Compact Disc Problem: Average Angular Acceleration

[sukreth]

Homework Statement

The inner and outer radii of a compact disc are 25 mm and 58 mm. As the disc spins inside a CD player, the track is scanned at a constant linear speed of 1.25m/s. The maximum playing time of a CD is 74.0 min. 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.

Homework Equations

v=wr
angular acceleration=rw^2

The Attempt at a Solution

I tried to integrate wr somehow, but w is not constant.

 

[G01]

You don't need to integrate.

First of all, $$\omega$$, is the angular frequency. Your second formula is wrong. It should read;

$$a=r\omega^2$$ where a is the linear acceleration.

Either way, all you should need to solve this is the first formula.

You know that v=1.25m/s.

Can you use this, and your first formula to find the angular acceleration? If so, it should just be an algebra problem to find the average angular acceleration.

 

[ogb p]

I also tried to use the fact that the linear speed is constant to find the angular speed, but I got stuck.

To solve this problem, we can use the relationship between linear speed and angular speed, v=wr, where v is the linear speed, w is the angular speed, and r is the radius. We also know that the maximum playing time of a CD is 74.0 min, which is equivalent to 4440 seconds.

First, we can find the angular speed of the disc by dividing the linear speed by the outer radius, as the track is scanned at a constant linear speed of 1.25m/s and the outer radius is 58 mm.

w = v/r = (1.25 m/s) / (0.058 m) = 21.55 rad/s

Next, we can use the equation for average angular acceleration, angular acceleration = (final angular speed - initial angular speed) / time, to find the average angular acceleration during the 74.0-min playing time.

angular acceleration = (0 - 21.55 rad/s) / (4440 s) = -0.00486 rad/s^2

Since the direction of rotation is defined as positive, the negative sign indicates that the disc is slowing down during its playing time. This makes sense as the disc starts at rest and gradually slows down as it reaches the end of the playing time.

Therefore, the average angular acceleration of a maximum-duration CD during its 74.0-min playing time is -0.00486 rad/s^2.