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. (b) 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?

Verified step by step guidance

1. Calculate the circumference of the inner and outer circles of the CD using the formula for the circumference of a circle, C = 2\pi r, where r is the radius. Use 25.0 mm for the inner radius and 58.0 mm for the outer radius.
2. Assume the track spirals linearly from the inner radius to the outer radius. Calculate the average radius by taking the average of the inner and outer radii.
3. Use the average radius to estimate the average circumference of the spiral track.
4. Knowing the linear speed of the track scanning (1.25 m/s) and the maximum playing time (74.0 minutes), convert the playing time into seconds and then calculate the total length of the track by multiplying the linear speed by the total playing time in seconds.
5. The result from the previous step gives the length of the track if it were stretched out in a straight line.

Key Concepts

Linear Speed

Linear speed refers to the distance traveled per unit of time in a straight line. In the context of a compact disc, it is the speed at which the laser scans the track of the disc. Understanding linear speed is crucial for calculating the total distance covered by the laser as it moves from the inner to the outer radius of the disc over a given time period.

Circumference of a Circle

The circumference of a circle is the distance around it, calculated using the formula C = 2πr, where r is the radius. For a compact disc, the track spirals outward, meaning the circumference increases as the radius increases. This concept is essential for determining the length of the track at various points from the inner to the outer radius.

Total Length of the Spiral Track

The total length of the spiral track on a compact disc can be calculated by integrating the circumferences of concentric circles from the inner radius to the outer radius. This involves summing the circumferences at infinitesimally small increments of radius, which gives the total length of the track when stretched out. This concept is key to solving the problem of finding the total length of the CD track.