11/7/2014HW-8: Ch. 19 - Kinetics of an ERO-2D (Impulse-Momentum) HW-8: Ch. 19 - Kinetics of an ERO-2D (Impulse-Momentum) Due: 11:59pm on Sunday, July 20, 2014 To understand how points are awarded, read the Grading Policy for this assignment. Principle of Impulse and Momentum Learning Goal: To be able to solve problems involving force, moment, velocity, and time by applying the principle of impulse and momentum to rigid bodies. The principle of impulse and momentum states that the sum of all impulses created by the external forces and moments that act on a rigid body during a time interval is equal to the change in the linear and angular momenta of the body during that time interval. In other words, impulse is the change in momentum. The greater the impulse exerted on a body, the greater the body’s change in momentum. For example, baseball batters swing hard to maximize the impact force and follow through to maximize the impact time. This principle holds true for both linear and angular impulse and momentum. For a rigid-body’s planar motion, the equations for the linear impulse and momentum in the x–y plane are given by m(v Gx ) m(v Gy ) 1 1 + ∑∫ + ∑∫ t2 t1 t2 t1 F x dt = m(v Gx ) 2 F y dt = m(v Gy ) 2 Similarly, the equation for the principle of angular impulse and momentum about the z axis, which passes through the rigid-body’s mass center G , is given by IG ω1 + ∑ ∫ t2 t1 M G dt = I G ω2 Part A - Angular velocity of the pulley The pulley shown has a moment of inertia 2 IA = 0.900kg ⋅ m , a radius r = 0.300m , and a mass of 20.0 kg. A cylinder is attached to a cord that is wrapped around the pulley. Neglecting bearing friction and the cord’s mass, express the pulley’s final angular velocity in terms of the magnitude of the cord’s tension, T (measured in N), 2.00s after the system is released from rest. Use the principle of angular impulse and momentum. Express your answer numerically in radians per second to three significant figures. Hint 1. How to approach the problem 1. Draw a free-body diagram of the pulley showing all the forces and couple moments that http://session.masteringengineering.com/myct/assignmentPrintView?assignmentID=1182016 1/20 11/7/2014 HW-8: Ch. 19 - Kinetics of an ERO-2D (Impulse-Momentum) produce impulses on the pulley. 2. Express the final angular velocity, ω2 , in terms of T by applying the principle of angular impulse and momentum, which states that the final angular momentum, adding the initial angular momentum, IA ω1 IA ω2 , is obtained by , and the angular impulses of moment \itM A during the time interval. Hint 2. Complete the free-body diagram of the pulley Complete the free-body diagram of the pulley by adding the forces that act on it. Draw the reactions at A ending at point A and pointing in the positive x and y directions. Draw the other vectors starting at the dots on the pulley’s circumference. The starting or ending point and orientation of your vectors will be graded. The length of your vectors will not be graded. ANSWER: This content requires Adobe Flash Player 10.0.0.0 or newer. Hint 3. Identify what is needed to apply the principle of angular impulse and momentum Which of the following statements are relevant when applying the principle of angular impulse and momentum to the pulley? Check all that apply. ANSWER: The initial angular momentum of the pulley is zero. The pulley’s angular momentum is the product of the pulley’s moment of inertia and the angular velocity. The pulley’s angular momentum is the product of the pulley’s mass and the angular velocity. The angular impulse is determined by time integration of the moments about point A during the 2.00s interval. The final angular momentum of the pulley is zero. Hint 4. Angular impulse generated by the tension What is the angular impulse generated by the tension in terms of the tension’s magnitude, T ? ANSWER: 2.00T 0.300T 0.600T ANSWER: http://session.masteringengineering.com/myct/assignmentPrintView?assignmentID=1182016 2/20 0. The angular impulse is determined by time integration of the moments of all external forces and the applied couple moments. ANSWER: This content requires Adobe Flash Player 10.667 T radians/s All attempts used. is obtained by adding the initial linear momentum. correct answer displayed A change in the angular momentum of a rigid body is caused by an angular impulse acting on the body. The starting point and orientation of the vectors will be graded.com/myct/assignmentPrintView?assignmentID=1182016 3/20 .(vB )2 .11/7/2014 HW-8: Ch. determine the final velocity of the cylinder of mass m = 12. Relate the cylinder’s final velocity with the pulley’s final angular velocity using rigid-body kinematics. Draw the vectors starting at the black dots.0 or newer. and the impulses exerted by the tension and the weight during the time interval. Hint 3.masteringengineering.0kg that is attached to the pulley.Principle of linear impulse and momentum For the same system. Express the tension in the cord in terms of the final velocity of the cylinder Which of the following is the correct expression for the tension’s magnitude? ANSWER: http://session. Part B . How to approach the problem 1. Express your answer to three significant figures and include the appropriate units. Solve the simultaneous equations to calculate the final velocity of the cylinder. The length of the vectors will not be graded. 19 . 3. Draw the free-body diagram of the cylinder showing all the forces that produce impulses on the cylinder. Hint 1. Hint 2. by applying the principle of linear impulse and momentum.Kinetics of an ERO-2D (Impulse-Momentum) ω2 = 0. Complete the free-body diagram of the cylinder Complete the free-body diagram of the cylinder by adding the forces that act on it. which states that the final linear momentum. m(vB )2 .0. 2. m(vB )1 . 4. Express the magnitude of the tension in the cord in terms of the final velocity. Identify the final velocity of the cylinder Which is the correct expression for the final velocity of the cylinder? ANSWER: (v B ) = (v B ) = (v B ) = 2 2 2 9.masteringengineering.7 m s All attempts used. 19 .com/myct/assignmentPrintView?assignmentID=1182016 m/s 4/20 .81m + (vB )2 ( T = m + (v B )2 ( m t m t ) ) T = 9. correct answer displayed A change in the linear momentum of a rigid body is caused by a linear impulse acting on the body.81mr I +mr 2 ANSWER: (v B ) 2 = 10. Part C . Relate the angular velocity of the pulley and the velocity of the cylinder Which of the following is the relationship between the angular velocity of the pulley and the velocity of the cylinder? ANSWER: (v B )2 = r ω2 (v B )2 = r ω2 (v B )2 = (v B ) 2 ω2 r = ω2 Hint 5.Kinetics of an ERO-2D (Impulse-Momentum) T = m − (v B )2 ( m t ) T = 9.Principle of angular impulse and momentum applied to the entire system kg http://session. The linear impulse is determined by integrating the external forces with respect to time.81m − (vB ) ( 2 m t ) Hint 4.11/7/2014 HW-8: Ch.81mtr I +mr mtr 2 2 2 I +mr 2 2 9. 300m .0 or newer. Draw the impulse and momentum diagrams of the system showing all the forces and couple moments that produce impulses on the pulley and blocks. between block A and the surface is 0.00s .11/7/2014 HW-8: Ch.0m/s in 8.900kg ⋅ m2 and a radius of 0. Drag the appropriate labels to their respective targets. μ .250. Hint 3. Identify the equation of angular impulse and momentum Which of the following is the correct equation for angular impulse and momentum? ANSWER: http://session. Assume that the pulley rotates about a frictionless bearing.00 to 16.0 kg has a moment of inertia of 0. 2. Express your answer to three significant figures and include the appropriate units.0kg block A to change its velocity from 8.com/myct/assignmentPrintView?assignmentID=1182016 5/20 .0. Apply the principle of angular impulse and momentum to the pulley-block system and determine the block’s mass. ANSWER: This content requires Adobe Flash Player 10. Label the impulse and momentum diagram Label the impulse and momentum diagram.0. How to approach the problem 1.Kinetics of an ERO-2D (Impulse-Momentum) Determine the mass of block B necessary to cause the 30. Hint 1. Apply the principle of angular impulse and momentum to the entire block-pulley system shown.masteringengineering. 19 . The coefficient of friction. Hint 2. The pulley of mass 20. 81μ m A tr) r(vB1 − vB2 +9.81μ m A tr) r(vB1 − vB2 +9.81t) (I ω2 −I ω1 )+ m A r(vA2 − vA1 )+(9.81μ m A t) (vB1 − vB2 +9. determine the angular velocity of the bag immediately after it has been hit.Kinetics of an ERO-2D (Impulse-Momentum) (I ω2 −I ω1 )+ m A r(vA2 − vA1 )−(9.masteringengineering. eliminates the need to include the reactive impulses that occur at the connections because they are internal to the system. Assume the counterclockwise rotation as positive.81t) (I ω2 −I ω1 )+ m A (vA2 − vA1 )+(9. correct answer displayed Applying the principle of impulse and momentum to an entire system of connected bodies. Express your answer with the appropriate units.9 kg All attempts used. angular momentum ) + (∑ syst. angular ) = (∑ impulse O1 O(1−2) syst. rather than to individual bodies. 19 .11/7/2014 HW-8: Ch.com/myct/assignmentPrintView?assignmentID=1182016 6/20 . http://session.81t) ANSWER: mB = 12.81t) (I ω2 −I ω1 )+ m A r(vA2 − vA1 )+(tr) r(vB1 − vB2 +9. angular momentum ) O2 Problem 19. The equation for the principle of angular impulse and momentum may be written in symbolic form as (∑ syst.16 Part A If the boxer hits the 75 kg punching bag with an impulse of I = 20 N ⋅ s . http://session. about which the bag appears to rotate. and its center of gravity is located at the four wheels has a weight of 105lb and a radius of gyration about its center of gravity of 1 f t . find the location d of point . ANSWER: d = 6.11/7/2014 HW-8: Ch.328 rad s Correct Part B Also.Kinetics of an ERO-2D (Impulse-Momentum) ANSWER: ω = 0.masteringengineering. Each of 7/20 . B Express your answer with the appropriate units. 19 .com/myct/assignmentPrintView?assignmentID=1182016 G .25×10−2 m Correct Problem 19.26 The body and bucket of a skid steer loader has a weight of 1970lb . Treat the bag as a uniform cylinder. The wheels roll without slipping. a center of mass at G .30m .Kinetics of an ERO-2D (Impulse-Momentum) Part A If the engine supplies a torque of M = 105lb ⋅ f t to each of the rear drive wheels. 19 .4 ft s All attempts used. From a video taken of the collision it is observed that the pole was given an angular velocity of 62rad/s when AC was vertical. Express your answer with the appropriate units.masteringengineering. and a radius of gyration about an axis perpendicular to the plane of the pole assembly and passing through G of kG = 2.11/7/2014 HW-8: Ch. The pole has a mass of 175kg . correct answer displayed Problem 19. which is designed to break away from its base with negligible resistance.com/myct/assignmentPrintView?assignmentID=1182016 8/20 . starting from rest. ANSWER: v = 26. http://session.29 • The car strikes the side of a light pole. determine the speed of the loader in t = 13s . 11/7/2014 HW-8: Ch.655f t .15 The 1.4 kN⋅s Correct Problem 19. 19 . 9/20 .com/myct/assignmentPrintView?assignmentID=1182016 G of kG = 0. ANSWER: I = 16.masteringengineering. Express your answer with the appropriate units.22lb tennis racket has a center of gravity at G and a radius of gyration about http://session.Kinetics of an ERO-2D (Impulse-Momentum) Part A Determine the horizontal impulse which the car exerts on the pole at the instant AC is essentially vertical. then the angular momentum of a system of connected rigid bodies is conserved about the system's center of mass or about a fixed point.com/myct/assignmentPrintView?assignmentID=1182016 Ws lb 10/20 . they collide. Part A Which of the following scenarios demonstrate the conservation of either linear or angular momentum? Check all that apply. at the same velocity and. ANSWER: rP = 1. linear momentum ) = (∑ 1 syst. it spins faster. the result is one ball of putty with zero velocity. the linear momentum of the system is conserved. this relationship is expressed as (∑ syst. the horizontal force exerted by the racket on the hand is zero. A penny is dropped from the top of a building and its velocity increases as it falls due to the acceleration from gravity. Correct Part B Wr lb http://session. linear momentum ) 2 and is called the conservation of linear momentum. ANSWER: From opposite sides of a room. angular momentum ) 2 and is called the conservation of angular momentum. Express your answer with the appropriate units.. two identical balls of putty move toward each other. A parent pushes a merry-go-round and.masteringengineering. without friction. angular momentum ) = (∑ 1 syst.11/7/2014 HW-8: Ch.Kinetics of an ERO-2D (Impulse-Momentum) Part A Determine the position P where the ball must be hit so that 'no sting' is felt by the hand holding the racket. If the sum of all the linear impulses acting on a system of connected rigid bodies is zero. 19 . this relationship is expressed as (∑ syst. If the sum of all the angular impulses (created by the external forces that act on the system) is negligible or zero. eventually.43 f t Correct ± Conservation of Momentum Learning Goal: To be able to describe the motion of rigid bodies by applying the conservation of linear and angular momenta. i. An ice skater tucks in her arms during a spin and her angular velocity increases. Mathematically.e. consequently. Mathematically. Find an expression for the initial angular momentum What is HA1 .52 2 slug ⋅ f t /s http://session. g. the R ? . .com/myct/assignmentPrintView?assignmentID=1182016 11/20 . and the initial velocity of the projectile. = 8. of weight The length of the rod is sphere is R d1 Wr = 9. 19 . and then derive an expression for the angular momentum of the system after impact in terms of the system's angular velocity. Find the initial angular momentum What is HA1 .00f t and the radius of the = 0. Express your answer in terms of Wp . d1 . the initial angular momentum of the system about point A in terms of the following variables: the weight of the projectile.11/7/2014 HW-8: Ch.250f t . How to approach the problem Consider the projectile and the pendulum to be part of the same system. these impulses can be omitted from the analysis. AB. Because the projectile exerts an impulse on the pendulum that is equal to but opposite in direction of the impulse that the pendulum exerts on the projectile. acceleration due to gravity. Without any external forces acting on the system. Hint 1. immediately after the projectile strikes the sphere? Express your answer numerically in radians per second to three significant figures. Hint 2. the length of the rod. ANSWER: HA1 = Wp g v1 (d 1 + R) ANSWER: HA1 = 93. What is ω. the angular velocity of the pendulum.500lb strikes the center of the sphere at a velocity of v 1 = 730f t/s and becomes embedded in the center of the sphere. d1 . A projectile of weight Wp = 0. the initial angular momentum of the system about point A? Express your answer numerically in slug-squared feet per second to four significant figures.masteringengineering. .40lb and a wooden sphere of weight Ws = 27. the angular momentum around point A is conserved. Find the angular momentum of the projectile about point A immediately before impact. the radius of the sphere.Kinetics of an ERO-2D (Impulse-Momentum) A pendulum consists of a slender rod. Hint 1.4lb . and R g v1 v1 . Wp . Express your answer in terms of d1 . 19 . g? Consult your textbook for the moment of inertia of a sphere that rotates about its center. Wr . d1 . d1 . ANSWER: Ir = 1 Wr 3 g d1 2 ANSWER: Ir = 6. the rod's weight. Hint 1. Hint 1. and g. . the total moment of inertia of the system after impact? Express your answer in slug-squared feet to four significant figures. Find the sphere's moment of inertia What is Is . Find an expression for the sphere's moment of inertia What is Is .228 2 slug ⋅ f t Hint 2. Find the rod's moment of inertia What is Ir . and the acceleration due to gravity. R Ws . ANSWER: http://session. . Find the total moment of inertia of the system after impact What is Itot . Use the parallel-axis theorem to transfer the moment of inertia of the sphere to the axis about A.com/myct/assignmentPrintView?assignmentID=1182016 12/20 . the sphere's moment of inertia about point A.11/7/2014 HW-8: Ch.Kinetics of an ERO-2D (Impulse-Momentum) Hint 3. g? Consult your textbook for the formula of a slender rod's moment of inertia about its end.masteringengineering. the sphere's weight. this is done by adding the product of the sphere's mass and the square of the distance between the sphere's center and point A to the moment of inertia about an axis through the center of mass. Hint 1. the rod's moment of inertia about point A. Find an expression for the rod's moment of inertia What is Ir . the sphere's moment of inertia about point A? Express your answer numerically in slug-squared feet to four significant figures. R Ws . and g. Wr . the sphere's radius. in terms of the following variables: the rod's length. the rod's moment of inertia about point A? Express your answer numerically in slug-squared feet to four significant figures. in terms of the following variables: the rod's length. and the acceleration due to gravity. Express your answer in terms of d1 . the projectile's moment of inertia about point A. R Wp . ANSWER: http://session. and g. g? Express your answer in terms of d1 . and the initial angular momentum.masteringengineering. .11/7/2014 HW-8: Ch. Express your answer in terms of HA1 and Itot . the angular velocity of the system after impact. the sphere's radius.94 2 slug ⋅ f t Hint 3.22 2 slug ⋅ f t Hint 4. ANSWER: Ip = Wp g (R + d 1 ) 2 ANSWER: Ip = 1. Hint 1. 19 . d1 . Find an expression for the projectile's moment of inertia What is Ip . of the system. in terms of the following variables: the rod's length. R Wp .057 2 slug ⋅ f t ANSWER: Itot = 65. in terms of the system's total moment of inertia. Find the projectile's moment of inertia What is Ip . and the acceleration due to gravity. Itot . the projectile's moment of inertia about point A after impact? Express your answer numerically in slug-squared feet to four significant figures. Find an expression for the angular velocity What is an expression for ω.com/myct/assignmentPrintView?assignmentID=1182016 13/20 . the projectile's weight. .Kinetics of an ERO-2D (Impulse-Momentum) Is = 2 Ws 5 g 2 R + Ws g 2 (R + d 1 ) ANSWER: Is = 57. HA1 . 11/7/2014 HW-8: Ch.masteringengineering. use conservation of energy: T1 + V1 = T 2 + V2 where T1 is the initial kinetic energy. the maximum angle measured from the vertical that the pendulum will swing. In this situation. Find What is T1 T1 . Hint 1. How to approach the problem After impact. Finding the kinetic energy of the system after impact Immediately after impact. ANSWER: T1 = 67. and ω is the angular velocity of the system. Set the initial potential energy of the system to zero and express the final potential energy term as a function of the maximum angle θ . To solve for the maximum angle the pendulum will swing. found in Part B to be 65. the kinetic energy of the system immediately after the projectile's impact? Express your answer numerically in foot-pounds to four significant figures. Hint 2.com/myct/assignmentPrintView?assignmentID=1182016 14/20 . 19 . and solve for θ .43 rad/s All attempts used. after the projectile impacts the pendulum? Express your answer numerically in degrees to three significant figures.Kinetics of an ERO-2D (Impulse-Momentum) ω = HA 1 Itot ANSWER: ω = 1. T2 is the final kinetic energy. the kinetic energy of the system is due to the rotational motion about point A: T1 = 1 2 2 IAω where IA is the moment of inertia of the system.43rad/s . V1 is the initial potential energy. the kinetic energy of the system immediately after impact . and the final conditions are at the maximum angle. Hint 1.22slug ⋅ f t2 . found in Part B to be 1. the initial conditions are immediately after impact.04 f t ⋅ lb http://session. correct answer displayed Part C What is θ . and V2 is the final potential energy of the system. the energy of the pendulum and lodged projectile is conserved. d1 Wr . and θ . the sphere's weight. R Express your answer in terms of Ws . d1 . Find an expression for the rod's final potential energy Find an expression for Vr . impact. in terms of Wp . At the peak of the pendulum's below point A. Hint 1. the rod's final potential energy. . Finding the change in height of the projectile The projectile originates at a distance d1 the projectile is (d1 + R) cos(θ) +R below point A. below point A. At the peak of the pendulum motion. the sphere's center is +R (d1 + R) cos(θ) below point A. The change in height of the sphere is the difference in these quantities. and θ . . ANSWER: Vp = Wp ( d1 + R)(1 − cos(θ)) Hint 4. and θ . d1 . The change in height of the rod is the http://session.11/7/2014 HW-8: Ch. and θ . d1 . the rod's . d1 . . 19 . the . R Hint 1. the projectile's weight. the sphere's final potential energy. R Hint 1. the maximum angle the pendulum sweeps after R Express your answer in terms of Wp . The change in height of the projectile is the difference in these quantities. length of the rod. in terms of Ws . the radius of the sphere. and θ . the length of the rod. Finding the change in height of the sphere The sphere's center originates at a distance d1 motion. Express your answer in terms of Wr . the projectile's final potential energy. and θ . Find an expression for the projectile's final potential energy Find an expression for V2 . the maximum angle that the pendulum sweeps after impact. d1 . Finding the change in height of the rod The rod's center of mass originates at a distance d1 /2 below point A. At the peak of the pendulum motion.Kinetics of an ERO-2D (Impulse-Momentum) Hint 3. the radius of the sphere.com/myct/assignmentPrintView?assignmentID=1182016 15/20 . in terms of the following variables: weight. the rod's center is (d1 /2) cos(θ) below point A. Find an expression for the sphere's final potential energy Find an expression for Vs . the maximum angle the pendulum sweeps after impact.masteringengineering. ANSWER: Vs = Ws ( d1 + R)(1 − cos(θ)) Hint 5. the length of the rod. each having a mass of 30 kg.masteringengineering.11/7/2014 HW-8: Ch. sit at the edge of the merry-go-round which rotates at ω = 2 rad/s. ANSWER: Vr = Wr d1 2 (1 − cos(θ)) ANSWER: θ = 41. the merry-go-round has a mass of 180 kg and a radius of gyration kz . Express your answer with the appropriate units.Kinetics of an ERO-2D (Impulse-Momentum) difference in these quantities.4 degrees Correct Problem 19. measured relative to the merry-go-round. Neglect friction and the size of each child.6 m Part A Determine the angular velocity of the merry-go-round if A jumps off horizontally in the −n direction with a speed of 2 m/s. = 0. 19 .43 rad s http://session. Excluding the children. ANSWER: ωz = 2.41 • Two children A and B.com/myct/assignmentPrintView?assignmentID=1182016 16/20 . measured relative to the merry-go-round? Express your answer with the appropriate units. The block can slide freely along the two vertical guide rods. determine the maximum height attained by the 50lb block D.com/myct/assignmentPrintView?assignmentID=1182016 17/20 . Part A Just before the impact the hammer is gripped loosely and has a vertical velocity of 75 f t/s . ANSWER: h = 4.masteringengineering.5. Express your answer with the appropriate units.Kinetics of an ERO-2D (Impulse-Momentum) Correct Part B What is the merry-go-round's angular velocity if B then jumps off horizontally in the −t direction with a speed of 2 m/s.11/7/2014 HW-8: Ch.99 f t http://session. 19 .96 rad s Correct Problem 19. The plank is initially in a horizontal position. ANSWER: ωz = 2. If the coefficient of restitution between the hammer head and the plank is e = 0.50 The rigid 30-lb plank is struck by the 15-lb hammer head H . Kinetics of an ERO-2D (Impulse-Momentum) Correct Problem 19. sliding on a smooth horizontal surface with a velocity of 12 f t/s . A 2-lb block. ANSWER: to the left to the right Correct Part B Determine the magnitude of the velocity of the block immediately after the collision. strikes the rod at its end B.11/7/2014 HW-8: Ch. The coefficient of restitution between the block and the rod at B is e = 0.36 ft s Correct http://session.masteringengineering.8. Express your answer with the appropriate units.54 The 4-lb rod AB hangs in the vertical position. ANSWER: vb = 3.com/myct/assignmentPrintView?assignmentID=1182016 18/20 . 19 . Part A Determine the direction of the velocity of the block immediately after the collision. 11/7/2014 HW-8: Ch. Assume the counterclockwise rotation as positive. it is at rest. Express your answer with the appropriate units.55 The pendulum consists of a 10-lb sphere and 4-lb rod.com/myct/assignmentPrintView?assignmentID=1182016 19/20 . http://session.9 rad s Correct Problem 19. strikes the target at A and becomes embedded in it. A 23g bullet.masteringengineering. traveling at 615m/s . Part A Determine the angular velocity of the target after the impact. Initially.47 The target is a thin 5kg circular disk that can rotate freely about the z axis. ANSWER: ω = 24. 19 .Kinetics of an ERO-2D (Impulse-Momentum) Problem 19. determine the angle θ of rebound after the sphere strikes the floor. ANSWER: θ = 35. 19 . http://session.3 ∘ Correct Score Summary: Your score on this assignment is 96. You received 8 out of a possible total of 9 points. Express your answer with the appropriate units. Take e = 0.7%.com/myct/assignmentPrintView?assignmentID=1182016 20/20 . plus 0.Kinetics of an ERO-2D (Impulse-Momentum) Part A If it is released from rest when θ = 90 ∘ .76.masteringengineering.71 points of extra credit.11/7/2014 HW-8: Ch.