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PHY103 Princeton University Fall 2023 Learning Guide 6 : Rolling and Rotation Problem I: Rotational Inertia The rigid body shown in the figure con- sists of four 10 kg spheres connected by four light rods. Treat the spheres as point particles and neglect the mass of the rods. B C A 1 m 4 m 1. Which is greater, the rotational inertia about axis A or the rotational inertia about axis B ? What are their exact values? Key 19 2. Use the parallel-axis theorem to calculate the rotational inertia about axis C . Check your answer by calculating the rotational inertia about axis C from the definition. Key 24 Problem II: Incline Roll A solid cylinder of mass m and radius R rolls down an incline of angle θ , as shown. R θ a 1. Assuming that the cylinder rolls without slipping, find the acceleration down the slope. If you need help, look at Helping Questions 1 through 3. Key 50 2. What is the smallest value of µ s that is needed to keep the cylinder from slipping? Consult Helping Questions 4 and 5 as needed. Key 2 1

Problem III: Double Wheel A wheel having moment of inertia I = 2 kg-m 2 turns freely on a horizontal axis. Two ropes are wrapped around the wheel, one at radius r 1 = 0 . 1 m, which is connected to mass m 1 = 3 . 1 kg, and a second at radius r 2 = 0 . 25 m, which is connected to mass m 2 = 1 . 2 kg. See the sketch. You may neglect the mass of the ropes and assume that the system is released from rest. 2 r 1 r 2 m 1 m 1. Which way does the wheel rotate? Key 41 2. How long does it take for the wheel to undergo one full rotation? Key 5 Consult Helping Questions 6 through 10 as needed. Problem IV: Cylinder Unwind A string wrapped around a solid cylinder of mass M and radius R is pulled vertically upward to prevent the cylinder from falling as it unwinds the string (i.e., the center-of-mass of the cylinder does not move). M R 1. What is the tension in the string? If you disagree with the key, use Helping Question 11. Key 23 2. If the cylinder is initially at rest, how much string is unwound after a time t ? If, after a good effort, you’re stuck, use Helping Questions 12 and 13. Key 51 2

Problem VIII: Truck Roll A bowling ball rests on the back of a flat-bed truck a distance d from its back edge. The truck begins to move forward with constant acceleration. How far does the truck go before the ball rolls off its back? You should assume that the ball rolls without slipping. Key 16 d Consult Helping Questions 25 through 29 as needed. Problem IX: Pool Ball A pool cue strikes a pool ball which is sitting on a level pool table with a coefficient of kinetic friction µ k . The ball is given an initial speed of v 0 with no spin. How fast is the ball moving when it begins to roll without slipping? Even though the center-of-mass frame is a non-inertial reference frame , τ = Iα still holds in this frame. If you need more hints, use Helping Questions 30, 31, and 32. Key 28 4

Problem X: The Nutcracker The “ 1 2 -20” bolt shown to the right is a standard item in mechanical assemblies. The “1/2” means that the diameter of the bolt is 1/2 inch (about 12.5 mm), while the “20” means that there are 20 threads per inch. Although bolts like this are typically used to hold pieces of metal together, they can also be used as small “jacks.” The idea is that by applying a torque to the bolt, one can gener- ate a very large force along the axis of the bolt. A whimsical application might be a nutcracker, as shown in the figure to the left. The bolt passes through a threaded hole on the upper plate, which is held a fixed distance from the lower plate on which a nut is placed. As the bolt is turned it presses on the nut. F Nut Calculate the linear force, F , resulting if a person applies a force of 22 pounds to the end of a 12-inch wrench that is used to turn the bolt. Assume that the force acts at right angles to the wrench and ignore friction. If you need help, look at Helping Questions 33 through 35. Key 17 5

Helping Questions 1. Draw a free body diagram for the cylinder. Take care to indicate the point of applica- tion of the forces. Key 13 2. What are the corresponding force and torque equations? Key 14 3. What is the relationship between the angular acceleration, α , and the linear accelera- tion, a ? Key 18 4. Is there a friction equation? Why or why not? Key 32 5. When the cylinder is just about to slip, what is true about µ s ? Key 26 6. Indicate the forces (tensions) and accelerations for the masses and the wheel. Key 3 7. What are the free body diagrams for each object? Key 59 8. What are the corresponding equations? Key 54 9. Count equations and unknowns. Are additional equations needed? If so, what are they? Key 22 10. What is α ? Key 30 11. Is the cylinder’s center of mass moving? Then what must be the sum of the external forces on the cylinder? Key 47 12. What is the angular acceleration of the cylinder? Key 4 13. What is the rotational analogue of the translational kinematic formula ∆ x = v 0 t + 1 2 at 2 ? Key 31 14. Draw force (free-body) diagrams for each block and for the pulley. Call the tension in the horizontal part of the string T 1 and the tension in the vertical part of the string T 2 . What is the torque τ on the pulley in terms of T 1 and T 2 ? What are T 1 and T 2 in terms of µ k , m, g, and a , the acceleration of the blocks? Key 38 15. Can you relate the torque on the pulley to its angular acceleration? Can you relate the angular acceleration to the linear acceleration of the blocks? Now can you solve for the acceleration a ? Key 40 7

16. What did you get for the acceleration a ? What kinematic formula would be appropriate to get the distance y ? Key 29 17. What is the change in potential energy? Key 39 18. What is the change in kinetic energy? Key 25 19. How much mechanical energy is lost to friction? Key 52 20. What is a convenient point about which to take as the axis of rotation? Why is it a good choice? Key 44 21. What is the free body diagram for the rod immediately after the line is cut? Key 42 22. What is the resulting torque equation? Key 9 23. What is the relationship between the angular acceleration α and the accelerations a r and a c of the right end and center of the rod, respectively? Key 55 24. What is Newton’s 2nd Law for the center of mass motion? Key 10 25. If the truck moves forward with acceleration a with respect to the ground and the bowling ball accelerates backward with respect to the truck bed with acceleration a r , what is the acceleration, a ′ , of the ball with respect to the ground? Key 1 26. Draw a free-body diagram for the bowling ball. Key 35 27. What equations result? Key 20 28. What is the moment of inertia of a solid sphere Key 34 29. What is a r and how is it related to d ? Key 6 30. What is the condition required for rolling without slipping? Key 46 31. What are the forces on the ball? Then what is v ( t )? Key 12 32. What are the torques on the ball? Then what is ω ( t )? Key 36 33. How much work is done by a force F acting through a distance ∆ x ? Key 21 34. How much work is done by a torque τ acting through an angle ∆ θ ? Key 53 35. How far does the tip of the bolt advance if the wrench is used to turn the bolt by one full rotation? Key 7 8

ANSWER KEY 1. a ′ = a − a r 2. µ s = 1 3 tan θ 3. 2 r 1 r 2 m 1 m 2 α a 1 a T 1 T 2 4. 2 g/R 5. t = s 2∆ θ α = 16 . 4 s 6. a r = 5 7 a and d = 1 2 a r t 2 , where t is the time that the ball rolls before falling off the truck. 7. 1/20 inch 8. y = 1 2 m 2 g − m 1 µ k g m 1 + m 2 + I/R 2 ! t 2 9. τ : mg L 2 = Iα = mL 2 3 α 10. mg − T = ma c 11. v = v u u t ( m 2 − m 1 µ k ) ( m 1 + I/R 2 + m 2 ) 2 gy 12. − µ k mg ; v ( t ) = v 0 − µ k gt 13. R f mg N θ y x a 14. x : mg sin θ − f = ma y : N − mg cos θ = 0 τ : fR = mR 2 2 α 15. τ = Iα. 16. s = a 2 t 2 = a 2 14 d 5 a = 1 . 4 d 17. 33,200 pounds (about 15 metric tons) 18. α = a/R 10

19. I A > I B ; I A = 160 kg · m 2 and I B = 10 kg-m 2 20. x : f = ma ′ = ma − ma r τ : fR = Iα = 2 5 mR 2 a r R 21. W = F ∆ x 22. There are three equations and five un- knowns: α, a 1 , a 2 , T 1 , & T 2 . The two ad- ditional required equations are: a 1 = αr 1 and a 2 = αr 2 23. Mg 24. I C = 320 kg · m 2 25. K f − K i = 1 2 m 1 v 2 + 1 2 Iω 2 + 1 2 m 2 v 2 26. Friction is at its maximum value, in which f = µ s N , which now is an equation. 27. T = mg − ma c = mg 4 28. v = v 0 (1 + I/mr 2 ) = 5 7 v 0 29. a = m 2 g − m 1 µ k g m 1 + m 2 + I/R 2 ; y = v 0 y t + 1 2 at 2 30. α = g ( m 2 r 2 − m 1 r 1 ) I + m 1 r 2 1 + m 2 r 2 2 = − 0 . 047 rad / s 2 31. ∆ θ = ω 0 t + αt 2 / 2 32. No. Since the cylinder is rolling with- out slipping, the force involved is static friction, where f ≤ µ s N . That is an inequality and not an equation. 33. a = αr. 34. I = 2 5 R 2 35. +x N mg f R α a’ 36. µ k mgr ; ω ( t ) = µ k mgrt/I 37. τ = Tr. 38. τ = R ( T 2 − T 1 ) – into the page; 39. U f − U i = − m 2 gy 40. τ = Iα ; yes : a = Rα 11

2 2 a 2 +y r 1 r 2 α a 1 T 1 T 2 m 1 m 1 T 1 m 2 g +y T m 13