Consider again the CD in Problem 60. As was explained in Example 8.2, the data on the CD a...

Consider again the CD in Problem 60. As was explained in Example 8.2, the data on the CD arc encoded in a long spiral “track,” where the spacing between each turn of the track is 1.6 µm and the inner and outer radii of the program area are about 25 mm and 58 mm, respectively. Estimate the length of this spiral.

Problem 60

Consider the CD in Example 8.2. This CD player is turned on at t = 0 and very quickly starts to play music so that the CD has an angular velocity of 500 rpm just a few seconds after t = 0. The CD then plays for 1 hour, during which time it has a constant angular acceleration and a final angular velocity of 200 rpm. Estimate the total distance traveled by a point on the edge of the CD. Assume the CD has a radius of 6.0 cm.

Example 8.2

Angular Acceleration of a Compact Disc

A compact disc used to play music does not spin with a constant angular velocity. Instead, it spins most rapidly when music is being read from regions nearest the rotation axis (the inner “tracks” near the center of the disc) and slowest when music is played from regions near the edge of the disc (the outer “tracks”). If the angular velocity decreases uniformly from ω = 500 rpm to 200 rpm as the CD is scanned from the innermost to the outermost track, what is the angular acceleration? Assume the CD contains 60 minutes of music.

RECOGNIZE THE PRINCIPLE

We are given that the angular velocity decreases uniformly, which means that the angular acceleration is constant; hence, α = αave. We can compute α from the change in ω during the 60 minutes it takes to play the CD.

SKETCH THE PROBLEM

Figure 8.6 shows the tracks on a CD. These tracks form one very long ,spiral- that begins near the center of the disc and spirals out to the edge.

Figure 8.6 Example 8.2. The information on a compact disc is stored in a very long spiral “track.”

IDENTIFY THE RELATIONSHIPS

The angular velocity ω decreases from 500 rpm to 200 rpm as the CD player moves from the inside to the outside tracks. Because α is constant, we can write Equation 8.5 with a time interval that is not infinitesimally small. The angular acceleration is then given by

SOLVE

The initial angular velocity is ωi = 500 rpm and the final angular velocity is ωf = 200 rpm, which leads to

We now need to convert the units and express the answer in rad/s2:

What does it mean?

The information on a CD is stored in a very “tight” spiral (Fig. 8.6), so at any particular point on the CD, the path containing the musical information is extremely close to being a perfect circle. During one revolution, the length of each of these approximately circular paths is just the circle’s circumference, a distance smaller near the center of the CD than for regions near the edge. The amount of music stored in a circular track is proportional to the length of a track, so a circle near the center contains less music than a circle near the edge. To compensate for this difference, the CD player spins faster when reading music from near the center since the CD player must read (and also play) music at a constant rate.