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 $\mathrm{CD}$ player, the track is scanned at a constant 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 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 $\mathrm{CD}, r_{0}$ is the inner radius of the spiral track. If we take the rotation direction of the CD to be positive, $\beta$ must be positive so that $r$ increases as the disc turns and $\theta$ increases. (a) When the disc rotates through a small angle $d \theta$ , the distance scanned along the track is $d s=r d \theta .$ Using the above expression for $r(\theta),$ integrate $d s$ to find the total distance $s$ scanned along the track as a function of the total angle $\theta$ through which the disc has rotated. (b) since the track is scanned at a constant linear speed $v,$ the distance $s$ found in part (a) is equal to $v t$ . Use this to find $\theta$ as a function of

time. There will be two solutions for $\theta$ ; choose the positive one, and explain why this is the solution to choose. (c) Use your expression for $\theta(t)$ to find the angular velocity $\omega_{z}$ and the angular acceleration $\alpha_{z}$ as functions of time. Is $\alpha_{z}$ constant? (d) On a CD, the inner radius of the track is 25.0 $\mathrm{mm}$ , the track radius increases by 1.55$\mu \mathrm{m}$ per revolution, and the playing time is 74.0 min. Find the values of $r_{0}$ and $\beta,$ and find the total number of revolutions made during the playing time. (e) Using your results from parts (c) and (d), make graphs of $\omega_{z}$ $($ in $\operatorname{rad} / \mathrm{s})$ versus $t$ and $\alpha_{z}$ $($ in $\operatorname{rad} / \mathrm{s}^2)$ versus $t$ between $t=0$ and $t=74.0$ min.