Average angular acceleration of a CD

[cycam]

Homework Statement

Compact Disc. A compact disc (CD) stores music in a coded pattern of tiny pits 10^-7 m deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 mm and 58.0 mm, respectively. As the disc spins inside a CD player, the track is scanned at a constant linear speed of 1.25 m/s.

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

alpha = (w_out - w_in)/ t

The Attempt at a Solution

I found the angular speeds of the inner and outer radii, but I'm not sure how to apply it to this problem. When I plug in the values into the equation above, I get -0.006 rad/s^2 from [(21.55-50)rad/s]/ [(74min)(60s/min)], which seems to be the incorrect answer.

Any help would be appreciated :] Thanks in advance.

 

[anathema]

Thank you for your question! I can help you with your problem. First, let's review the given information. We know that the CD stores music in a spiral track with inner and outer radii of 25.0 mm and 58.0 mm, respectively. The track is scanned at a constant linear speed of 1.25 m/s. We also know that the playing time of the CD is 74.0 minutes.

To find the average angular acceleration of the CD, we can use the equation alpha = (w_out - w_in)/t, where alpha is the angular acceleration, w_out is the final angular speed, w_in is the initial angular speed, and t is the time interval. We can calculate the angular speeds using the formula w = v/r, where v is the linear speed and r is the radius.

So, for the inner radius, we have w_in = (1.25 m/s) / (0.025 m) = 50 rad/s. And for the outer radius, we have w_out = (1.25 m/s) / (0.058 m) = 21.55 rad/s. Plugging these values into the equation, we get alpha = (21.55 - 50) rad/s^2 / (74 min * 60 s/min) = -0.006 rad/s^2. Your answer is correct, but it is negative because we took the direction of rotation to be positive. If we take the opposite direction as positive, we would get a positive value for alpha.

I hope this helps you understand the problem better. Keep up the good work!



Scientist

 

[m2003]

The average angular acceleration of a CD can be calculated by dividing the change in angular velocity by the time it takes to change. In this case, the angular velocity changes from 21.55 rad/s at the inner radius to 50 rad/s at the outer radius, over a period of 74 minutes (or 4,440 seconds). Therefore, the average angular acceleration of the CD can be calculated as follows:

alpha = (w_out - w_in)/t
= (50 rad/s - 21.55 rad/s) / 4,440 s
= 0.006 rad/s^2

This means that the CD has an average angular acceleration of 0.006 rad/s^2 during its 74-minute playing time. This value is positive, indicating that the direction of rotation is consistent with the direction assumed in the problem (counter-clockwise). It is important to note that this is an average value and the actual angular acceleration may vary at different points on the disc due to factors such as friction and imperfections in the disc.