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. (a) What is the angular speed of the CD when the innermost part of the track is scanned? The outermost part of the track?

Verified step by step guidance

1. Identify the relationship between linear speed (v), angular speed (\(\omega\)), and radius (r) using the formula \(v = r \omega\).
2. Solve for angular speed (\(\omega\)) by rearranging the formula to \(\omega = \frac{v}{r}\).
3. Calculate the angular speed for the innermost part of the track using the radius of 25.0 mm (convert this to meters by dividing by 1000).
4. Calculate the angular speed for the outermost part of the track using the radius of 58.0 mm (again, convert this to meters).
5. Compare the angular speeds at the innermost and outermost parts to understand how the angular speed changes as the radius of the track changes.

Key Concepts

Angular Speed

Angular speed is a measure of how quickly an object rotates around a central point, typically expressed in radians per second. It is calculated by the formula ω = v/r, where ω is the angular speed, v is the linear speed, and r is the radius from the center of rotation. In the context of a compact disc, as the disc spins, different points along the track have varying distances from the center, affecting their angular speeds.

Linear Speed

Linear speed refers to the distance traveled per unit of time along a path. In the case of a compact disc, the linear speed is constant at 1.25 m/s, meaning that regardless of the radius of the track being scanned, the speed at which the laser reads the data remains the same. This constant linear speed is crucial for determining the varying angular speeds at different radii.

Radius of the Track

The radius of the track on a compact disc is the distance from the center of the disc to the point being scanned. In this scenario, the inner radius is 25.0 mm and the outer radius is 58.0 mm. The radius directly influences the angular speed; as the radius increases, the angular speed decreases when the linear speed is held constant, illustrating the inverse relationship between these two quantities.