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 1.39 m/s.

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

(b) Find the required angular speed at the end of the recording, where the spiral has a radius of 5.25 cm.

(c) A full-length recording lasts for 70 min, 31 s. Find the average angular acceleration of the disc.

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

(e) Find the total length of the track.

Please give equation and answer. Thank YOu !

Explanation / Answer

ω = v/R
v = 0.90 m/s
(a) R = 2.30 cm = 0.023 m, ω = 1.39/0.023 = 60.43 rad/s
(b) R = 5.25 cm = 0.0525 m, ω = 1.39/0.0525 = 26.47 rad/s
(c) Δt = 70 mn 31 s = 4231 s
. .Δω = 26.47- 60.43 = -33.96 rad/s
Angular acceleration:
α = Δω/Δt = -33.96 / 4231 = -0.0080 rad/s ²
(d) Angular displacement:
θ = -(1/2)αt ² = (0.0080/2)×4231 ² = 71605 radian
(e) Total length of the track:
L = vΔt =1.39 × 4231 = 5881.09 m

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