Q: A uniform rod of length $$L$$ and mass $$M is $$hinged at one end and held horizontally by a cord at the other end, which makes an angle $$\theta$$ with the rod. A mass $$m$$ hangs from the end. Which equation would you use to find the tension $$T in $$the cord?

$$T = \frac{(Mg\frac{L}{2} + mgL)}{L \sin \theta}$$

Question 1

A solid sphere, a hollow sphere, and a block are released from rest at the top of identical ramps of height $$h. $$The block slides without friction, while both spheres roll without slipping. Which object reaches the bottom with the greatest speed?

Accepted Answer

The block, because it does not have to share its energy with rotational motion.

Question 2

An ice skater is spinning with arms extended. She pulls her arms in close to her body. What happens to her rotational velocity $$\omega$$ and rotational inertia $$I$$?

Accepted Answer

Rotational velocity increases, rotational inertia decreases.

Question 3

A child walks from the edge to the center of a rotating platform. The system is isolated. What happens to the angular velocity of the platform?

Accepted Answer

It increases, because the rotational inertia decreases.

Question 4

A wheel rotates through $$10.0$$ radians in $$2.5$$ seconds as it comes to rest with constant angular acceleration. What was its initial angular velocity $$\$$omega_0$$?

Accepted Answer

$$8.0\ \mathrm{rad/s}$$

Question 5

A block of mass $$m$$ slides up a frictionless incline, while a sphere of mass $$m$$ and radius $$R$$ rolls up an identical incline without slipping. Both start with the same speed $$v. $$When they have risen to height $$h$$, which statement is true?

Accepted Answer

The block has a greater speed because the sphere's energy is shared between translational and rotational motion.

Question 6

Four forces are applied to a hexagonal board nailed at its center. Two $$4\ \mathrm{N}$$ forces act at opposite vertices, and two $$2\ \mathrm{N}$$ forces act at the other two opposite vertices. What is the net torque on the board?

Accepted Answer

Zero, because the torques cancel each other out.

Question 7

A wrench has five equal forces $$V$$, $$W$$, $$X$$, $$Y$$, and $$Z$$ applied at different points and angles. Which two forces produce the same clockwise torque?

Accepted Answer

V and X

Question 8

A motorcycle stuntman in midair increases the rear wheel's spin using the throttle. What happens to the rotation of the rider-motorcycle system?

Accepted Answer

The front of the motorcycle rotates upward.

Question 9

A uniform rod of length $$L$$ and mass $$M is $$hinged at one end and held horizontally by a cord at the other end, which makes an angle $$	heta$$ with the rod. A mass $$m$$ hangs from the end. Which equation would you use to find the tension $$T in $$the cord?

Accepted Answer

$$T = \frac{(Mg\frac{L}{2} + mgL)}{L \sin 	heta}$$

Question 10

A disk rotates with angular velocity $$\omega$$ and a ladybug walks from the edge toward the center. What happens to the disk's angular velocity if no external torques act?

Accepted Answer

It increases, because the system's rotational inertia decreases.

Question 11

A meter stick is held horizontally with a $$500\ \mathrm{g}$$ mass at the $$50\ \mathrm{cm}$$ mark and a $$1000\ \mathrm{g}$$ mass at the $$100\ \mathrm{cm}$$ mark (Case A). In Case B, the positions of the masses are swapped. In which case is the torque about the $$0\ \mathrm{cm}$$ end greater?

Accepted Answer

The torque is the same in both cases.

Question 12

A thin hoop of radius $$r$$ rolls without slipping at $$v_0$$ and then up a ramp, coming to rest. Which energy bar chart best represents the system from the start to the end?

Accepted Answer

Initial: $$K_{trans}$$ and $$K_{rot}$$; Final: $$U_g$$ only.

Question 13

A bullet of mass $$m is $$shot at a hinged rod and sticks to it. Which conservation law is most appropriate to use immediately before and after the collision?

Accepted Answer

Conservation of angular momentum about the hinge.

Question 14

A disk with rotational inertia $$I$$ rotates with angular velocity $$\$$omega_0$$. A torque $$	au$$ is applied in the opposite direction. What is the angular acceleration $$\alpha$$?

Accepted Answer

$$\alpha = -\frac{	au}{I}$$

Question 15

A student derives $$\omega = \frac{I v_0$}{m_{disk} L^2$}$$ for the post-collision angular speed of a rod after a disk sticks to it. Why is this equation not physically plausible?

Accepted Answer

The units do not match; $$\omega$$ should have units of $$\mathrm{rad/s}$$, but this expression does not.

Question 16

A disk-shaped platform rotates with constant angular velocity. A student wants to measure the frictional torque. Which quantities should be measured?

Accepted Answer

Angular acceleration and rotational inertia.

Question 17

Two pulleys of radii $$r$$ and $$2r$$ are attached and rotate together. A mass $$m$$ hangs from each, with the larger mass on the larger pulley. Which statement is correct about the net torque and angular acceleration?

Accepted Answer

The net torque is not necessarily zero; the angular acceleration depends on the net torque and the system's moment of inertia.

Question 18

A solid sphere of mass $$m$$ and radius $$r$$ rolls without slipping up a ramp of mass $$2m$$ and angle $$	heta. In $$Trial 1, the ramp is free to slide; in Trial 2, the ramp is fixed. In which trial does the sphere reach a greater height?

Accepted Answer

Trial 2, because the ramp cannot move and all energy goes into raising the sphere.

Question 19

A disk is spinning with angular velocity $$\$$omega_0$$. Oil is added to the axle at t$$ = $$t_1/2$, reducing friction. Which equation best models the torque $$	au$$ as a function of time for t$$ \u003e $$t_1/2$?

Accepted Answer

$$	au = C_1$ \left(1 - \frac{t}{t_1$}ight)$$

AP Physics 1 Rotational Motion & Torque Review – Step-by-Step Study Guidance

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