- Compact Disc. A compact disc (CD) stores music in a coded...

- Compact Disc. A compact disc (CD) stores music in a coded pattern of tiny pits $10^{-7} \mathrm{~m}$ 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 $25.0 \mathrm{~mm}$ and $58.0 \mathrm{~mm}$, respectively. As the disc spins inside a CD player, the track is scanned at a constant linear speed of $1.25 \mathrm{~m} / \mathrm{s} .$ (a) What is the angular speed of the CD when the innermost part of the track is scanned? The outermost part of the track? (b) The maximum playing time of a $\mathrm{CD}$ is $74.0 \mathrm{~min}$. What would be the length of the track on such a maximum-duration CD if it were stretched out in a straight line? (c) What is the average angular acceleration of a maximumduration CD during its $74.0$-min playing time? Take the direction of rotation of the disc to be positive.

Key Concepts

Spiral Geometry in Data Storage

The geometry of a spiral is used in compact discs to maximize the storage capacity within a limited physical area. Understanding the spiral layout is significant because it implies that while the track continuously winds outward, the data is stored in a continuous curve, and its radial variation affects the relationship between linear and angular speeds, as well as the effective total length of the track.

Distance-Time Relationships in Rotational Systems

This concept connects the linear distance traveled along a rotating path with the time elapsed and the constant linear speed (distance = speed × time). It is particularly important in determining the total length of a spiral track on a rotating disc, as the track length corresponds to the path that the reading mechanism would have followed if the disc’s linear speed is maintained constantly over the playing time.

Rotational Kinematics and Angular Acceleration

Rotational kinematics involves the study of the angular displacement, angular speed, and angular acceleration of rotating objects. Angular acceleration is defined as the rate of change of angular speed over time. This concept is crucial when analyzing scenarios where the angular speed of a rotating disc changes (as it does when moving from the inner to the outer track in a CD) or when computing the overall change in rotation rate over a given time period.

Linear and Angular Speed Relationship

In rotational motion, the linear (tangential) speed at a point on a rotating object is directly proportional to both the angular speed and the radial distance from the axis of rotation (expressed as v = r × ?). This key concept allows one to determine the angular speed when the linear speed and the effective radius are known, and it is essential in analyzing how different radial positions (inner versus outer parts) on a disc yield different angular speeds despite a constant linear scanning speed.

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