Rotational Motion
Studio Physics I
Part I-Velocity and Acceleration in Rotations
The solid disk shown at the right is is spinning counter-clockwise.
At a certain instant in time, three points on the disk (A,B and C)
are located as shown.
B
A
C
Figure 1
1. Copy this picture onto your paper and draw arrows at each of the three points which
indicate the direction of the linear velocity of each point at this moment.
2. How does the time needed for point A to complete one revolution compare to the time
needed for point B to complete one revolution? How does the time needed for point A to
complete one revolution compare to the time needed for point C to complete one
revolution?
3. How does the distance traveled by point A in completion of one revolution compare to
the distance traveled by point B in completion of one revolution? How does the distance
traveled by point A in completion of one revolution compare to the distance traveled by
point C in completion of one revolution?
4. Based on your answers to questions 11 and 12 above, how do the linear speeds of
points A, B and C compare? Explain your reasoning in complete sentances using the idea
that average velocity is a distance to time ratio.
5. Figure 2 shows the disk a short time after the moment pictured in Figure 1. During
this time, the disk has rotated through a half of one revolution. For each of the three
points, draw an arrow indicating the direction of the linear velocity at the point now.
C
Figure 2
A
B
6. How do the linear velocities of each point compare between this later moment in
time and the first moment in time, pictured in Fig. 1? Discuss both the magnitude and the
direction of the velocity. Is the linear velocity of a given point (say point A) on the disk
changing with time? Why or Why not?
 1999-2001 K. Cummings
7. Is there one single linear velocity that applies to all points on this disk at all times?
Explain your reasoning in complete sentences.
8. Suppose the disk shown in Fig. 1 and Fig. 2 completes one rotation in 10 seconds.
For each of the three points A,B and C, find the change in angle  (measured in radians)
during 1 second.
9. What is the rate of change in the angle of point A on the disk? What is the rate of
change in the angle of point B on the disk? What is the rate of change in the angle of
point C on the disk?
The rate of change in the angle that you calculated in question 9 above is the
magnitude of the angular velocity vector . In order to determine the direction of the
angular velocity vector, we must use the right hand rule. This is done by curling the
fingers of your right hand in the direction of motion for the point or object in question.
Your extended thumb then points in the direction of the angular velocity vector.
B
A
C
Figure 3
10. If the disk shown above is rotating counter-clockwise, what is the direction of the
angular velocity of point A? What about point B?
11. If the disk shown above is rotating clockwise, what is the direction of the angular
velocity of point A? What about point C?
12. Is there one single angular velocity that applies to all points on this disk at all times?
Explain your reasoning in complete sentances.
13. If the angular velocity of a disk is changing, then disk has an angular acceleration .
How do you think that the direction of angular velocity is related to the direction of
angular acceleration if the disk is speeding up? (Are the directions the same? Opposite?
Or what?) How do you think that the directions of angular velocity and acceleration are
related if the disk is slowing down? State your answer using complete sentences.
14. Suppose that the disk shown above in Fig 3 is moving as described below, what is
the direction of its angular acceleration?
a) Disk is rotating clockwise and speeding up?
b) Disk is rotating counter-clockwise and slowing down?
c) Disk is rotating counter-clockwise and speeding up?
d) Disk is rotating clockwise and slowing down?
 1999-2001 K. Cummings
Part II- Introduction to Rotational Inertia
15. Consider the system of point particles shown. Each “particle” has the same mass
(50 g) and is the same distance (12 cm) from the center point (which is marked with an X).
There is a rod connecting the masses, but it is so light that it can be ignored. Calculate the
object’s rotational inertia for a rotation about the center point. (Hint: The rotational inertia for
an object made up of point masses is calculated as I=mr2)
.
X
16. The locations of the point masses are now changed. They are pushed toward the center of
the rod. See the diagram below. The masses are now each 2 cm from the center point. What
is the object’s rotational inertial about the center point now? (mass is still 50 g).
X
17. Your two answers above should not be the same, even though we consider the same
amount of mass in both cases. Discuss how the distribution of mass for an object with large
rotational inertia compares to the distribution of mass for one with small rotational inertiaassuming we keep the total mass of the object constant.
We are now going to investigate the concept of rotational inertia experimentally. You will
need a file called rotational inertia.mbl You can get it from the course web page.Go to the
activity listing and scroll to the bottom. Remember to transfer the file with internet
explorer, not netscape. On your work table is a rod with two movable masses that are
attached to a rotational motion sensor. Configure the masses so that they are very close to
the center of the rod (sort of like the figure shown in question # 16 above). Be sure to
completely tighten the screws on the masses. If they fly off the rod, someone could get hurt.
Open the LoggerPro file, click collect, and then give the rod as hard a push as you can.
Observe what happens. (The rod slows down because the rotational motion sensor is not
frictionless.)
18. Note on your activity sheet the maximum velocity the object has (right after the push)
and how long it takes for it to reach a zero velocity. Repeat this measurement -doing your
best to reproduce the same force you used the first time. (That is, push as hard as you can).
Note the maximum velocity and time to reach zero velocity for each of the trials. Given the
two trials that you have made, what is the approximate average of the maximum velocity you
can produce when you push the object as hard as you can. About how long does it take on
average for the velocity to reach zero?
19. Now reconfigure masses so that the masses are out toward the ends of the rod(sort of like
the figure in question #15 ). Theoretically, does the rotational inertia of the object increase,
decrease or stay the same as a result of this change?
20. Now repeat the experiment with the masses in their new positions. To do this,
click collect, give the rod as hard a push as you can. Observe what happens. Note
on your activity sheet the maximum velocity the object has (right after the push) and
 1999-2001 K. Cummings
how long it takes for it to reach a zero velocity. Repeat this measurement -doing your
best to reproduce the same force you used the first time. (That is, push as hard as you
can). YOU DO NOT HAVE TO WAIT FOR THE VELOCITY TO REACH
ZERO THE SECOND TIME-IT IS JUST TOO BORING. Note the maximum
velocity. Given the two trials that you have made, what is the approximate average of
the maximum velocity you can produce when you push the object as hard as you can.
About how long does it take on average for the velocity to reach zero? How do these
values compare to what you found for the rod with the masses in toward the center?
21. Rotational inertia plays the same role in rotational motion as mass plays in linear
motion. Based on F=ma, for a given force, how do the accelerations of a larger mass
object and smaller mass object compare? (That is, which one would have the larger
acceleration)?
22. Discuss how your results for the experiments with the two configurations of the
masses are consistent with the idea that for a given force a larger rotational inertial is
associated with a smaller angular acceleration. Discuss BOTH the maximum (initial)
velocity and the time it takes for the object to stop rotating. (A rotation producing
force is called a torque).
 1999-2001 K. Cummings