Study Guide: Rotational Motion
Test Date: Wednesday, February 8th Types of Questions: 20 MC and 3 FRQ
Points: 100
Topics: Center of Mass, Inertial Mass vs Gravitational Mass, Torque, Rotational Inertia, Rotational Kinematics (motion),
Rotational Dynamics (torque and motion), Rotational Energy, Angular Momentum
1. A cue ball moving 2 m/s hits an 8 ball which is at rest. Both balls fly off in
different directions at different speeds. Both balls have the same mass.
a. Mark the location of the center of mass of the two-ball system
before the collision.
b. Mark the location of the center of mass after the collision.
c. In what direction does the center of mass move after the
collision?
d. Which is true about the velocity of the center of mass (vcm) after
the collision?
____ vcm increases _____ vcm decreases ___vcm stays the same.
Explain.
2. A student sets a cart on an incline of 30° and attaches a force sensor to it so it does not slide down. He records the
reading on the sensor. After doing some trig and setting up a net force equation, the student comes up with the
mass of the object. Did he measure the inertial mass or the gravitational mass of the cart? Explain.
3. A door (seen from above in the figures below) has hinges on the left hand side. Which force produces the largest
torque? The magnitudes of all forces are equal.
4. What amount of weight must be added in order for the beam pictured above to balance?
Answer on another piece of paper:
5. A 0.5 kg meter stick is pivoted at the 30 cm mark. A mass of 1.5 kg is placed at the 5cm mark, a mass of 1.2 kg is
placed at the 80 cm mark, and an upward force of 10 N is applied at the 90 cm mark. Determine the net torque
around the pivot point and state the direction of rotation. (Hint: Sketch a picture first.)
6. A solid cylinder and a hollow cylinder both of mass M and radius R roll down an incline plane. Which will reach the
bottom first? Explain.
7. Which will have a larger rotational inertia: A hollow sphere of mass M with radius 2R or a hollow sphere of mass 2M
and radius R? Explain.
8. A wheel has a rotational inertia of 3.00 kgm2. It is subjected to a 3.50 Nm of torque in the clockwise direction for 5
seconds.
a. Sketch the three rotational motion graphs. Assume the wheel started from rest.
b. Solve for the angular acceleration of the wheel.
c. What is the final angular velocity of the wheel?
d. What angle did the wheel rotate through in this time period. State your answer in radians, degrees, and
revolutions.
9. List the three formulas that connect translational motion and rotational motion. Under what circumstances can you
use the three translation-rotation connector formulas?
10. A large ball with a mass of 5 kg and a radius of 45 cm is rolling and completes three full rotations without slipping in
8 seconds (at a constant rate).
a. How far did the ball’s center of mass move?
b. What is the angular velocity of the ball?
11. Two identical balls are released from rest from the top of two different ramps. One ramp is frictionless and the
other is not.
a. Which ball has the fastest translational velocity at the bottom of the ramp? Explain.
b. Write a conservation of energy equation for each ramp.
c. Solve each equation for v, the velocity of the center of mass at the bottom of the ramp. Does your answer
confirm what you said in a?
12. Suppose the large disk of mass 2 kg, radius 0.2 m and initial angular velocity of 50
rad/s and a small disk with mass 4 kg, radius 0.1m and initial angular velocity of
200 rad/s are pushed into one another. (I = ½ Mr2)
a. Find the common final angular velocity after the disks collide.
b. Is kinetic energy conserved?
13. A door that is 1 meter wide, has a mass of 15 kg, and is hinged at one side so it can rotate without friction
about a vertical axis. A bullet having mass 10 grams and speed 400 m/s is fired into the door, in a direction
perpendicular to the plane of the door, and embeds itself at the exact center of the door. Find the angular
velocity of the door just after the bullet embeds itself. ( Idoor = 1/3 (ML 2) )
Rotational Motion: Multiple Choice Practice
14. An ice skater is spinning about a vertical axis with arms fully extended. If the arms are pulled in closer to the body, in which of
the following ways are the angular momentum and rotational kinetic energy of the skater affected?
Angular Momentum
Rotational Kinetic Energy
(A) Increases
Increases
(B) Increases
Remains Constant
(C) Remains
Constant Increases
(D) Remains Constant
Remains Constant
15. A wheel with rotational inertia I is mounted on a fixed, frictionless axle. The angular speed
to
A)
ωf in a time interval T.
𝜔𝑓
𝑇
B)
𝐼𝜔𝑓2
𝑇
C)
𝐼𝜔𝑓2
𝑇2
ωof the wheel is increased from zero
What is the average net torque on the wheel during this time interval?
D)
𝐼𝜔𝑓
𝑇
16. Multiple Correct. A disk sliding on a horizontal surface that has negligible friction collides with a rod that is free to move and
rotate on the surface, as shown in the top view above. Which of the following quantities must be the same for the disk-rod system
before and after the collision? Select two answers.
I. Linear momentum
II. Angular momentum
III. Kinetic energy
(A) Linear Momentum
(B) Angular Momentum
(C) Kinetic Energy
(D) Mechanical Energy
17. A particle of mass m moves with a constant speed v along the
dashed line y = a. When the x-coordinate of the particle is xo, the
magnitude of the angular momentum of the particle with respect to
the origin of the system is
(A) zero (B) mva (C) mvxo (D)√x0 + a
FRQ Practice: Rotational Motion
FRQ #1 Dynamics: Two masses m1 and m2 are connected by light cables to the perimeters of two cylinders of radii r1 and
r2, respectively, as shown in the diagram above. The cylinders are rigidly connected to each other but are free to rotate
without friction on a common axle. The moment of inertia of the pair of cylinders is I = 45 kg•m 2
Also r1=0.5 meter, r2=1.5 meters, and m1=20 kilograms.
a. Determine m2 such that the system will remain in equilibrium.
The mass m2 is removed and the system is released from rest.
b. Determine the angular acceleration of the cylinders.
c. Determine the tension in the cable supporting m1
d. Determine the linear speed of m1 at the time it has descended 1.0 meter.
FRQ#2 Rolling: A large sphere rolls without slipping across a horizontal surface. The sphere has a constant translational
speed of 10 meters per second, a mass m of 25 kilograms, and a radius r of 0.2 meter. The moment of inertia of the
sphere about its center of mass is I = 2mr2/5. The sphere approaches a 25° incline of height 3 meters as shown above
and rolls up the incline without slipping.
a. Calculate the total kinetic energy of the sphere as it rolls along the horizontal surface.
b. i. Calculate the magnitude of the sphere's velocity just as it leaves the top of the incline.
ii. Specify the direction of the sphere's velocity just as it leaves the top of the incline.
c. Neglecting air resistance, calculate the horizontal distance from the point where the sphere leaves the incline to the
point where the sphere strikes the level surface.
d. Suppose, instead, that the sphere were to roll toward the incline as stated above, but the incline were frictionless.
State whether the speed of the sphere just as it leaves the top of the incline would be less than, equal to, or greater than
the speed calculated in b. Explain briefly.
FRQ #3: Dynamics A uniform solid cylinder of mass m1 and radius R is mounted on frictionless bearings about a fixed axis
through O. The moment of inertia of the cylinder about the axis is I = ½m1R2. A block of mass m2, suspended by a cord
wrapped around the cylinder as shown above, is released at time t = 0.
a. On the diagram below draw and identify all of the forces acting on the cylinder and on the block.
b. In terms of m1, m2, R. and g, determine each of the following.
i. The acceleration of the block
ii. The tension in the cord