Instructions

This page is designed to get you started using the applet. The applet should be open. The step-by-step instructions on this page are to be done in the applet. You may need to toggle back and forth between instructions and applet if your screen space is limited.

Observing the Motion

Data

Vectors

Observing the Motion

To be able to visualize the motion shown by the applet, imagine a spool made of an inner cylinder (grey) rigidly connecting two outer disks (blue). The spool moves so that its inner cylinder is rolling without slipping on a horizontal surface (salmon-colored). A point (black) drawn on the outer disk facing you is revolving with the disk.

Exercise 1.

RESET the applet. Make sure the spool support is set to "Bottom".

Set the three sliders as follows:

• angular speed (w): 1.00 rad/s
• center-to-point distance (a): 0.25 m
• inner radius of spool (r): 0.15 m

The outer radius of the spool, R, is fixed at 0.50 m.

Select the Trace toggle button , and PLAY the motion.

After the spool has moved some distance, click the REVERSE button , then PLAY again, and then PAUSE . You should be able to construct a looped path as in Figure 1 below.

Figure 1

Exercise 2. REWIND the applet. Set the angular speed to 2.00 rad/s. Then PLAY or REVERSE the motion. You should be able to observe that the revolving point traces out the same path as before, but that the motion is twice as fast.

Exercise 3. RESET the applet, and select "Top" as the the spool support. Restore all other settings to those used in Exercise 1:

• angular speed (w): 1.00 rad/s
• center-to-point distance (a): 0.25 m
• inner radius of spool (r): 0.15 m

PLAY or REVERSE the motion with the path displayed. The new path is a mirror image of the previous path, the previous path being reflected in a horizontal mirror plane. The new path is illustrated in Figure 2 below.

Figure 2

Exercise 4.

RESET the applet, and select "Bottom" for the spool support.

Set the three sliders as follows:

• angular speed (w): 1.00 rad/s
• center-to-point distance (a): 0.25 m
• inner radius of spool (r): 0.50 m

PLAY and REVERSE the motion with the path displayed. You should be able to construct a wavy instead of a looping path, as illustrated in Figure 3 below.

Figure 3

Exercise 1. RESET the applet, and make sure the spool support is set to "Bottom".

Display the two data boxes by selecting the Data button .

Figure 4

Duplicate the constants (not the time elapsed) displayed in the data box on the left in Figure 4 by setting the three sliders as follows:

• angular speed (w): 1.00 rad/s
• center-to-point distance (a): 0.41 m
• inner radius of spool (r): 0.25 m

Exercise 2. PLAY or REVERSE the motion. Try to duplicate (approximately) the coordinate values in the data box on the right in Figure 4. The point on the spool would be oriented as in Figure 5 below. All coordinates in the data box on the right are represented in the applet window at a scale of 2 pix = 1 cm.

Figure 5

• (X,Y)-coordinates. These are the coordinates of the center of the spool. The X-coordinate has its origin at the left edge of the applet window. The X-coordinate of the midpoint of the applet window will depend on the width of the window. The Y-coordinate of the center of the spool is chosen to be 0, whether you are working with the "Bottom" or "Top" setting of the spool support.
• (x,y)-coordinates. These are the coordinates of the revolving point relative to the Lab frame, which is fixed relative to the applet window. The origin of the (x,y)-coordinates coincides with that of the (X,Y)-coordinates.
• (x',y')-coordinates. These are the coordinates of the revolving point relative to the frame in which the spool is at rest, i.e., relative to the frame moving along with the spool. The (x',y')-axes are fixed in orientation, however. They are not rotating with the spool. The origin of the (x',y')-coordinates is at the center of the spool.

  The x, x', and X values increase going to the right and the y, y', and Y values going upward.

  Working with the values in the data box on the right in Figure 4 or with those displayed by your applet, verify that the three sets of coordinates satisfy the following addition rules:

  x = x' + X (1a)

  y = y' + Y (1b)

  E.g., using the data from Figure 4, Equation (1a) amounts to

  2.41 m = -0.22 m + 2.63 m (2)

  which adds up correctly.

  

  Vectors

  Exercise 1. Continuing from the previous section, display the vector panel by selecting the Vectors button .

  

  Figure 6

  On the vector panel, as illustrated in Figure 6 above, select all three velocity vectors, velocity of revolving point relative to lab frame, velocity ' of revolving point relative to spool frame, and velocity of spool frame relative to lab frame.

  Then drag so that its tail-end coincides with the tip of '. This positions the three vectors so that one can see that the Galilean velocity addition theorem,

  = ' + , (3)

  is satisfied. Figure 7 below illustrates how the three velocities combine when the spool is at the same point as in Figure 5.

  

  Figure 7

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