A digital audio compact disc carries data along a continuous spiral track from t

Question

A digital audio compact disc carries data along a continuous spiral track from the inner circumference of the disc to the outside edge. Each bit occupies 0.6 µm of the track. A CD player turns the disc to carry the track counterclockwise above a lens at a constant speed of 1.45 m/s.

(a) Find the required angular speed at the beginning of the recording where the spiral has a radius of 2.30 cm.
rad/s

(b) Find the required angular speed at the end of the recording, where the spiral has a radius of 5.00 cm.
rad/s

(c) A full-length recording lasts for 69 min, 3 s. Find the average angular acceleration of the disc.
rad/s2

(d) Assuming that the acceleration is constant, find the total angular displacement of the disc as it plays.
rad

(e) Find the total length of the track.
km

Explanation / Answer

a) use, w1 = v/r1

= 1.45/0.023

= 63 rad/s

b) w2 = v/r2

= 1.45/0.05

= 29 rad/s

c) time taken, t = 69*60 + 3

= 4143 s

angular acceleration, alfa = (w2 - w1)/t

= (29 - 63)/4143

= -0.0082 rad/s^2

d) angular displacement, theta = w1*t 0.5*alfa*t^2

= 63*4143 + 0.5*(-0.0082)*4143^2

= 1.906*10^5 rad

e) length of the track = theta*r (here r is average radius)

= 1.906*10^5*(0.023 + 0.05)/2

= 6957 m

= 6.957 km

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