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 1.21 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 5.35 cm. (c) A full-length recording lasts for 61 min, 29 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.

Explanation / Answer

v = r

v = 1.21 m/s constant speed

inner radius = 2.30 cm = .0230 m

a) = v/r = 1.21/ .0230 = 52.6086 rad/s


b) angular speed at the end of the recording, where the spiral has a radius of 5.35cm.

= v/r = 1.21/0.0535 = 22.616 rad/s

c)
Average Angular Acceleration
• The average angular acceleration of an
object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change:

= / t
Unit: rad/s²

total time = 61 min 29 s = 61x60 + 29 = 3689 s

= (22.616 - 52.6086)/3689 = -29.99/3689 = -0.008130 rad/s²

d) since is constant average = (22.616 + 52.6086)/2 = 37.6123 rad/s

t = 3689 s
= total angular displacement

=> = /t => = x t = 3689 x 37.6123 = 138751.77 radians

e) the total length of the track

distance = speed x time = 1.21 m/s x 3689 = 4463.69 m or 4.46369 km

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.