Let\'s begin with the angular acceleration of a compact disk (CD). To play music

Question

Let's begin with the angular acceleration of a compact disk (CD). To play music the CD must rotate at high speed while a laser reads data encoded in a spiral pattern on the disk. The disc has radius R=6.0cm=0.060m; when data are being read, it spins at 7200 rev/min. What is the CD's angular velocity in radians per second? How much time is required for it to rotate through 90? If it starts from rest and reaches full speed in 4.0 s, what is its average angular acceleration?

SOLUTION

SET UP We make a sketch showing the data (Figure 1) . We need to be careful to use consistent units; we’ll choose radians, meters, and seconds.

SOLVE To convert the disc's angular velocity to radians per second, we multiply by (1 min/60 s) and by (2 rad/1 rev):

=7200rev1min=7200rev1min(1min60s)(2rad1rev)=754rad/s

From the definition of angular velocity (=/t), we find the time t for the disc to turn through 90 (/2rad):

t==/2rad754rad/s=0.0021s=2.1ms

From the definition of angular acceleration(av=/t), if the disc reaches a final angular velocity of 754 rad/s in 4.0 s (starting from rest), the average angular acceleration is

av=t=754rad/s4.0s=189rad/s2

REFLECT A typical circular (table or radial-arm) saw used in wood-working shops runs at 3600 rev/minand reaches full speed about 1 s after it is turned on. The radius of a typical blade is 12 cm. Note, however, that the radius is not needed to determine the angular acceleration calculation. The angular acceleration is about twice that for the CD.

Part A - Practice Problem:

An old-fashioned vinyl record is designed to turn at 33 rev/min. Find the average angular velocity.

Express your answer in radians per second to two significant figures.

*I got 3.5 rad/s.

Part B - Practice Problem:

Find the angular acceleration of the record if it spins through ten full rotations before coming to a stop when the record player is turned off.

Express your answer in radians per second squared to two significant figures. rad/s2

*Im having trouble with this one.

Explanation / Answer

for 10 full rotations the angular displacement is theta = 10*2*3.142 = 62.84 rad

final angular velocity is wf = 0 rad/s

initial angular velocity is wi = 33*2*3.142/60 = 3.5 rad/s

then Apply

wf^2-wi^2 = -2*alpha*theta

alpha = wi^2/(2*theta) = 3.5*3.5/(2*62.84) = 0.0974 rad/s^2

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