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 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. Step 1: Understand the relationship between linear speed and angular speed. The formula to relate linear speed (v) and angular speed (ω) is: v = ω * r, where r is the radius of the circular path. Rearrange this formula to solve for angular speed: ω = v / r.
2. Step 2: Convert the given radii from millimeters to meters. The inner radius is 25.0 mm, which is 0.025 m, and the outer radius is 58.0 mm, which is 0.058 m.
3. Step 3: Use the formula ω = v / r to calculate the angular speed at the innermost part of the track. Substitute the given linear speed (v = 1.25 m/s) and the inner radius (r = 0.025 m) into the formula.
4. Step 4: Similarly, use the same formula ω = v / r to calculate the angular speed at the outermost part of the track. Substitute the given linear speed (v = 1.25 m/s) and the outer radius (r = 0.058 m) into the formula.
5. Step 5: Compare the angular speeds at the inner and outer radii. Note that the angular speed decreases as the radius increases, since the linear speed is constant.

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 CD, as the disc spins, different points on the disc 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 any point on the disc moves at this speed as the disc rotates. This constant linear speed is crucial for determining the angular speed at different radii of the disc.

Radius of Rotation

The radius of rotation is the distance from the center of rotation to the point of interest on the rotating object. For a compact disc, 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 for a constant linear speed, illustrating the inverse relationship between these two quantities.