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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 V: Massive Pulley In the apparatus shown in the sketch, both blocks ac- celerate as a result of the force of gravity on m 2 . The coefficient of kinetic friction between m 1 and the table is µ k . The pulley has frictionless bearings, moment of inertia I , and radius R . The string does not slip on the pulley, so the pulley undergoes an angular acceleration. The blocks are initially at rest. At time t , through what distance y has m 2 moved? Use Helping Questions 14, 15, and 16 if you need to. Key 8 m 1 m 2 Problem VI: Mass Drop After the block of mass m 2 in the preceding problem has fallen a distance y from rest, it has speed v . Use the work-energy theorem to express v in terms of m 1 , m 2 , µ k , g, y, I, and R . Use Helping Questions 17, 18, and 19 if necessary. Key 11 Problem VII: Line Cut A uniform rod is held vertically by two strings of neg- ligible mass, as shown. Consider the situation imme- diately after the line on the right side is cut. 1. What is the linear acceleration, a r , of the free (right) end of the rod? Key 58 2. What is the acceleration, a c , of the middle of the rod? Key 48 3. What is the tension, T , in the left-hand string? Key 27 Cut Consult Helping Questions 20 through 24 as needed. 3

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