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 min. What would be the length of the track on such a maximum-duration $\mathrm{CD}$ if it were stretched out in a straight line? (c) 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 dise to be positive.

Key Concepts

Angular Acceleration

Angular acceleration is the rate of change of angular speed over time and is given by ? = ??/?t, where ?? is the change in angular speed and ?t is the change in time. This concept is central to rotational dynamics, particularly in situations where the angular speed is not constant, such as when a device speeds up or slows down over a period of time.

Spiral Geometry

Spiral geometry deals with curves that continuously change their radius as they wind around a central point. In contexts such as the compact disc, the track is arranged in a spiral where the radius increases steadily from the inner to the outer edge. Understanding spiral geometry is crucial when calculating quantities like total track length and the variation in angular speed along different radii.

Relationship between Linear and Angular Speed

This concept explains how an object's tangential (linear) speed on a circular path relates to its rotational (angular) speed. The key formula is v = ?r, where v is the constant linear speed, ? is the angular speed, and r is the radius of the circle. This relationship is fundamental in problems where an object moves in a circular path at a constant or variable speed, allowing conversion between linear and angular measures.

Arc Length in Circular Geometry

Arc length is the distance along the curved path of a circle, calculated using s = r?, where s is the arc length, r is the radius, and ? is the angle in radians. This concept is important when determining the total path length along a circular track or spiral, as it connects the geometry of the circle to the distance traveled.

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