CA 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?

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to find the total length of the track on the CD if it were stretched out in a straight line. The CD is scanned at a constant linear speed of 1.25 m/s, and the maximum playing time is 74.0 minutes. The length of the track can be calculated using the relationship between speed, time, and distance: \( \text{distance} = \text{speed} \times \text{time} \).
2. Step 2: Convert the playing time from minutes to seconds. Since there are 60 seconds in a minute, multiply the playing time (74.0 minutes) by 60 to get the total time in seconds: \( t = 74.0 \times 60 \).
3. Step 3: Use the formula \( \text{distance} = \text{speed} \times \text{time} \) to calculate the length of the track. Substitute the given linear speed (1.25 m/s) and the total time (calculated in Step 2) into the formula: \( \text{distance} = 1.25 \times t \).
4. Step 4: Simplify the expression to find the total length of the track. This will give the length of the track in meters, as the speed is in meters per second and the time is in seconds.
5. Step 5: Interpret the result. The calculated distance represents the total length of the spiral track on the CD if it were stretched out in a straight line. Ensure the units are consistent and the result is reasonable given the context of the problem.

Key Concepts

Linear Speed

Linear speed refers to the distance traveled per unit of time along a path. In the context of a compact disc, it is the speed at which the laser scans the track, which is given as 1.25 m/s. 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.

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 changes as the radius increases from the inner to the outer edge. This concept is essential for determining the length of the track at various points along the disc.

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.