AP Physics 1 6.3.C Angular Impulse & Momentum

[Maximum number: 6]

(7 points, suggested time 13 minutes)

A block of unknown mass is attached to a long, lightweight string that is wrapped several turns around a pulley mounted on a horizontal axis through its center, as shown. The pulley is a uniform solid disk of mass M and radius R. The rotational inertia of the pulley is described by the equation $I=\frac{1}{2} M R^{2}$ . The pulley can rotate about its center with negligible friction. The string does not slip on the pulley as the block falls.

When the block is released from rest and as the block travels toward the ground, the magnitude of the tension exerted on the block by the string is $F_{\mathrm{T}}$ .

Consider scenarios 1 and 2 at the end of time interval $\Delta t$ . In a clear, coherent paragraph-length response that may also contain equations and drawings, explain why the change in angular momentum of both pulleys is the same but the change in rotational kinetic energy is greater for the disk.

Figure 1

For indicating that the torque, $\tau$ , is the same for both pulleys

1 point

For indicating that the impulse, $\tau \Delta t$ , (or change in momentum $\Delta L$ ) is the same for both
pulleys because $\tau$ and $\Delta t$ are the same

1 point

For indicating that the rotational inertia, I, of the disk and hoop are different

1 point

For providing reasoning that because the rotational inertia, I, are different for the disk and
hoop, the kinematic quantities ( $\Delta \theta, \omega, \alpha$ ) are also different for the disk and hoop

1 point

For one of the following:
- Indicating that because $\Delta \theta$ is greater for the disk the work done on the disk is
greater

1 point

For a logical, relevant, and internally consistent argument that follows the guidelines
described in the published requirements for the paragraph-length response

1 point

Example Response
The rotational inertia, I, of the hoop is larger than the rotational inertia of the disk because the hoop's mass is all on the outside instead of distributed throughout like the disk. Equal forces are applied to both pulleys at the same distance, which means that the torques exerted on the pulleys will also be equal. Since the same torque is applied to both pulleys for the same time period, the change in angular momentum will be the same for the disk and hoop. The magnitude of the angular velocity for the hoop will be smaller than that of the disk since angular velocity is inversely proportional to the rotational inertia $\left(\omega=\frac{L}{I}\right)$ . Since kinetic energy is proportional to rotational inertia and the square of angular velocity $\left(K_{R}=\frac{1}{2} I \omega^{2}\right)$ , the difference in angular velocity more greatly affects the rotational kinetic energy. That means the disk will have a greater rotational kinetic energy than the hoop.

Total for part (b) for question 47 points

AP Physics 1: Algebra-Based 6.3 C Relate an Objects or Rigid Systems Change in Angular Momentum to the Angular Impulse Given to That Object or Questions

Exam points

AP Physics 1: Algebra-Based 6.3 C Relate an Objects or Rigid Systems Change in Angular Momentum to the Angular Impulse Given to That Object or Questions question 1