*. CALC On a compact disc (CD), music is coded in a pattern...

*. CALC On a compact disc (CD), music is coded in a pattern of tiny pits arranged in a track that spirals outward toward the rim of the disc. As the disc spins inside a $C D$ player, the track is scanned at a conAxis stant linear speed of $v=1.25 \mathrm{~m} / \mathrm{s}$. Because the radius of the track varies as it spirals outward, the angular speed of the disc must change as the $\mathrm{CD}$ is played. (See Exercise 9.20.) Let's see what angular acceleration is required to keep $v$ constant. The equation of a spiral is $r(\theta)=r_{0}+\beta \theta$, where $r_{0}$ is the radius of the spiral at $\theta=0$ and $\beta$ is a constant. On a CD, $r_{0}$ is the inner radius of the spiral track. If we take the rotation direction of the $\mathrm{CD}$ to be positive, $\beta$ must be positive so that $r$ increases as the disc turns and $\theta$ increases. (a) When the disc

*. CALC On a compact disc (CD), music is coded in a pattern of tiny pits arranged in a track that spirals outward toward the rim of the disc. As the disc spins inside a $C D$ player, the track is scanned at a conAxis stant linear speed of $v=1.25 \mathrm{~m} / \mathrm{s}$.
Because the radius of the track varies as it spirals outward, the angular speed of the disc must change as the $\mathrm{CD}$ is played. (See Exercise 9.20.) Let's see what angular acceleration is required to keep $v$ constant. The equation of a spiral is $r(\theta)=r_{0}+\beta \theta$, where $r_{0}$ is the radius of the spiral at $\theta=0$ and $\beta$ is a constant. On a CD, $r_{0}$ is the inner radius of the spiral track. If we take the rotation direction of the $\mathrm{CD}$ to be positive, $\beta$ must be positive so that $r$ increases as the disc turns and $\theta$ increases. (a) When the disc

Key Concepts

Angular Acceleration in Variable Radius Systems

Angular acceleration is the rate of change of angular velocity and becomes especially significant in systems where the effective radius varies during the motion. When maintaining a constant linear speed on a spiral path, the angular speed must be adjusted accordingly, and analyzing the necessary angular acceleration involves understanding how variations in radius affect the system’s rotational dynamics.

Spiral Geometry in Rotational Systems

Spiral geometry deals with curves in the plane that wind around a central point while continuously moving outward or inward. In the context of rotational systems, such as the tracks on a compact disc, the radius of the motion changes with the angle, which requires a modified approach to relate radial changes to rotational motion parameters.

Circular Motion Kinematics

Circular motion kinematics involves the analysis of objects moving along curved paths, particularly emphasizing quantities such as angular displacement, angular velocity, and angular acceleration. Understanding these quantities—and how they relate to their linear analogs—is essential when analyzing systems where the motion path or speed criteria change over time.

Relationship between Linear and Angular Velocity

In circular motion, there exists a fundamental connection between linear and angular velocity, expressed by the equation v = r?, where v is the linear speed, r is the radius, and ? is the angular speed. This concept is crucial when dealing with any system where a point on a rotating object must maintain a constant linear speed despite changes in radius.

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