AP Physics C: Mechanics Unit 5 Review: Torque and Rotational Dynamics

Derive an expression for the angular velocity of the disk at time $t = \frac{\pi}{2\omega}$. Express your answer in terms of M, R, $\tau_0$, $\omega$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. In a different scenario, the disk is initially at rest and a time-varying torque $\tau(t) = \tau_0\cos(\omega t)$ is applied to the disk about its axis of rotation, where $\tau_0$ is a positive constant and $\omega$ is a positive constant angular frequency. The torque is applied from time t = 0 to time $t = \frac{\pi}{2\omega}$. There is no string or hanging mass in this scenario.

While the system is accelerating, the magnitudes of the tension forces in the string on each side of the disk are $T_1$ (on the side with block 1) and $T_2$ (on the side with block 2).

Derive an expression for the magnitude of the linear acceleration $a$ of block 1. Express your answer in terms of $m_1$, $m_2$, $M_D$, $R$, $g$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Indicate whether the magnitude of the angular acceleration $\alpha$ of the disk in this scenario is greater than, less than, or equal to the magnitude of the angular acceleration of the disk in the original scenario. In a different scenario, block 2 is replaced with a block of mass $m_3 = 1.0$ kg, as shown in Figure 2. The system is again released from rest. The disk has the same mass $M_D = 2.0$ kg and radius $R = 0.30$ m as in the original scenario.

Draw and label free-body diagrams showing all the forces exerted on the disk and on the block while the block is falling. Use Figure 2 to draw each force vector starting on the dot and pointing in the direction of the force. The length of each vector should be proportional to the magnitude of the force. Clearly label each force using standard notation (e.g., $F_T$ for tension, $F_g$ for gravitational force). While the block is falling and before it hits the ground, the string exerts forces on both the disk and the block.

Derive an expression for the magnitude of the angular acceleration $\alpha$ of the disk while the block is falling. Express your answer in terms of M, m, R, g, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. The block falls with acceleration $a_y$ in the downward direction. The rotational inertia of the disk about its central axis is $I = \frac{1}{2}MR^2$.

Sketch a graph of the angular velocity $\omega$ of the disk as a function of time t from t = 0 to a time significantly after $t_1$ using Figure 3. Your graph should show the behavior both before and after the block hits the ground. Clearly indicate any discontinuities or changes in the relationship between $\omega$ and t. The angular velocity of the disk increases as the block falls. At time $t_1$, the block hits the ground and the string goes slack. After $t_1$, the disk continues to rotate freely about the axle with no friction.

Indicate which disk (Disk 1, Disk 2, or Disk 3) will have the greatest angular velocity immediately after the block hits the ground. If two or more disks will have the same angular velocity, indicate this explicitly. The experiment is repeated three times using different disks, each with radius R = 0.40 m. The same block of mass m = 0.50 kg falls the same distance h = 1.5 m in each trial. The properties of the three disks are shown in Figure 4.