8/17/2015Assignment 15 Assignment 15 Due: 11:59pm on Monday, August 17, 2015 To understand how points are awarded, read the Grading Policy for this assignment. Finding Torque A force F ⃗ of magnitude F making an angle θ with the x axis is applied to a particle located along axis of rotation A, at Cartesian coordinates (0, 0) in the figure. The vector F ⃗ lies in the xy plane, and the four axes of rotation A, B, C, and D all lie perpendicular to the xy plane. A particle is located at a vector position r ⃗ with respect to an axis of rotation (thus r ⃗ points from the axis to the point at which the particle is located). The magnitude of the torque τ about this axis due to a force F ⃗ acting on the particle is given by τ = rF sin(α) = (moment arm) ⋅ F = rF ⊥ , ⃗ where α is the angle between r ⃗ and F , r is the magnitude of r ⃗, F is the magnitude of F ⃗ , the component of r ⃗ that is perpendicualr to F ⃗ is the moment arm, and F⊥ is the component of the force that is perpendicular to r ⃗. Sign convention: You will need to determine the sign by analyzing the direction of the rotation that the torque would tend to produce. Recall that negative torque about an axis corresponds to clockwise rotation. In this problem, you must express the angle α in the above equation in terms of θ , ϕ , and/or π when entering your answers. Keep in mind that π = 180 degrees and (π/2) = 90 degrees . Part A What is the torque τ A about axis A due to the force F ⃗ ? Express the torque about axis A at Cartesian coordinates (0, 0) . Hint 1. When force is applied at the pivot point Does a force applied at a pivot point cause an object to rotate about that pivot? ANSWER: https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=3766134 1/15 located a distance b from the origin along the y axis. Hint 2. π.com/myct/assignmentPrintView?assignmentID=3766134 2/15 . A helpful figure The figure shows r ⃗ for this part of the problem. What is the value of α in terms of θ ? ANSWER: π−θ (π/2) − θ (π/2) + θ θ ANSWER: https://session. Finding r with respect to an axis The vector r ⃗ should be drawn from the axis B to the point where F ⃗ is being applied.masteringphysics. ⃗ Hint 1.) Express the torque about axis B in terms of F .8/17/2015 Assignment 15 τA = 0 Correct Part B What is the torque τ B about axis B due to the force F ⃗ ? (B is the point at Cartesian coordinates (0. b). ϕ . θ . Consider both the magnitude and direction of this vector. and/or other given coordinate data. π. What is the value of F⊥ ? ANSWER: F sin θ F cos θ Hint 2. Hint 1.masteringphysics. Look at the direction of F ⃗ in the figure. Clockwise or counterclockwise? Imagine a wheel of radius r with its axle passing through the axis of rotation (so that the particle at point A is on the rim of the wheel).com/myct/assignmentPrintView?assignmentID=3766134 3/15 . ANSWER: https://session. A helpful figure The figure shows r ⃗ for this part of the problem. a distance c along the x axis. and/or other given coordinate data. 0) .8/17/2015 Assignment 15 τB = bF sin( π 2 + θ) Correct Part C What is the torque τ C about axis C due to F ⃗ ? (C is the point at Cartesian coordinates (c.) Express the torque about axis C in terms of F . ϕ . θ . Which way do you think F ⃗ will "try" to turn the wheel: clockwise or counterclockwise? Note that only the component of F ⃗ that is tangent to the rim of the wheel (perpendicular to r ⃗) generates torque. Calculating Torques Using Two Standard Methods Learning Goal: To understand the two most common procedures for finding torques when the forces and displacements are all in one plane: the moment arm method and the tangential force method. https://session. torques that would cause counterclockwise rotation are considered to be positive. The tension in the cable is T . τ = rF sin(α) = (moment arm) ⋅ F = rF ⊥ . The other end of the pole is attached to a cable. Throughout the problem. and/or other given coordinate data. The purpose of this problem is to give you further practice finding torques in twodimensional situations.com/myct/assignmentPrintView?assignmentID=3766134 4/15 . In this case it is overkill to use the full cross product definition of the torque because the only nonzero component of the torque is the component perpendicular to the plane containing the problem. so that the pole makes an angle θ with respect to the wall. and making the following associations: r = d and sin(α) = sin(π − ϕ + θ) = sin(ϕ − θ) moment arm = r ⊥ = d sin(ϕ − θ) F ⊥ = F sin(ϕ − θ) . Consider a uniform pole of length L. ϕ . The pole is attached to the wall at point A.masteringphysics. π. θ . ANSWER: τD = dF sin(ϕ − θ) Correct You could have found τ D by using any of the three equations listed at the top of the page. There are two common methods for finding torque in a twodimensional problem: the tangential force method and the moment arm method. and the cable is horizontal. attached at its base (via a pivot) to a wall.) Express the torque about axis D in terms of F .8/17/2015 Assignment 15 τC = −cF sin(θ) Correct Part D What is the torque τ D about axis D due to F ⃗ ? (D is the point located at a distance d from the origin and making an angle ϕ with the x axis. Both of these methods will be illustrated in this problem. This perpendicular component of the force is called the tangential force.8/17/2015 Assignment 15 Tangential force method The tangential force method involves finding the component of the applied force that is perpendicular to the displacement from the pivot point to where the force is applied.masteringphysics. you calculate the torque using the equation τ = Ft d.com/myct/assignmentPrintView?assignmentID=3766134 5/15 . ANSWER: Ft = cos(θ)T Correct When using the tangential force method. the magnitude of the tangential force that acts on the pole due to the tension in the rope? Express your answer in terms of T and θ . https://session. Part A What is Ft . com/myct/assignmentPrintView?assignmentID=3766134 6/15 .8/17/2015 Assignment 15 where d is the distance from the pivot to the point where the force is applied. The sign of the torque can be determined by checking which direction the tangential force would tend to cause the pole to rotate (where counterclockwise rotation implies positive torque). about point A. imagine a line parallel to the force. The magnitude of the torque about the pivot is the product of the moment arm and force. To do this. due to the tension in the rope? Express your answer in terms of T . Express your answer in terms of L and θ . provided you do not change the direction of the force vector as you shift it. ANSWER: τ = T Lcos(θ) Correct Moment arm method The moment arm method involves finding the effective moment arm of the force. Part B What is the magnitude of the torque τ on the pole. and the sign of the torque is again determined by the sense of the rotation of the pole it would cause. L. and extending off to infinity in either direction. The moment arm of the force is the distance between the pivot and the tail of the shifted force vector. Part C Find Rm . For example.masteringphysics. ANSWER: https://session. consider the force due to tension acting on the pole. running through the point at which the force is applied. and θ . Shift the force vector to the left. It is generally most convenient to shift the force vector to a point where the displacement from it to the desired pivot point is perpendicular to its direction. the length of the moment arm of the force. You may shift the force vector anywhere you like along this line without changing the torque. This displacement is called the moment arm. so that it acts at a point directly above the point A in the figure. 8/17/2015 Assignment 15 Rm = cos(θ)L Correct To calculate the torque using the moment arm method. ANSWER: τ = cos(θ)T L Correct For this problem. the two methods of finding torque involved nearly the same of amount of algebra. about point A. Which method of finding the torque would be the easiest to use? ANSWER: https://session. This small bone has a length x. and θ . A normal force of magnitude N acts upward on the ball of her foot. and the angle between this bone and the Achilles' tendon is ϕ . The Achilles' tendon is attached to the back of the foot. Part E Suppose you were asked to find the torque about point P due to the normal force N in terms of given quantities.com/myct/assignmentPrintView?assignmentID=3766134 7/15 . both methods lead to the same final result. Of course. The tendon pulls on the small bone in the rear of the foot with a force F . L. where Rm is the moment arm perpendicular to the applied force. Now consider a woman standing on the ball of her foot as shown . Part D Find the magnitude of the torque τ on the pole. Express your answer in terms of T .masteringphysics. and either method could be used. The horizontal displacement between the ball of the foot and the point P is D. due to the tension in the rope. use the equation τ = F Rm . ANSWER: τN = −N D Correct Part G Suppose you were asked to find the torque about point P due to the force of magnitude F in the Achilles' tendon.masteringphysics. ϕ . Correct Part H Find τ F . the torque about point P due to the normal force. the torque about point P due to the force applied by the Achilles' tendon. Express your answer in terms of F . Express your answer in terms of N and any of the other quantities given in the figure. and x. The moment arm method must be used.8/17/2015 Assignment 15 tangential force method moment arm method Correct Part F Find τ N . Neither method can be used. Either method may be used.com/myct/assignmentPrintView?assignmentID=3766134 8/15 . Which of the following statements is correct? ANSWER: The tangential force method must be used. ANSWER: τF = sin(ϕ)F x Correct https://session. A monkey of mass 1. note that if a force has a line of action that goes through a particular point. SET UP the problem using the following steps: 1. IDENTIFY the relevant concepts The rigid body under consideration is the bar. Do not include forces exerted by this body on other bodies. including dimensions. Often.65 m from the other end.4 kg walks from one end of the bar to the other. You always need as many equations as you have unknowns. ∑ τ z = 0 . Find the tension T 1 in string 1 at the moment that the monkey is halfway between the ends of the bar. If you've done everything correctly. there are several equally good sets of force and torque equations for a particular problem. and adding the results. You can often eliminate unknown forces or components from the torque equation by a clever choice of point for your calculation.8 kg and length 3.masteringphysics. including the instant at which the monkey is halfway between its ends. Because there is no indication to suggest otherwise. Draw a freebody diagram showing the forces acting on the selected body and no others.1 Equilibrium of a Rigid Body. 2. Represent forces in terms of their components with respect to the axes you have chosen. ProblemSolving Strategy 11. each with its appropriate lever arm and sign. Be careful to show correctly the point at which each force acts. EXECUTE the solution as follows: 1. In choosing a point about which to compute torques. it is reasonable to assume that the bar remains at rest as the monkey walks from one end of the bar to the other. Draw a sketch of the physical situation.1 Equilibrium of a Rigid Body Learning Goal: To practice ProblemSolving Strategy 11. Also. String 1 is attached to the end of the bar.com/myct/assignmentPrintView?assignmentID=3766134 9/15 . Keep in mind that ∑ Fx = 0. the above strategy can be applied. using a different choice of origin. Thus.0 m is hung horizontally on two vertical strings. you may need to compute torques with respect to two or more axes to obtain enough equations. you can compute the torque of that force by finding the torque of each component separately. Depending on the number of unknowns. and select the body in equilibrium to be analyzed. Choose coordinate axes. The body doesn't actually have to be pivoted about an axis through the chosen point. Write equations expressing the equilibrium conditions. ∑ Fy = 0. you'll get the same answers using this new choice of origin as you did with your original choice. This means that the conditions for equilibrium hold at any time. SET UP the problem using the following steps https://session. A horizontal uniform bar of mass 2. 4. and specify a positive direction of rotation for torques. and string 2 is attached a distance 0. recall that when a force is represented in terms of its components. and ∑ τ z = 0 are always separate equations; never add x and y components in a single equation. 2. the torque of the force with respect to that point is zero. 3.1 Equilibrium of a Rigid Body IDENTIFY the relevant concepts: The first and second conditions for equilibrium are useful whenever there is a rigid body that is not rotating and not accelerating in space. EVALUATE your answer: A useful way to check your results is to rewrite the second condition for equilibrium.8/17/2015 Assignment 15 PSS 11. com/myct/assignmentPrintView?assignmentID=3766134 10/15 .8/17/2015 Assignment 15 Part A Which of the following diagrams correctly represents the forces acting on the bar at the moment described in the problem introduction? (Note that the forces are not necessarily drawn to scale. For example.) ANSWER: diagram A diagram B diagram C diagram D Correct Four vertical forces act upon the bar: the upward tension in both strings. In general. the choice of reference point for calculating torques is completely arbitrary. Part B One of the equilibrium conditions that should be applied in this problem requires that you write a torque equation for the bar. Note that whereas the weight of the bar is represented as a force acting at the center of gravity of the bar (which coincides with the center of mass the bar). the force exerted by the monkey acts at the point of contact between the monkey and the bar. Be sure to label all the forces in your diagram. Which of the following choices of reference point for calculating torques would lead to a torque equation in which the only unknown quantity is T 1 ? Hint 1. This may be clearer if you draw your own freebody diagram. which is a point on the top surface of the bar halfway between its ends. if a force has a line of action that goes through a particular point. The point of reference for calculating torques In equilibrium problems. the weight of the bar. https://session. the torque of the force with respect to that point is zero and its contribution would not appear in the torque equation ∑ τ z = 0 . and the downward force exerted by the monkey.masteringphysics. it helps to pick the point so as to simplify the calculations as much as possible. com/myct/assignmentPrintView?assignmentID=3766134 11/15 . but they might lead to more complicated mathematics. Other choices of reference point would also work. generally makes it easier to solve.8/17/2015 Assignment 15 ANSWER: the center of mass of the bar the point of attachment of string 1 the point of attachment of string 2 the end of the bar closest to string 2 Correct If you choose the point of attachment of string 2 as the reference point for calculating torques. the torque equation will not depend on T 2 . at the moment that the monkey is halfway between the ends of the bar. the (unknown) tension in string 2. Eliminating terms from the torque equation. choose a set of coordinate axes.masteringphysics. ANSWER: T1 = N EVALUATE your answer Part D This question will be shown after you complete previous question(s). EXECUTE the solution as follows Part C Find T 1 . and the lever arm of the force in this case would be zero. and specify a positive direction of rotation for torques. You did not open hints for this part. the magnitude of the force of tension in string 1. Express your answer in newtons using three significant figures. Forces on a Bridge https://session. particularly those involving unknown quantities. because the line of action of T 2 goes through the point of attachment of string 2. you must use the same point to calculate all the torques on a body. Now. Keep in mind that once you make your choice. com/myct/assignmentPrintView?assignmentID=3766134 12/15 . Hint 1. the vertical force that pier P exerts on the left end of the bridge. constructed of 11 beams of equal length L and negligible mass. Real bridges of this sort have steel rockers at the ends (check one out sometime). Find the sum of the torques The sum of the torques is _________. L. Hint 2. τQ about the right pier Q of the bridge. and find the torque about a pivot chosen to eliminate the torque from one of the unknowns. ANSWER: positive negative zero https://session. ANSWER: ∑ τQ = 2M gL − 3F P L Hint 3.8/17/2015 Assignment 15 A bridge. Express the vertical force at P in terms of M and g . supports an object of mass M as shown. A suggested origin for determining torques Find the sum of the torques Σ torque is positive. How to approach the problem View the entire bridge as the system.masteringphysics. Throughout this problem. Part A Find FP . use g for the magnitude of the acceleration due to gravity. g . these assure that the support forces on the bridge are vertical even when it expands or contracts thermally. Remember that counterclockwise Answer in terms of M . and FP . masteringphysics.. Two different methods One way to solve this problem is to add up the torques about the left pier P using the method analogous to Part A. what is the tension T in the horizontal segment directly above the point where the object is attached? If you find that the horizontal segment directly above the point where the object is attached is being stretched.e. If the segment is being compressed. Ftotal . indicate this with a negative value for T . ANSWER: FQ = 1 3 Mg Correct Part C Assuming that the bridge segments are free to pivot at each intersection point. the total force Ftotal is zero. Hint 1. and g .com/myct/assignmentPrintView?assignmentID=3766134 13/15 . M . Another is to use Newton's 2nd law to find the total vertical force on the bridge. ANSWER: F total = FP + FQ − M g Hint 2. Express the vertical force at Q in terms of M and g . Answer in terms of FP . indicate this with a positive value for T . nothing is moving).8/17/2015 Assignment 15 ANSWER: FP = 2 3 Mg Correct Part B Find FQ . Take the upward direction to be positive. FQ . the vertical force that pier Q exerts on the right end of the bridge. Express the tension in terms of M and g . https://session. Total force Because this is a statics problem (i. masteringphysics. Hint 2. Torque from tension Find the magnitude τ T of the torque about this pivot point due to the tension in the top member. See if you can find a rigid portion of the bridge and a wisely chosen pivot to which you can apply this powerful fact. ANSWER: https://session. with 60 ∘ angles. Hint 1. the sum of the torques acting on any portion of the bridge you choose is zero for any pivot point you may choose. (The triangle is equilateral.) Find the torques on this lefthand triangle (which can be considered a solid piece because of the connections). so that the force that the tension in this segment exerts on the bold triangle is directed to the right.com/myct/assignmentPrintView?assignmentID=3766134 14/15 . Hint 2.) Answer in terms of L and T . (This choice eliminates the torques due to the tensions in the beams that attach at point x. The moment arm must be found from trigonometry. One way to start Since the bridge and all segments of it are static. and FP . Forces required to find torque The only forces not passing through this pivot point are T and FP . Find the torque about a special point Consider the triangular portion shown in bold and let x be the pivot. Remember that counterclockwise torque is positive. Express the torque in terms of T . Assume that the horizontal segment above M is being stretched. L.8/17/2015 Assignment 15 Hint 1. 8/17/2015 Assignment 15 τT = T L⋅√3 2 ANSWER: ∑ τx = −T L⋅√3 2 − FP L Hint 3. Score Summary: Your score on this assignment is 75. This should be clear if you think of the bridge as being composed of the two shaded sections shown: The compressive load in the top beam prevents the load from falling into the river. You could then go to the Machinist's Handbook or an equivalent reference and find out what cross sectional shape of each segment would be sufficient to bear the maximum anticipated load times the required safety factor. >By repeating this process. https://session.0%. ANSWER: T = − 2 3 Mg . you can solve the equation you found in the previous hint for T and then substitute for FP using the expression you found in Part A.masteringphysics. you should be able to solve for the tension or compression of every segment in the bridge. You received 30 out of a possible total of 40 points. How to find T from the torque Since ∑ τ x must equal zero (for a statics problem).866 Correct The negative value of the tension shows that the segment is actually under a compressive load.com/myct/assignmentPrintView?assignmentID=3766134 15/15 .