How Long Is the Track on a Full-Length CD When Stretched Straight?

[Peach]

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.

The maximum playing time of a CD is 74.0 min. What would be the length of the track on such a maximum-duration CD if it were stretched out in a straight line?

Homework Equations

theta = (w_f + w_0)t
s = r*theta

w = angular velocity

The Attempt at a Solution

This is part 3 of the problem so I already have the angular velocity for the innermost and outermost of the disc--which I assume it is the initial and final angular velocity. Pls correct me if I'm wrong. I've used the eqns above to find theta and the length but I'm coming out with the wrong answer, 5.25km. I took the outer radius - inner radius = r. Can someone pls correct me? I don't know what I'm doing wrong.

Edit: Nvm. I was thinking way too complicated. ><;

 

[grant9076]

As the disc spins inside a CD player, the track is scanned at a constant linear speed of 1.25 m/s.

The maximum playing time of a CD is 74.0 min. What would be the length of the track on such a maximum-duration CD if it were stretched out in a straight line?

If it is being tracked at constant linear speed as you said, then you can simply multiply the linear speed by the playing time in seconds to get your answer.

 

[m2003]

I can provide a response to the content and offer a solution to the problem. The statement provides information about the physical structure and scanning mechanism of a compact disc. It also poses a question about the length of the track on a maximum-duration CD if it were stretched out in a straight line.

To solve this problem, we can use the formula s = r*theta, where s is the length of the track, r is the radius of the spiral track, and theta is the angle subtended by the track. We know that the angular velocity is constant, so we can use the formula theta = (w_f + w_0)t, where w_f and w_0 are the final and initial angular velocities, respectively, and t is the time.

First, we need to find the angular velocity of the CD at the innermost and outermost points. We can use the formula w = v/r, where v is the linear speed and r is the radius. Plugging in the values, we get w_inner = 1.25 m/s / 0.025 m = 50 rad/s and w_outer = 1.25 m/s / 0.058 m = 21.55 rad/s.

Next, we can calculate the time it takes for the CD to play at maximum duration, which is 74 minutes or 4440 seconds. So, using the formula theta = (w_f + w_0)t, we get theta = (21.55 rad/s + 50 rad/s) * 4440 s = 313,980 rad.

Finally, we can calculate the length of the track using the formula s = r*theta. Plugging in the values, we get s = 0.058 m * 313,980 rad = 18,220 m or 18.22 km. This is the length of the track if it were stretched out in a straight line.

In conclusion, the length of the track on a maximum-duration CD would be approximately 18.22 km. This solution can be verified by using the given information and applying the relevant equations. It is important to pay attention to units and use the correct formulas to arrive at the correct answer.