An audio CD carries musical data as a series of digital \"bits\" along a continu

Question

An audio CD carries musical data as a series of digital "bits" along a continuous spiral track from the inner curcumfrence (r=2.3cm) of the disk to the outer edge (r=5.8 cm). A CD player turns the disk to move the track counterclockwise above a lens at a constant linear speed of 1.3 m/s. The average recording lasts for 75 minutes.

1.What are the required angular speeds at the begining and the end of the recording?
2. What is the average angular acceleration of the discas it spins through the entire recording?
3. Assuming constant acceleration, how many revolutions does the CD make for the entire recording?

Explanation / Answer

Data: Radius of the inner circumference, r1 = 0.023 m Radius of the outer edge, r2 = 0.058 m Linear speed, v = 1.3 m/s Time, t = 75 min              = 75 * 60 s              = 4500 s Solution: (a) Angular speed at the beginning: 1 = v / r1       = 1.3 / 0.023       = 56.52 rad/s Angular speed at the end: 2 = v / r2       = 1.3 / 0.058       = 22.41 rad/s Ans: Angular speed at the beginning, 1 = 56.52 rad/s Angular speed at the end, 2 = 22.41 rad/s (b) Average angular acceleration: = ( 2 - 1 ) / t    = ( 22.41 - 56.52 ) / 4500    = - 7.58 x 10^-3 rad/s^2 Ans: Avg angular acceleration = - 7.58 x 10^-3 rad/s^2 (c) = [ ( 1 + 2 ) / 2 ] * t    = [ ( 56.52+ 22.41 ) / 2 ] * 4500    = 1.78 x 10^5 rad    = ( 7.976 x 10^4 / 2 ) rev    = 2.83 x 10^4 rev Ans: No. of rev = 2.83 x 10^4 rev

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