Acompact disc (cd) stores music in a coded pattern of tiny pits 10−7m deep. the pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 mm and 58.0 mm, respectively. as the disc spins inside a cd player, the track is scanned at a constant linear speed of 1.25 m/s. part a what is the angular speed of the cd when scanning the innermost part of the track? b what is the angular speed of the cd when scanning the outermost part of the track? c the maximum playing time of a cd is 74.0 min. what would be the length of the track on such a maximum-duration cd if it were stretched out in a straight line? d what is the average angular acceleration of a maximum-duration cd during its 74.0-min playing time? take the direction of rotation of the disc to be positive.

(a) 50 rad/s

The angular speed of the CD is related to the linear speed by:

where

is the angular speed

v is the linear speed

r is the distance from the centre of the CD

When scanning the innermost part of the track, we have

v = 1.25 m/s

r = 25.0 mm = 0.025 m

Therefore, the angular speed is

(b) 21.6 rad/s

As in part a, the angular speed of the CD is given by

When scanning the outermost part of the track, we have

v = 1.25 m/s

r = 58.0 mm = 0.058 m

Therefore, the angular speed is

(c) 5550 m

The maximum playing time of the CD is

And we know that the linear speed of the track is

v = 1.25 m/s

If the track were stretched out in a straight line, then we would have a uniform motion, therefore the total length of the track would be:

(d)

The angular acceleration of the CD is given by

where

is the final angular speed (when the CD is scanned at the outermost part)

is the initial angular speed (when the CD is scanned at the innermost part)

is the time elapsed

Substituting into the equation, we find