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 average angular acceleration of a maximum duration CD during its 74.0-min playing time? Take the direction of rotation of the disc to be positive.

Verified step by step guidance

1. Step 1: Understand the relationship between linear speed and angular velocity. The linear speed (v) is related to the angular velocity (ω) by the formula: $v=r\omega$, where r is the radius of the spiral track at any given point.
2. Step 2: Calculate the initial and final angular velocities. Use the inner radius (r₁ = 25.0 mm = 0.025 m) and outer radius (r₂ = 58.0 mm = 0.058 m) to find the initial and final angular velocities using the formula: $\omega =\frac{v}{r}$. Substitute v = 1.25 m/s for both cases.
3. Step 3: Determine the change in angular velocity. Subtract the initial angular velocity (ω₁) from the final angular velocity (ω₂) to find the change in angular velocity: $\Delta \omega =\omega$₂-ω₁.
4. Step 4: Convert the playing time from minutes to seconds. Since the playing time is 74.0 minutes, multiply it by 60 to convert it to seconds: $t=74.0\times 60$.
5. Step 5: Calculate the average angular acceleration. Use the formula for angular acceleration: $\alpha =\frac{\Delta }{\omega }$. Substitute the values for Δω and t to find the average angular acceleration.

Key Concepts

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time. It is a vector quantity that indicates how quickly an object is rotating and in which direction. In the context of a spinning disc, it can be calculated by determining the change in angular velocity as the disc plays music, which is influenced by the constant linear speed and the radius of the spiral track.

Linear Speed and Angular Velocity Relationship

The relationship between linear speed (v) and angular velocity (ω) is given by the formula v = rω, where r is the radius of the circular path. This relationship is crucial for understanding how the linear speed of the CD affects its angular motion. As the CD spins, the radius changes from the inner to the outer edge, impacting the angular velocity and, consequently, the angular acceleration.

Total Time of Rotation

The total time of rotation is the duration for which the disc spins, which in this case is 74.0 minutes. This time frame is essential for calculating the average angular acceleration, as it provides the time interval over which the change in angular velocity occurs. Converting this time into seconds is necessary for accurate calculations in physics.