Rotational Kinematics AP Physics C Mechanics FRQ – Exam Style Questions etc.
Rotational Kinematics AP Physics C Mechanics FRQ
Unit 5: Torque and Rotational Dynamics
Weightage : 10-15%
Question
3. A solid uniform disk is supported by a vertical stand. The disk is able to rotate with negligible friction about an axle that passes through the center of the disk. The mass and radius of the disk are given by \(M_{d}\) and R, respectively. The rotational inertia of the disk is \(I_{d} = \frac{1}{2}M_{d}R^{2}\). A string of negligible mass is draped over the disk and attached to the top of the disk at point P. One end of the string is connected to an unstretched ideal spring of spring constant k , which is fixed to the ground as shown in Figure 1.
A block of mass \(m_{B}\) is then attached to the string on the right side of the disk. The block is slowly lowered until the spring-disk-block system reaches equilibrium, as shown in Figure 2. In this equilibrium position, the disk has rotated clockwise through a small angle \(\Theta \).
Give all algebraic answers in terms of \(M_{d}\) , R, k , \(\Theta \), and physical constants, as appropriate.
(a) Derive an expression for the mass \(m_{B}\) of the block.
(b) At time t = 0, the string on the right side of the disk is cut and the block falls to the ground. On the circle below, which represents the disk, draw and label the forces (not components) that act on the disk immediately after the string is cut and the block is falling to the ground. Each force should be represented by an arrow that starts on and is directed away from the point of application.
(c) Derive an expression for the angular acceleration \(\alpha \) of the disk immediately after the string is cut.
(d) At t = \(t_{1}\), the disk has rotated and point P is again directly above the axle. Sketch a graph of the magnitude of the angular velocity \(\omega \) of the disk as a function of time t from t = 0 to t = \(t_{1}\).
(e) The disk is adjusted on the support so that the axle does not pass through the center of mass of the disk. The block is again hung on the right side of the disk and the spring-disk-block system comes to equilibrium, as shown in Figure 3. The axle does not exert a torque on the disk. For each force on the disk, indicate whether the magnitude of the torque about the axle caused by that force increases, decreases, or stays the same relative to part (b).
▶️Answer/Explanation
3(a) Example Response
For indicating that the sum of the torques on the disk equals zero
\(\sum \tau _{on\\\ disk} = 0\)
\(\tau _{g} = \tau _{s}\)
OR
For indicating that the sum of the forces equals zero
\(\sum F = 0\)
\(F_{g} = F _{s}\)
For correctly substituting the expressions for the forces
\(F_{g}R = F_{s}R\)
\(m_{B}gR = k\Delta xR\)
\(m_{B}g = k\Delta x\)
OR
\(F_{g} = F _{s}\)
\(m_{B}g = k\Delta x\)
For correctly substituting for △x
\(m_{B}g = kR\Theta \)
\(m_{B} = \frac{kR\Theta }{g}\)
3(b) Example Response
Scoring Notes:
- Scoring Note: Examples of appropriate labels for the force due to gravity include: \(F_{G}\), \(F_{g}\), \(F_{grav}\), W , mg , Mg , “grav force”, “F Earth on block”, “F on block by Earth”, \(F_{Earth\\\ on\\\ Block}\), \(F_{E\\\ ,\\\ Block}\). The labels G and g are not appropriate labels for the force due to gravity. \(F_{n}\), \(F_{N}\), N , “normal force”, “ground force”, or similar labels may be used for the normal force, which can be used instead of \(F_{axle}\). \(F_{spring}\), \(F_{s}\), \(T_{spring}\), T, “string force,” or similar labels may be used for the tension force exerted by the string.
- A response with extraneous forces or vectors can earn a maximum of two points.
3(c) Example Response
For indicating that the net torque is due only to the force exerted on the disk by the tension in the rotational form of Newton’s second law
\(\tau _{s} = I_{a}\alpha\)
For correctly expressing the torque on the disk by the tension in terms of the spring force, which is equal to the tension, and the lever (moment) arm
\(F_{s}R = I_{d}\alpha\)
For correctly substituting for \(F_{s}\)
\(-k\Delta xR = I_{d}\alpha \)
For correctly substituting \(I_{d}\) and \(\Delta x\), or an expression for \(\Delta x\) consistent with part (a)
\(-k(R\Theta )R = \frac{1}{2}M_{d}R^{2}\alpha \)
\(\alpha = -\frac{2k\Theta }{M_{d}}\)
Scoring Note: The negative sign is not necessary to earn this point.
3(d) Example Response
Scoring Note: Any part of the graph beyond t1 is not considered in scoring
3(e) Example Response
The counterclockwise torque due to the tension caused by the spring must increase to counteract the increase in clockwise torques due to the force due to gravity of the disk and tension caused by the force of gravity due to block to keep the disk in equilibrium.
Scoring Notes:
- A response that references the torque due to the force at the axle staying the same can earn all 3 points.
- A response that references the torque due to the force on the axle changing, or any additional torques can earn a maximum of 2 points.