On a compact disc (CD), music is coded in a pattern of tiny pits arranged in a track that spirals outward toward the rim of the disc. As the disc spins inside the a CD player, the track is scanned at a constant linear speed of v=1.25m/s. Because the radius of the track varies as it spirals outward, the angular speed of the disc must change as the CD is played. Let's see what angular acceleration is required to keep v constant. The equation of a spiral is r(theta)=r((sub zero)+beta(theta), where r(sub zero) is the radius of the spiral at theta=0and beta is a constant. On a CD, r(sub zero) is the inner radius of the spiral track. If we take the rotation direction of the CD to be positive, beta must be positive so that r increases as the disc turns and theta increases.

(a) When the disc rotates through a small angle dtheta, the distance scanned along the track is ds=r dtheta. Using the above expression for r(theta), integrate ds to find the total distance s scanned along the track as a function of the total angle theta through which the disc has rotated.

(b) Since the track is scanned at a constant linear speed v, the distance s is found in part (a) is equal to vt. Use this to find theta as a function of time. There will be two solutions for theta; choose the positive one, and explain why this is the solution to choose.

(c) Use your expression for theta(t) to find the angular velocity omega(sub z) and the angular accleration alpha(sub z) as functions of time. Is alpha(sub z) constant?

(d) On a CD, the inner radius of the track is 25.0mm, the track radius increases by 1.55um per revolution, and the playing time is 74.0 min. Find r(sub zero), beta, and the total number of revolutions made during the playing time.

(e) Using your results from parts (c) and (d), make graphs of omega(sub z) (in rad/s) versus t and alpha(sub z) (in rad/s^2) versus t between t=0 and t=74.0 min.