no longer operating. There is no friction and (net) gravity is negligible. The collar has mass m D = 1 kg.48: ˆ k D ˆ ˆ ı D C u z Bead slides on rigid bent rod. Discuss whether this maneuver can be done in terms of momentum.) [For aliens and those less patriotic.Problems for Chapter 12 12.1) 12.4 m A O 1m problem 12.45 A bifilar pendulum is formed of a rod of length a and mass m.157. 12. θ and z are cylindrical coordinates that are aligned with the cylinder and mark the position of the cosmonaut relative to the cylinder. hung by three wires a distance below the ceiling.49 A thin circular disk of mass m and radius R is mounted on a thin light axle. A thin uniform disk D of mass m = 2 kg and radius r = 0. the same problem arises in the case of a trampoline jumper who decides to change direction in mid-air.g.2) ˆ e3 A L R B L a 12. Rod CO is welded to the shaft and has negligible mass. Find the absolute angular velocity and acceleration of the disk.1) m = 2 kg r = 0. is mounted on a massless semi-circular gimbal as shown in the figure. The disk spins about its own axis at a constant rate ω = 60 rpm with respect to the shaft (or bar CO). in turn.blue. What is the frequency of small ‘torsional’ oscillations about a vertical axis though the rod’s mid-point 0? (Filename:summer95f. ∗ A O z 0. What does she predict for the location of the cosmonaut (the values of θ and z) at an arbitrary later time t? You can assume that both the cylinder and the astronaut were stationary in the same Newtonian (inertial) frame. The axle. (Filename:pfigure. z = z 0 .46 A trifilar pendulum is formed of a thin horizontal disk of radius a and mass m.44 A collar slides on the rigid rod ABC that has a 30◦ bend at B as shown in the figure.44: 12.blue. The collar is currently at point D on the rod. coincident with its axis of symmetry.48 See also problem 12. and z = z 0 . Describe as best you can why these motions differ. as shown in the figure. but you are not allowed to throw anything. a) Assume a mechanism not shown constrains the collar to slide out along the rod at the constant rate u = 1 m/s (u is the distance along the rod traveled per unit time).5) x θ y problem 12.49: (Filename:pfigure.1) m 12. At the instant shown in the picture. Compute the vertical components of the reaction forces at A and B. What is the acceleration of the collar? 12. or other dynamical concepts. The . You wish to reorient your body in order to face the central government buildings so as to salute your national leaders before reentering the atmosphere.2 m B problem 12. What is the net force that acts on the collar when it is 1 m from B? b) Assume that the contact between the collar and the rod is frictionless and that no forces act on the collar besides those due to the rod (e.2 m 12. which passes through the center of the disk D.58. A nearby astronaut notes that at time ˙ ˙ ˙ ˙ t = 0: θ = θ0 . and rotates at the constant rate ω1 = 2 rad/s. (You are free to move your arms and legs. m D = 1 kg ω1 y B is mounted on a massless shaft AB as shown in the figure. The cylinder has radius a. The rod ABC is held by fixed hinges that are aligned with the z-axis. Rod in yz-plane at instant shown. The turntable spins at a ˆ constant speed about e3 . The shaft rotates at a constant angular speed = 120 rpm about the positive x-axis.47 Sometimes thrown footballs (and frisbees too!) wobble and sometimes they spin smoothly.5m 30o 1m ω x C 0. energy. Is it possible?] 2b b O a g 12.blue. 105 ω1 = 2 rad/s = const. The attachment points on the ceiling form an equilateral triangle.blue. a) If the support wires are vertical what is the period of oscillation? b) How big should the triangle on the ceiling be to minimize the period of oscillation? c) To maximize the period of oscillation? µ B A ˆ er ˆ e2 ˆ ˆ The er -e3 plane (side view) µ ˆ er problem 12.42: (Filename:pfigure. A uniform disk of mass m = 2 kg and radius R = 0. Will the reaction forces change with time? ˆ e3 g ˆ e1 problem 12. at a constant rate = 2 rad/s.7 Circles on circles and other special 3-D motions 12.50 Steady precession of a spinning disk. supported by two filaments of length and separation 2b. The raˆ ˆ dial unit vector er = cos( t)e1 + sin( t)e2 ˆ is fixed to the turntable and aligned with e1 at time t = 0.45: (Filename:pfigure.127.68.39.42 A cosmonaut who is otherwise free to move in 3-dimensional space finds himself stuck to the surface of a giant stationary cylinder with slippery magnets. The axle is supported by bearings A and B attached to a turntable. assume that u = 1 m/s (because of some initial conditions not discussed here) but is not necessarily constant in time. The gimbal rotates about the vertical x-axis. while the disk spins about its axle at a constant speed µ relative to the turntable (via motors not shown).25 m is rigidly mounted on a massless axle AB which is supported by frictionless bearings at A and B. neglect gravity). θ = θ0 . The wire attachments are equally spaced on the perimeter of the disk. 90.56: d x ˆ r ˆ ı ˆ k wheel z Rear end view y (Filename:pfigure. even though the wheel is rolling. Ignore gravity. and I x y = I x z = I yz = 0. width b.52: forces acting between the wheel and the horizontal surface. The radius of the circle that the left rear wheel travels on is R. The sphere pivots freely about O.55 A car drives counter-clockwise in circles at constant speed V .blue. The stick revolves around O at a rate of ω. and height h rolls without slip along a road which runs into the page(along the z-axis. write the body’s angular velocity ω as well as the angular velocity of the moving coordinate system relative to inertial space.5m x ˆ k ˆ ıO ˆ O r C y ˆ ı ˆ ˆ k problem 12. Assuming that the bicycle is vertical. Point O is at the height of the center of the wheel. Just when the wheel is rolling due north a pebble of mass m.) a) Using an axis system attached as shown (z lies along the shaft.51: y (Filename:pfigure. Does the right or left wheel experience the larger force? ˙ ψ cord z φ θ d y problem 12. (All products of inertia are zero. Use the coordinates given in the sketch. determine the total angular velocity and angular acceleration of the wheel of the bike in the sketch (the rest of the bicycle is not shown). H O . x is out of the page and y is chosen to produce a righthanded system).53 In problem 12.106 axle AB is spinning at a constant rate ω = 120 rpm with respect to the gimbal. [Hint: several intermediate calculations are probably required.90. what is the force that the tread causes on the pebble? Resolve this force into east.52 Rolling disk presses down. y v 12. which is stuck in the tread.1) problem 12. the axle is aligned with the zaxis and the disk D is in the x y-plane. What is the vertical reaction on the bottom of the wheel? ∗ CONTENTS wheel axle.mom) problem 12.54: h 12. (Filename:pfigure. Compute the contact forces acting on each wheel normal to the road in the following cases: a) The vehicle moves at constant speed v along a straight road. A rigid disk (idealized as flat) with radius R and mass m is rigidly attached (welded) to a rigid massless stick with length L which is connected by a ball-and-socket joint to the support at point O. circular track of radius r = 150 ft at a constant tangential speed of v = 22 ft/s on 27 in diameter wheels.s94h13p4) 12. Note: The moments of inertia of a solid sphere of mass m and radius a about any axis through the center is (2/5)ma 2 . due to symmetry.) The inertial properties about the vehicle center of mass are cm cm cm cm cm cm I x x . north and upwards components. Make additional assumptions as needed to get an answer to the problem.blue.1) 12. The radius of the wheel is r .50: (Filename:pfigure. b) What is the sphere’s inertia tensor about O? c) What is the top’s angular momentum about O? d) Compute the tension in the cord. Assume that the mass of each disk is m. d) Compute the precession rate for the slow precession case (ω ). The top of the rod is held by a horizontal cord. H O . you may assume there are no frictional 12.) At this instant.51 A cyclist rides a flat.Izz .blue. z R ω R B r D 0.54 A vehicle of mass m.29. At the instant shown. It is set spinning with a ˙ constant rate φ and precessing with a constant ˙ rate ψ. c) Calculate the rate of change of angular ˙ momentum about O. is passing directly above the .56 A child constructs a spinning top from a solid sphere with radius a and mass m attached to a massless rod of length 2d.] 12.25m 12. b) Write the disk’s angular momentum about O. b) The vehicle moves at constant speed v along a circular road (radius ρ) curving to the left.57 A thin disk of mass m and radius R is connected by a massless shaft of length L to a free swivel joint at O. a) Write the disk’s inertia tensor about O.I yy .110. Say gravity is negligible and. find the angular momentum and rate of change of angular momentum of the two disks about point O. Specify the axes that you have chosen. At the instant of interest: a) What is the angular velocity ωD of the disk? ∗ b) What is the angular acceleration of the disk? ∗ c) What is the net torque required to keep the motion going? ∗ d) What are the reaction forces at the bearings A and B? ∗ problem 12. The wheel rolls on a horizontal surface. The disk spins about the shaft at rate ω and proceeds in a steady slow precession (rate ) about the vertical.2) x b problem 12.ang. (You should assume that the wheel axle is horizontal.53: (Filename:twodisks. ball and socket O z ω L R B weld y x (Filename:s92f1p7) A D m = 2kg R = 0. in the same coordinate system. that the wheel rolls without slip and that the wheel deformation and gravity are negligible. located at x = a = 2 ft and z = b = 1 ft and representing a point in the inner ear of a passenger. 12. approximately.62 Foucoult pendulum 12.1) 12. The picture is not accurate if the precession rate is not much smaller than the rotation rate. state clearly how it moves). problem 12. Why does this simplification happen to work? 12.59 An amusement park ride consists of two cylinder-shaped cars of length = 8 ft and radius r = 3 ft mounted at opposite ends of a long arm of length 2h = 20 ft which is pivoted at its center as shown in the figure. In terms of parameters you define. Each car can also rotate about its own axis.s94h14p4) y O z x h O ω1 12.64 The path of an air mass sliding on a frictionless earth .blue. What feature picks out the solutions that are observed with real coins? 12.blue. This rotation about the y axis is called precession. Make other simplifying approximations so that you can do the arithmetic without a calculator. respectively. The picture is a reasonable approximation if the top is rotating much faster than it is precessing. This axis is in turn (so to speak) rotating about the y axis at a constant rate. Car A. Find the vectors v P and a P .59: (Filename:pfigure. is undergoing absolute velocity and acceleration v P and a P .5 rad/s in the direction shown. a) If the top is precessing as well as spinning what is the direction of the angular velocity vector? b) What is the direction of the angular momentum vector of the top? Is it parallel to the top’s axis? Is it parallel to the angular velocity vector? c) How does the angular momentum vector differ from that shown in the figure? d) Does this difference affect any predictions about the precession rate of a spinning top? e) Assume the top is on a pedestal and is horizontal as it precesses.63 Estimating the error of using the earth as a Newtonian frame 12. presently at the top of its flight. spinning at constant rate about its symmetry axis.57: (Filename:pfigure. Here are some questions to help you see why. as seen by a ground observer.8 Motion relative to the earth 12.107.58: x (Filename:pfigure.Problems for Chapter 12 y ˙ H 107 y L ω θ O H b a P z r ω2 x m R problem 12. Radius of earth ≈ 4000 miles ≈ 2 × 107 feet ≈ 6 × 106 meters.122. The angular momentum vector thus changes in time as the top precesses and the angular momentum vector also precesses. In this case the correct precession rate is predicted by the naive/approximate theory that the angular momentum is just that due to spin and that this vector precesses with the top axis. The point P. The vertical arm is rotating about axis O − O at the rate ω1 = 0. (If you are using a moving coordinate system to determine v P and a P .61 Estimate the sideways force (Pounds or Newtons) required for a person to walk a straight marked path over the North pole. is spinning (much to the discomfort of its occupants) on its own axis as shown at a constant rate of ω2 = 5 rad/s relative to the vertical arm. (hard question) The picture schematically shows what is going on when a spinning top is in steady precession.60 buzzing coin If you drop a round coin it will often go round and round progressively faster before buzzing to a stop.58 Spinning and precessing top. The top is.1) O z problem 12. what is the frequency of this buzzing as a function of the tip angle? This problem is more subtle than a straightforward calculation as there are many solutions for a coin rolling at a given tip angle in circles. The arm is constrained to rotate about the horizontal axle O − O. 23 ) (a)T AB = 30 N. and (c) ¨ ¨ m y + 2ky − 2k 0 2y 2 = F(t) ¨ ¨ 5. 11). m θ 2 sin θ 4. F1 F2 ≈ 1.2π ≈ 1. Mg ˆ ˆ c) F C = 1−2 cos θ sin θ ı + (cos θ − 2) . (b) T AB = 300 N. 4. T2 = (N − 1)mg. we will find that this twodegree-of-freedom system has two natural frequencies.74 ) N = 1000 N (ˆ + + k). 3 2. (c)T AB = 17 4. ˆ ˆ ˆ 2. F1 F2 c) = eµθ = e0. and algebra. 2.(same as (7)) ˆ 2. 2. √ ˆ ˆ 2. ı ˆ ry = r · = 5. i.87. 5.56a) m1 0 3k m. c) 1 (−2. c) ω1 = 5.83a) n = b) d = 1. 1 ˆ ˆ ı 3 (2ˆ + 2 + k). 0 0 +y 3 3 2.94 ) (0.2. (b) m x + kx = F(t). i) If the initial position D is more than 0 + 2mg/k. ω2 = x1 ¨ x2 ¨ k m.58) a B = x B ı = m1B [−k4 x B − k2 (x B − x A ) + c1 (x D − ı x B ) + k3 (x D − x B )]ˆ.. Needs somewhat involved trigonometry.77 ) d = 2 .55a) Two normal modes. 34 50 d) AO B = 34.108 ANSWERS Answers to *’d problems ˆ 2.21a) F2 = Ro −Ri sin φ 0 m2 + b) For Ro = 3Ri and µ = 0. ˙ ı 5.39a) period= 2πk = 0.3 ) Suprise! This pendulum is in equilibrium for all values of θ .68 ) No partial credit. sin b) T = mg = 2Mg 2 cos θθ−1 .22 ) h max = e2 h. F = −(kx + 5.75 ft c) period= 2 2h g + m k π + 2 tan−1 mg 2kh ≈ 1.41 ) LHS of Linear Momentum Balance: ˆ b x)ˆ + (N − mg) .17a) M = R R sin +r = 1+2 cos θ .5 m. −0. ˙ . ¨+ kx =0 f) x m ˆ ˆ g) x(t) = D − N·m. ı 50 1 ˆ ˆ ˆ b) λ O A = √ (3 + 5k). cos θ sin θ b) T = mg = 2Mg 1+2 cos θ . b) x 2 = const ∗ x1 = const ∗ (A sin(ct) + B cos(ct)). and in general Tn = (N + 1 − n)mg 4.69 ) To get chicken road sin theta. Each mass will undergo simple harmonic motion at one of the two natural frequencies only if the initial displacements of the masses are in the fixed ratio associated with that frequency. m 2 sin θ e) tan ψ = M = 1+2 cos θ .34 ) (a) m x + kx = F(t). Ro +Ri sin φ F1 4. h) period=2π m k. then the spring is in compression for part of the motion.14. ¨ ˆ ˙ 5.5 cos θ) ft.45 deg . m R sin θ 2 sin θ 4.4 m) 4. N·m.5 sin θ) ft. 2 sin θ ˆ ˆ ˆ ˆ c) F C = Mg − 2 cos θ +1 ı + (where ı and are aligned with the horizontal and vertical directions) sin θ d) tan φ = 2+cos θ . TN = (1)mg. 19.5 ) x(3 s) = 20 m N −k2 x1 0 k1 + k2 = −k2 k2 + k3 x2 0 b) If we start off by assuming that each mass undergoes simple harmonic motion at the same frequency but different amplitudes.8 ) Yes. Associated with each natural frequency is a fixed ratio between the amplitudes of each mass. A floppy spring would buckle. e) F1x = 0 f) r D O × F 1 = g) Mλ = h) Mλ = 140 √ 50 140 √ 50 + mg k cos k m t +( 0 + mg k ) 100 60 ˆ √ − √ k ˆ 34 34 N·m.e. geometry. (3 sin θ − 1.25g) TE H = 0 as you can find a number of ways.33 ) r x = r · ı = (3 cos θ + 1.where const = ±1. 5. F 2 = √ N (4ˆ + 3 + 5k).96 s m b) maximum amplitude=0. √ 5 26 2 4. 5.81a) λ O B = √1 (4ˆ + 3 + 5k).18a) M = R cos θ −r = 2 cos θ −1 . 34 5 7 ˆ ˆ ˆ ˆ ı c) F 1 = √ N (3 + 5k). its weight is exactly balanced by the upwards force of the spring at this constant position x.37b) mg − k(x − 0 ) = m x k c) x + m x = g + km0 ¨ e) This solution is the static equilibrium position. when the mass is hanging at rest.64 s.2 ) T1 = N mg. 63a) a bike = MpR fc . a B = +m √ g ˆ ˆ − 2(4m 1 +m 2 ) (2m 2 − 3m 2 )λ2 . (b) a A = 2F ı . ˆ ı 5. where λ1 is parallel to the slope that mass m 1 travels along. ˆ ı b) a(5 s) = (6ˆ + 120 ) m/s2 .132a) System of equations: x ˙ y ˙ vx ˙ vy ˙ = = = = vx vy b 2 vx vx + v 2 y m b 2 −g − v y vx + v 2 y m − 5. You need to know the angular momenta of the particles relative to the center of mass to complete the calculation.66b) TE H = 0 c) (RC x − T AB )ˆ + (RC y − ı TG D ˆ ˆ √ )k = m a = 10 Nk. Simple superposition just doesn’t work. 5. ˆ 6. ı 2 (1−µ)mg cos θ tipping if N A = 2 ga b) max( a bike )= a+b+2R f . pointing down ˆ and to the left. b) v1 = (m+m B ) v B . w deceleration of car: acar = −µg. where ı is parallel to the m ˆ m ˆ ground and points to the right. the rear of the car would flip over the front.72a) v (5 s) = (30ˆ + 300 ) m/s. deµgw celeration of car: acar = − 2(w−µh) . r = g(sin φ − µ cos φ) t2 ˆ ˆ d) v = g sin φt ı . where x is the distance ¨ measured from the unstretched position of the center of the pulley.155 ) v A = 6. √ ˆ b) a A = (4m 1g 2 ) (2m 2 − 3m 2 )λ1 . b) The cart undergoes simple harmonic motion for any size oscillation. i) Reaction at rear wheel is negative.9) 6.18a) a A = 5F ı .51d) Normal reaction at rear wheel: Nr = 2(hµ+w) .15 ) ratio of the acceleration of A to that of point B. ˆ ˆ 5. a B = 25F ı .. aA 6. mgw e) Normal reaction at rear wheel: Nr = mg − 2(w−µh) . ˆ ˆ ˆ ˆ 5. m ˆ m ˆ F F F ˆ ˆ ˆ (c) a A = 2m ı .33 ) T AB = + g) 3 6. In actuality. 6. i. 0]. 2 4k m 1− A mg .65a) TB D = 92. a B = 4F ı . mgw normal reaction at front wheel: N f = 2(w−µh) .52a) a = g(sin φ − µ cos φ)ˆ. no tipping c) No if µ < 1 since cos θ > 0 for 0 < θ < π . k mA √ 5 39 28 m(a y = m(x ı + y ). > 0.73 ) r (t) = x0 + u 0 − u 0 cos( t) ı + (y0 + v0 t) .28a) a A = − 9kd ı . m 2 +m B m A A m+m B m v2 − B 1 2 2 m AvA. w normal reaction at front wheel: N f = mg(w/2+µh) . mA ˆ b) v = 3d 6. B Pt n vt N . 5. 0.3 ) Tn = (m+m B )2 2 vB m m B kδ 2 . √ 6. Not allowing for rotation of the car in the x y-plane gives rise to this impossibility.59 ) a B = x B ı = m1B [−k4 x B − c1 (x B − x A ) + (k2 + k3 )(x D − x B )]. mgw 6.ANSWERS ¨ ˆ ˙ ˙ 5. ¨ˆ ¨ ˆ 5. where ı is parallel to the ı slope and pointing downwards b) a = g sin φ 2 ˆ c) v = g(sin φ − µ cos φ)t ı . 6. 1 5.131 ) Equation of motion: xˆ ˙ ˙ ˆ −mg − b(x 2 + y 2 ) √ı + y ˆ 2 ˙ ˙ 2 x +y ˙ ˙ t t 109 6.39 ) Can’t solve for T AB .9a) a B = m B +m B mA m b) T = 2 m AA m BB g. −m A 6.60a) v = d m .81 ) v = 2t m/s2 ı + e s m/s .6 lbm · ft/s2 .13 ) (a) a A = a B = where ı is parallel to the ground and pointing to the right. (1−µ)mg cos θ ˆ ( − µˆ ). +m 6.. 1±4 17 ]. g) No. 1.54a) R A = ˆ 6.(Here 2 µ = 0. 2 TG D √ ) ˆ 2 + (TH E + RC z + . µgw deceleration of car: acar = − 2(hµ+w) .143) No.66a) One normal mode: [1. a B = 4m ı . r = g sin φ t2 ı g F ˆ m ı. and λ2 is parallel to the slope that mass m 2 travels along.154a) v0 = m (mv B + m B v B + m A v A ).56 ) braking acceleration= g( 1 cos θ − sin θ ). Car stops at same rate for front or rear wheel skidding if h = 0. Car stops more quickly for front wheel skidding. m 2 c) (1) Eloss = 1 m v0 − 2 Eloss = 1 m 2 5. 6. pointing down and to the right. m 5. a = 2 m/s2 ı + e s m/s2 .64 ) TE F = 640 2 lbf. a B = F ˆ − m ı.22 ) angular frequency of vibration ≡ λ = 64k 65m . f) Normal reaction at rear wheel: Nr = mg(w/2−µh) .36 ) ax > 2 g 6.e. normgw mal reaction at front wheel: N f = mg − 2(hµ+w) . b) The string will go slack if ω > 6. √ b) TG H = 5 61 lbm · ft/s2 .27a) m x + 4kx = A sin ωt + mg. − (m + m B )v 2 . 2 k 6. (d) a A = m ı . information which is not given. ˆ 6. h) No reaction at rear wheel.62a) ω = 2k .118 ) T3 = 13 N 5. √ b) The other two normal modes: [0. a B = 81. F L 6. 71a) FL = 1 m tot g. RC z = 5 N. 4 ı 4 7.35a) 2mg.51 ) a t = 20 in/s2 .47a) The velocity ˆ ˆ of departure is v dep = k( m ) − 2G R .72 ) sideways force = FB ı = wma ı . ¨ 7. ˆ er ˆ et v a = = ˆ −ˆ ı 2πr ˆ = − ı τ 4π 2r ˆ = − 2 . θ = α ˙ f) Tmax = 30N 2 7.3 ) ˆ ı • one solution is r = R(cos(ωt)ˆ + sin(ωt) ) with R and ω constants.41b) The solution is a simple multiple of the person’s weight.110 d) e) RC x − T AB = 0 TG D RC y − √ = 0 2 TG D RC z + √ + T E H = 5 N 2 TG D −TE H + √ − RC z = 0 2 TG D −TE H − √ + RC z = 0 2 TG D − √ + RC x − RC y = 0 2 √ Mcm = ( TG D − TH E − RC z )ˆ + (RC z − ı 2 ˆ √ ˆ TH E ) + (T AB + RC − RC − TG D )k = 0 x y ANSWERS TG D √ 2 − and for θ = 210◦ . ı ˆ c) a P = −4 m/s2 . τ2 d) Tension is enough. 7. 0. (H O ) I I = 0. ˆ 0.43a) θ = −(g/L) sin θ ˙ d) α = −(g/L) sin θ. F net = −mg . 6.g.3 N 7.85 kW c) 750 rev/min d) 100 N·m.014 N·m · sk. ˆ c) Position-A: (H O ) I = 0.012 N·m · sk. 7. b) v P = −2 m/sˆ.012 N·m · sk. ˆ ˆ 6.0080 N·mk. where λr is a unit vector pointing in the direction of the rack.. 7. ˆ er ˆ et v a 2 2 √ 3 1 ˆ ˆ = − ı− 2 √ 2 1 3 ˆ ˆ ı− = 2√ 2 πr 3πr ˆ ˆ + ı = − τ τ √ 2 2 3π r 2π 2r ˆ ˆ = ı + 2 .33b) T = 0.2 m 7. T AB = 5 N.65 ) ωmin = 10 rpm and ωmax = 240 rpm 7.04π 2 kg·m/sk c) H O =√ O c) F = mw m tot (2T ˆ − D − 2FD ) − T + FD ı + ˆ ı (m w g − FL ) and M = (bFL − am w g)ˆ + mw ˆ (bFD − cT ) + a m tot (2T − D − 2FD ) .(H O ) I I = ˆ ˆ Position-B: (H ) I 0.52 lbf = 2.. ˆ ˆ 7.27b) (H O ) I = 0.] 7.31 ) 0 = 0. b) Just before leaving the tube the net force on the pellet ˆ is due to the wall and gravity. ˆ er ˆ et v a for θ = 90◦ .17 ) F = 0. F net = −mg − | v dep |2 ˆ m R ı . Just after leaving the tube. RC y = 5 N. • general solution is any motion where speed = v = | v | is a constant.√ b) ω = 99g/r c) r ≈ 1 m (r > 0. tot c) T = 4mπ r . the net force ˆ on the pellet is only due to gravity. TG D = √ N.16π 4 N . e. k−mωo 7. ˙ ˙ ˆ 7. 7.012 N·m · sk. g) Find moment about C D axis.85 kW b) 7. O ˆ (H ) I I = 0.38 ) (b) θ + 2L sin θ = 0 7. ˙ R b) v = v0 e−µθ . 2 ˆ b) a P = m1 [2(T − FD ) − D] ı . ( MC = r cm/C × ˆ ˆ m a cm ) · λC D .76a) 7. where λC D is a unit vector in the direction of axis C D. a n = −250 in/s2 ı . 2 TE H = 0 N. where is perpendicular to the curved end of the tube. down and to the right.24b) For θ = 0◦ . • another solution is ˆ ˆ r = [At ı + Bt 2 ]/ 1 + 4B 2 t 2 /A2 .44a) v = −µ v . τ = = ˆ ı ˆ 2πr ˆ = τ 4π 2r ˆ = − 2 ı. 2 7. ˆ ˆ d) v r = 1 m/sλr .29 ) r = kro 2 . 2 τ τ T AB 10 f) RC x = 5 N.79a) v P = 2 m/s. τ ˆ d) r = [ 22 − v cos( π t )]ˆ + [ 22 + v sin( π t )] .98 m) ¨ 3g 7. 2 √ . ANSWERS e) No force needed to move at constant velocity. 7.80a) Pin = 7.33kilo-watts b) 500 rpm c) Mout = 140 N·m 7.84a) α B = 20 rad/s2 (CW) b) a = 4 m/s2 (up) c) T = 280 N. 7.86a) FB = 100 lbf. c) vright = v. 7.94b) (b) T = 2.29 s (e) T = 1.99 s (b) has a longer period than (e) does since in (b) the moment of inertia about the center of mass (located at the same position as the mass in (e)) is non-zero. ¨ 7.98a) θ = 0 rad/s2 . ¨ b) θ = sin θ (Dk − mg). m ¨ 7.99b) −F(t) cos φ − mg sin φ + Tm = −m 2 φ. ˆ ı 7.100a) (a) F = 0.33 Nˆ − 0.54 N . 2 7.101a) T (r ) = mω (L 2 − r 2 ) 2L b) at r = 0; i.e., at the center of rotation √ c) r = L/ 2 7.103 ) Izz = 0.125 kg · m2 . 7.104a) 0.2 kg · m2 . b) 0.29 m. 7.105 ) At 0.72 from either end 7.106a) (Izz )min = m 2 /2, about the midpoint. b) (Izz )max = m 2 , about either end 7.107a) C b) A A B c) Izz /Izz = 2 √ C d) smaller, r gyr = Izz /(3m) = 2 O O 7.108a) Biggest: Izz ; smallest: I yy = I xO . x 3 2 O O O = 3m 2 . b) I x x = m = I yy . Izz 2 d) r gyr = . gL ( M+ m )+K 2 7.113a) ωn = 2 . ( M+ m ) L 2 +M R2 3 gL ( M+ m )+K 2 . Frequency higher than in (a) b) ωn = ( M+ m ) L 2 3 cm 7.114a) Izz = 2m 2 b) P ≡ A, B, C, or D √ c) r gyr = / 2 O 7.115 ) I xO = I yy = 0.3 kg · m2 . x 2m b hx/b 2 O (x + y 2 ) d y d x. 7.117a) Izz = bh 0 0 7.128a) Point at 2L/3 from A b) mg/4 directed upwards. 2π 1 d 7.129c) T = √ 12(d/ ) + g) d = 0.29 2 2 ˆ ı 7.132a) Net force: F net = −( 3mω L )ˆ − ( mω L ) , Net mo2 2 ment: M net = 0. g/ 111 b) Net force: F net = −( 3mω L )ˆ + (2mg − ı 2 ˆ Net moment: M net = 3mgL k. 2 7.133 ) Tr ev = 2M O 2 π 3F R p . 2 mω2 L ˆ 2 ) , 7.139 ) period = π 2m . k ˆ ˆ ¨ k =? rad/s2 k (oops). 7.141a) α = θ b) T = 538 N. 7.142 ) v B = 2gh[m B −2m A (sin θ+µ cos θ )] 4m A +m B +4m C KC RC 2 . 7.143 ) a m = 0.188 m/s2 ˆ 8.13a) r P/O = 0.5 m (cos φ)ˆ + sin φ . ı ˆ b) vP = (0.5ω sin φ k) m/s, 2 sin φ ) m/s2 . ˆ a P = (−0.5ω 8.23a) TD B = 0.9 N. b) F A = −0.3 Nˆ, F C = −0.6 Nˆ ı ı c) Torquemotor = 0. 8.25a) Angles φ for steady circular motion are solutions to 10 tan φ = 8 + 40 sin φ. b) φ = [1.3627 rad, −0.2735 rad, −1.2484 rad, −2, 9813 rad], four solutions, found numerically. Graphically, the solutions are the interns of the curves f 1 (φ) = 10 tan φ and f 2 (φ) = 8 + 40 sin φ between −π and π. 8.28b) Period of revolution= 2π , Period of simple penduω lum (for small oscillations)= 2π L , where is the g length of the pendulum. c) T = mω2 L (strange that φ drops out!), v r ock = H ˆ ˆ −ωL sin φ ı , a r ock = −ω2 L sin φ , ddt x = −mωL 2 sin φ cos φ. 8.29 ) φ = cos−1 ( ωgL ). 2 8.32 ) (a) T = −69.2 N, (b) 1.64 rad/s < ω < 3.47 rad/s 8.36 ) x = −0.554 m ˙ ˆ 8.37b) H = − 1 mω2 2 sin φ cos φ k. 1 ˆ c) F A = ( 12 mω2 2 /d sin φ cos φ) , F B = −F A . 1 ˆ d) F A = ( 12 mω2 2 /d sin φ cos φ + 1 mg) , F B = 2 −F A . √ 2 8.40 ) T = 2mω R . 6 8.57a) ω × ([I G ] · ω) = 0 kg·m2 /s2 . b) ω × ([I G ] · ω) = −25ˆ kg·m2 /s2 . ı 8.66b) ω2 = 2L 3g φ , Tr ev = 2π . cos ω O 12 8.67a) ω2 = 8.68a) 3g n L(2n+1) cos φ . 4 [I O ] = m L 2 0 0 0 2 −2 0 −2 , 2 in terms of the coordinate system shown. ˆ ˆ b) H O = 2m L 2 ω(− + k). ˙ = −2m L 2 ω2 ı . ˆ c) H O ˆ d) F = 0, MO = −2m L 2 ω2 ı . 112 8.69c) R O = − mω 8.70a) a D = My cm IY Y 2 bh(b2 −h 2 ) ANSWERS ˆ k = −R C . ˆ 9.38a) α Disk = − 4 rad/s2 k. 3 8 2ı. b) a A = 3 m/s ˆ ı 9.40a) aC = (F/m)(1 − Ri /R O )ˆ. ˆ b) (−Ri /R O )F ı . ˙ 9.45a) speed v = R θ . 3 3 12(b2 +h 2 ) 2 √ 2dc √ b) B = − c2 +d 2 cm M y IY Z ˆ ı. ˆ ı. c) R A = −4.56 Nˆ, R C = −13.439 Nˆ, Mz = 0. ı ı 8.90b) M Az is always zero. c) No. See problems 8.68 and 8.37 for counter-examples. ˙ ˙ ˆ 9.1 ) plot(2): θ (t) = br (t) and v = θ er + θbθ et . bˆ 9.1 ) plot(2): x = r cos(br ) and y = r sin br or x = θ θ b cos(θ ) and y = b sin θ ˆ ˆ ˆ 9.12 ) v P = ω1 L(cos θ − sin θ ı ) + (ω1 + ω2 )r ı , a P = 2 ˆ ˆ ˆ −ω1 L(cos θ ı + sin θ ) + (ω1 + ω2 )2 . 9.17a) ˆ et = t 2t s ˆ ˆ sı+e ı 8.79a) H O = (−0.108ˆ + 0.126k ) kg·m2 /s b) H O is in the x z plane at the moment of interest and at an angle α ≈ 41◦ , clockwise from the z -axis. c) H O does not change in magnitude (m, R, ω, β are constants). H O rotates with the disk; i.,e., it changes direction. ˙ d) H O = the rate of change of H O . The tip of H O goes in a circle. At the instant of interest, the change will ˙ ˆ ˆ be in the or direction. Thus, MO = H O will be in the√ˆ direction. ˙ ˆ ˆ 8.82a) H cm = 9 3 N·m = 0.122 N·m 128 ˙ = −26.89 N·m ˆ b) H A cm 2IY Y c2 +d 2 4 ts2 + e 4t m s3 2 2 2t s at = + e s m/s2 2t s 2 2t 4 ts2 + e 2t an ˆ en = = ˆ (2 − 2 ts )e s m/s2 ı + (4 ts2 4 ts2 + e ˆ ˆ e s ı − 2 ts 4 ts2 + e 2 2 2t s 2t t 2 2t s ρ = (4 ts2 + e s ) (2 − 2 ts )e s t 9.47 ) vo = 2 3 . ˆ 9.53 ) Accelerations of the center of mass, where ı is parallel ˆ to the slope and pointing down: (a) a cm = g sin θ ı , 2 1 ˆ ˆ (b) a cm = 3 g sin θ ı , (c) a cm = 2 g sin θ ı , (d) a cm = 2 ˆ g sin θ ı . So, the block is fastest, all uniform disks 3 are second, and the hollow pipe is third. 9.65 ) h = 2L/3] ı ı 10.11 ) (g) v P = 8 m/sˆ, a P = 0; (h) v P = 3 m/sˆ, a P = 0. ı ˙ ˆ 10.12 ) a b = −(s 2 A + aω2 + 2ω D vb )ˆ − a ω D k. D 10.17b) One case: a P = a O , where P refers to the bug; the bug stands still on the turntable, the truck accelerates, and the turntable does not rotate. c) One case: a P = a O + a P/D , where D is the turntable; the bug accelerates along a straight line on the turntable, the truck accelerates, and the turntable does not rotate. t ˆ − 4 ts )e s m/s2 10.18a) ω ˙ˆ O A = θ k. ¨ˆ b) α O A = θ k. ˙ ˙ ˆ ˆ c) v A = −R θ sin θ ı + R θ cos θ . ˙ ¨ ¨ d) a A = −(R θ 2 cos θ + R θ sin θ )ˆ + (R θ cos θ − ı ˙ ˆ R θ 2 sin θ ) . ˙ e) ω AB = √−R θ cos θ 2 . f) θ = (2n + 1) π , n = 0, ±1, ±2, . . .. 2 ˆ ˆ 10.21 ) ω D E = − ω k = −1.5 rad/sk 2 2k ˆ 10.22 ) α D E = 3.5 rad/s ˙ˆ 10.29 ) One situation: v = R θ eθ ;the case of a particle constrained to move in a circle. ¨ˆ 10.30 ) One situation: a = R e R ; the case of no rotation ˙ = θ = 0). ¨ (θ ˆ 10.31 ) ω D = (ω1 + ω2 )k. 10.33a) ˆ et = t 2t s ˆ ˆ sı+e 2 −R 2 sin 0 b) The energy lost to friction is E f ric = 6 . The energy lost to friction is independent of µ for µ > 0. Thus, the energy lost to friction is constant for given ˙ m, R, and θ0 . As µ → 0, the transition time to rolling → ∞. It is not true, however, that the energy lost to friction → 0 as µ → 0. Since the energy lost is constant for any µ > 0, the disk will slip for longer and longer times so that the distance of slip goes to infinity. The dissipation rate → 0 since the constant energy is divided by increasing transition time. The energy lost is zero only for µ = 0. 9.46 ) V = 2 m/s. √ ˙ m R2 θ 2 2g R θ m F ˆ 9.22a) a cm = m ı . 6F ˆ b) α = − m k. c) a A = 4F ı . m ˆ d) F B = F ı 2ˆ F ˆ 9.25a) a = m . Fd ˆ b) α = R2 k. m 9.34a) Fout = lb. b) Fout is always less than the Fin . 3 7 2 4 ts2 + e 2 2t s ANSWERS 4t m s3 113 + e s m/s2 2 2t s 2 2t at = 4 ts2 + e 2t an ˆ en = = ˆ (2 − 2 ts )e s m/s2 ı + (4 ts2 4 ts2 + e ˆ ˆ e s ı − 2 ts 4 ts2 + e 2 2 2t s 2t t 2 2t s ρ = (4 ts2 + e s ) (2 − 2 ts )e s t m 2 ¨ 11.9a) R(t) − ωo R(t) = 0. d 2 R(θ) dθ 2 ˆ ˆ 11.45a) −k(x ı + y ) = m(x ı + y ). ¨ˆ ¨ ˆ ˙ ˆ ¨ ˆ b) −kr er = m(¨ − r θ 2 )er + m(r θ + 2˙ θ )et . r r˙ ˆ d c) dt (x y − y x) = 0. ˙ ˙ d t ˙ d) dt (r 2 θ ) = 0. ˆ − 4 ts )e s m/s2 ˆ ˆ e) dot equation in (a) with (x −y ı ) to get (x y −y x) = 0 ¨ ¨ which can be rewritten as (c); dot equation in (b) with ¨ ˆ r˙ et to get (r θ + 2˙ θ ) = 0 which can be rewritten as (d). f) 1 m(x 2 + y 2 ) + 1 k(x 2 + y 2 ) = const. ˙ ˙ 2 2 ˙ g) 1 m(˙ 2 + (r θ )2 ) + 1 kr 2 =const. r 2 2 ˙ˆ ˙ ˆ ¨˙ h) dot equation in (a) with v = x ı + y to get m(x x + y y ) + k(x x + y y) = 0 which can be rewritten as (f). ¨˙ ˙ ˙ i) x = A1 sin(ωt + B1 ), y = A2 sin(ωt + B2 ), where k ω = m . The general motion is an ellipse. j) Yes. Consider B1 = B2 = 0, A1 = 1, and A2 = 2. 11.46a) 0.33 mˆ ı b) v C = 1.82 m/sˆ. ı c) T ≈ 240 N. 11.52b) b) − R(θ) = 0. c) R(θ = 2π ) = 267.7 ft, R(θ = 4π) = 1.43 × 105 ft. d) v = 2378.7 ft/s e) E K = (2.83 × 106 ) × (mass)( ft/s)2 . ˙ ¨ 11.10a) R = R0 ω2 − 2µ R0 ω. b) φ = sin−1 11.19a) ω− = 4 rad/s, where the minus sign ‘-’ means ‘just before leaving’. b) ω+ = 4 rad/s, where the plus sign ‘+’ means ‘just after leaving’. c) vr = 3.841 m/s, vt = 2.4 m/s. d) torque=0.072 N·m ˆ 11.20 ) F = −0.6 N . √ 34 11.22a) TBC = m 13 (2ax + g). 1 ˆ ı b) a = 34 [(25ax + 15g)ˆ + (15ax − 25g) ] c) TBC = m √1 (5ax + 3g). 34 ˆ 11.28a) H O = m L 2 ω1 + 1 mr 2 (ω1 + ω2 ) k. 2 ˙ = 0. b) H O ˆ c) H C = 1 mr 2 (ω1 + ω2 )k. 2 ˙ = 0. d) H C ( √ 1 1+µ2 −µ)2 +1 . ¨ ˙ R − Rθ 2 ˙˙ ¨ (Izz + m R )θ + 2m R R θ 2 = 0 = 0 c) The second equation in part (b) can be rewritten in the d ˙ form dt Izz + m R 2 )θ = 0. The quantity inside the derivative is angular momentum; thus, it is conserved and equal to a constant, say, (H/o )0 , which can be found in terms of the initial conditions. (Ho )2 0 ¨ d) R − R = 0. O e) E 0 = f) The bead’s distance goes to infinity and its speed approaches a constant. The turntable’s angular velocity goes to zero and its net angle of twist goes to a constant. 11.53a) ˙ θ ω ˙ ˙ R v ˙ 11.55 ) v2x = = = = = 2 (Izz +m R 2 )2 1 ˙2 1 2 m R + 2 ω(H/o )0 . ˆ ı 11.29 ) F = m a = −109.3 Nˆ − 19.54 N . 3g ¨ 11.31b) θ = − 2L cos θ. ˆ ¨ ˙ c) From the diagram, we see ω AB = −θ(t = 0)k = 3g 3 ˆ ˆ 2L cos θ0 , a G = 4 g cos θ0 [sin θ0 ı − cos θ0 ]. 3 ˆ d) R A = 4 mg sin θ0 cos θ0 ı , R B = mg[1 − 3 ˆ g cos2 θ0 ] . 4 ˆ e) a B = 3 g sin θ0 cos θ0 ı . 2 f) At θ = 11.37 3g θ0 θ0 ˆ 2 , ω AB = L (sin θ0 − sin 2 )k, a A = ˆ −3g 1 cos2 θ20 + sin θ20 (sin θ0 − sin θ20 ) . 2 2 poundals ) There are many solution methods. −8π ω − v ω2 R , v2 y = v2x n+1 tan φ, n 2m Rvω Izz + m R 2 2g(a−b) tan φ to the left. (Note: 1 poundal = 1 ft · lbm · s−2 .) (Mc +6Mw )v 2 0 . 11.40a) = k b) Tangential forces point toward the wall! c) = 2 (Mc +4Mw )v0 . k 2 refers to the top v1 x = wedge and 1 refers to the lower wedge. √ ˆ 4ˆ 11.56a) F = m(g + a0 ) 43 ı + 3 . b) For a0 = −g, the acceleration of the mass is exactly vertical; d0 , v0 , and t could be anything. 11.57a) angular speed = 8.8 rad/s. b) displacement = 0.5 ft. ¨ 11.68a) For point mass: φ = (g/L) sin φ − (ahand /L) cos φ. 1 (1+ n + n+1 tan2 φ) n v2x − n , v1 y = 0, where ˆ 12. M = 0. m ˙ = (96ˆ + 48 ) rad/s2 ˆ b) ω ı 2 mvo r ˙ ˙ ˆ g) H cm = H cm = − 2R et .52 ) N = ω m R k 2 L h) Mcm = − F bug = m bug a bug .39 ) A y = −3. where y(t) is the vertical displacement of your hand and φ(t) is the angle of the broom from the vertical. F ˆ 12. φ2 = −3g/2 . R r ˆ d) total torque.33 ) ωD = −ω0 ω1 ı + ω0 ω2 − ω1 ω2 k. ω1 h 2 ˆ 12. 2 b) This first-order approximation show that stable standing requires R > L .38a) H /G = 1 mr 2 ω2 ı + 1 mr 2 ω1 + mω0 (L 2 + 1 r 2 )k. ı 2 R ı + ωD − 1 ω2 D k.48 ) ωD = 2π(2ˆ + ) rad/s and α D = 8π 2 k rad/s2 ˆ 12. Show your result with appropriate plots.114 b) There are many correct solutions. ˆ H /G = 2 mr 0 1ˆ 0 2ˆ 1 2 12. e) M = 0.70 ) φ1 = 0. ˆ ˆ ˆ 2 d) a B = − ˆ ˆ ˆ 12. ˆ M /G = 2 mr 0 1ˆ 0 2ˆ 1 2 ˆ 12. ANSWERS ˆ ˙ b) H /E = 1 mr 2 ω2 ı + 1 mr 2 ω1 + 1 mr 2 ω0 k. H /E = ˆ 4 ˆ 4 2 1 ˙ 2 (−ω ω ı + ω ω − ω ω k). 2 mvo r ˆ 2R et .14a) a G = m k.30a) v B = − √ k. vo ˆ 12.10 ) Net force: F = m(−omega0 L) .26a) ωD/F = k + ω .71a) φ + 3(R − L ) L 2 φ = 0.12a) ωcar = R k.69e) φ − y +g sin φ = 0.16 N. g ¨ 11. Test your solution with a computer simulation.16 N ˆ ˆ ı 12.732g k. ¨ ¨ 11.15a) a B = 4F k. Net moment: 1 2 (−ω ω ı + ω ω − ω ω k). 2 4 1 ˙ 2 (−ω ω ı + ω ω − ω ω k). 2 vo vg vo ˆ ˆ i) a bug = − R er + r et . ˆ ˆ 12.16 N.7i) −(cos θ ı + sin θ ). R = L is neutrally stable 2 2 (like a wheel). 2 ˆ 12. ¨ ¨ 11. ı 4F ˆ c) a B = m k. 1+ r 2 h . ˆ ˆ 12. B y = 3.11 ) a P = 0. 2 ˆ 12. ˆ 2 c) v B = R + 1 ωDˆ. ˆ H /G = 2 mr 0 1ˆ 0 2ˆ 1 2 .27 ) ωD/F = ω2 ı + ω1 + ω0 k. where er is parallel to the axle r ˆ of the wheel and moves with the car and k is perpendicular to the ground. h +r 2 v2 = hr .50a) ωD = (2ˆ + 4π k) rad/s ı ˆ b) α D = −8π rad/s2 ˆ c) M = − π N·m 2 d) R A = π Nˆ. ˆ ˆ 4 ˆ 12. ˆ ˙ ˆ ˆ 12. A z = Bz = −63. ı b) α D/F = − ωˆ. ˆ b) ωtir e/car = − vo er .36b) a P = −( ˙ +2r ωs )ˆ +(2r ωc − 2 ) −r (ωc + ı 2 ˆ ωs )k. R B = −R A 2 ı 2 ˆ 12. b) α C/F = − c) ω1 = √ ω2 2 d) α C/F F h 2 +r 2 ω1 ω2 1+ r 2 h ˆ k. 2 2 1 ˆ o ˆ f) H cm = mv4 r − r er + R k . 3F ˙ ˆ b) ω = m L (ˆ − ). ˆ c) ωtir e = vo k − vo er .
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