Rotational Motion
Studio Physics I
The wheel shown at the right is is spinning counter-clockwise.
At a certain instant in time, three points on the wheel (A,B and C)
are located as shown.
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? Explain your reasoning in complete sentances.
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? Explain your reasoning in complete sentances.
4. Based on your answers to questions 2 and 3 above, how do the linear speeds of points A, B and C compare? Explain your reasoning in complete sentances.
5. Figure 2 shows the wheel a short time after the moment pictured in Figure 1. During this time, the wheel 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.
Figure 2
6. How do the linear velocities 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 velocity of a point on the rim of this wheel changing? Why or Why not?
7. Is there one single linear velocity that applies to all points on this wheel at all times? Explain your reasoning in complete sentences.
8. Suppose the wheel 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 Dq during 1 second.
9. What is the rate of change in the angle of point A on the wheel? What is the rate of change in the angle of point B on the wheel? What is the rate of change in the angle of point C on the wheel?
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.
Figure 3
10. If the wheel shown above is rotating counter-clockwise, what is the direction of the angular velocity of point A? What about point B?
11. If the wheel 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 wheel at all times? Explain your reasoning in complete sentances.
13. If the angular velocity of a wheel is changing, then wheel has an angular acceleration a. How do you think that the direction of angular velocity is related to the direction of angular acceleration if the wheel is speeding up? How do you think that the directions of angular velocity and acceleration are related is the wheel is slowing down? State your answer using complete sentences.
14. Suppose that the wheel shown above in Fig 3 is moving as described below, what is the direction of its angular acceleration?
a) Wheel is rotating clockwise and speeding up?
b) Wheel is rotating counter-clockwise and slowing down?
c) Wheel is rotating counter-clockwise and speeding up?
d) Wheel is rotating clockwise and slowing down?
e) Wheel is rotating clockwise and the angular velocity is not changing?
15. What is the relationship between the linear velocity of a point on this wheel and the angular velocity of the same point? Derive this relationship by following the steps below.
a) Copy the picture of the wheel above onto your paper. Draw an arrow from the center of the circle to the current location of point A. This arrow is a position vector for point A.
b) Now suppose that a time period Dt pass and that the wheel has rotated clockwise a short distance. Put a point on your diagram of the wheel that shows the new location of point A. Draw the position vector to the new location of point A. Sketch the path taken by point A as it moving between these two locations. Is the path a straight line or an arc? Label this path Ds. Label the change in angular position for point A between the two position vectors Dq.
c) Think back to your study of arcs and circles in junior high school math class and state the relationship between arc length s, radius r and angle q. What then is the relationship between the path length traveled by point A Ds, the change in angular position of point A Dq and the radius of the wheel r?
d) What is a simple definition of linear velocity? Use this definition, along with the your answer to part c and the fact that this movement occurred in a time period Dt to derive the relationship between the linear velocity of point A and the angular velocity of point A.
e) What does the relationship you derived in part d imply about how the linear speeds of points on the wheel will vary with the distance of the point form the center of the wheel? Is this consistent with your answers to question # 4?
ã 1999 K. Cummings