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Angular speed is an important concept when studying rotating objects like a compact disc (CD). It describes how quickly an object rotates around a fixed axis and is usually measured in radians per second (rad/s). When a CD is spinning inside a player, different parts of the disc spin at different angular speeds. This happens because the CD spirals outward from the center. The equation to find angular speed, \(\omega\), is given by the formula \(v = r \omega\), where \(v\) is the linear speed and \(r\) is the radius from the center of rotation. - **For the innermost radius**, the distance to the center is smaller. With the CD's constant linear speed of \(1.25 \, \text{m/s}\), the angular speed is calculated as \(\omega = \frac{v}{r_{\text{in}}} = \frac{1.25 \, \text{m/s}}{0.025 \, \text{m}}\).- **For the outermost radius**, the distance is larger, giving a different angular speed: \(\omega = \frac{v}{r_{\text{out}}} = \frac{1.25 \, \text{m/s}}{0.058 \, \text{m}}\).This variation ensures that the laser reads the CD at a constant linear speed, despite changes in angular speed.

Linear speed refers to the rate at which an object travels along a path, which in the context of a compact disc is the speed at which the track is scanned by the laser. For a CD, this speed is maintained as constant at \(1.25 \, \text{m/s}\) throughout the duration of its playtime.In calculating the track's **total length**, we multiply the linear speed by the total playing time. The formula used is:\[ \text{Length} = \text{time} \times \text{speed} = 74.0 \times 60 \, \text{sec} \times 1.25 \, \text{m/s} \]This calculation demonstrates that the length of the track, if it were to be unrolled in a straight line, is directly proportional to both the speed and the time the CD can play, which showcases how uniformly the disc is read at a constant speed.

Average angular acceleration provides insight into how the rate of rotation changes over time for the CD. It is a measure of how quickly the angular velocity of an object varies, expressed in radians per second squared (rad/s²).The formula to calculate average angular acceleration, \(\alpha_{\text{avg}}\), is:\[ \alpha_{\text{avg}} = \frac{\Delta \omega}{\Delta t} \]where \(\Delta \omega\) is the change in angular speed, and \(\Delta t\) is the time duration. For the given situation of a maximum-duration CD:- **\(\Delta \omega\)** is the difference in angular speeds at the inner and outermost parts of the CD, calculated previously.- **\(\Delta t\)** is the total playing time of \(74.0 \, \text{minutes}\) converted to seconds, which is \(74.0 \times 60 \, \text{seconds}\).Average angular acceleration helps us understand how the CD player maintains the necessary speed to ensure that the music plays effectively throughout its duration.