**AP Physics –
Rotational Mechanics Examples**

1. An
old record player turntable is turned on. It is noted that the disk turns 0.75
revolutions as it goes from rest to 33.3 rpm, clockwise. (a) Determine the
disk’s final angular velocity in radians per second. (b) Determine the disk’s
angular acceleration.

Now suppose the turntable is switched off and the disk takes 2.00 seconds to
come to a stop. (c) Determine the angular acceleration. (d) Determine the
angular displacement that occurs during the slow down.

2. Refer to the figure below. (a)
Determine the net torque about point C. (b) Determine the net torque about
point B. (c) Find the point along line BC about which net torque is zero.

3. Suppose two arbitrary point masses are located at arbitrary points in a vertical plane. Show that gravity’s torque about an origin is equal to the cross product of the position of the center of mass and the total weight of the two masses. Would this result work for a larger collection of masses?

4. Suppose
a force *F* acts on a mass *m*. Show that the torque about an
arbitrary origin divided by the angular acceleration equals *mr*^{2}.
This quantity is called “moment of inertia”.

5. Masses of 3.0 kg and 2.0 kg are attached to the ends of a rod of length 0.800 m that has negligible mass. (a) Determine the moment of inertia about the rod’s center. (b) Determine the moment of inertia about the 3.0 kg mass. (c) Determine the moment of inertia about the 2.0 kg mass.

6. Find
the moment of inertia of a thin rod of mass *M* and length *L* about
an axis that passes perpendicularly through its center. Repeat for an axis
that passes perpendicularly through its end.

7. Find the moment of inertia of a thin ring rotating about an axis that passes perpendicularly through its center.

8. Find the moment of inertia of a thin disk rotating about an axis that passes perpendicularly through its center.

9. Find the moment of inertia of a thin disk rotating about an axis that passes along its diameter.

10. Use the parallel axis theorem to determine the moment of inertia of a thin rod rotating about an axis that passes perpendicularly through one end.

11. Use the parallel axis theorem to determine the moment of inertia of a thin rectangular plate rotating about an axis that passes perpendicularly through its center.

12. A certain gyroscope has a solid disk of mass 195 g and radius 3.80 cm which turns with very little friction on an axle of radius 0.10 cm. The gyroscope is set into motion by wrapping 40.0 cm of string around the axle and pulling the end with a constant force of 5.0 N. (a) Determine the rate of angular acceleration. (b) Determine the angular speed that results from this action.

13. Two unequal masses, *m*_{1}
and *m*_{2}, are attached to a string that passes over a pulley
with moment of inertia *I* and radius *r*. Determine the linear
acceleration of the masses. Determine the angular acceleration of the pulley.

14. Determine the acceleration
of a bicycle in terms of the tangential force, *F*, applied to the pedal,
as a function of the following: *m* = mass of each wheel, *M* = mass
of frame and rider, *R* = radius of each wheel, *r*_{1} =
radius of front sprocket, *r*_{2} = radius of rear sprocket, *l*
= length of pedal’s moment arm. Assume bearings are frictionless, rolling
resistance is negligible, and that the chain, sprockets, and spokes have
negligible mass. Hint: analyze torques and rotation of each wheel and the
front chain sprocket and the linear acceleration of the entire system of
masses.

Answers:

1. a.
3.49 rad/s

b. 4.6 rad/s^{2}, CW

c. 1.74 rad/s^{2}, CCW

d. 3.49 rad CW

2. a.
125 Nm, CW

b. 233 Nm, CCW

c. 2.50 m upward from pt. C

3.

4.

5. a.
0.48 kg m^{2}

b. 1.3 kg m^{2}

c. 1.9 kg m^{2}

6. * _{}*,

7. *I*
= *MR*^{2}

8. _{}

9. _{}

10. * *

11.

12. a. 35.5 rad/s^{2}

b. 169 rad/s

13. _{} , _{}

14. _{}