A compact disc (CD) player varies the rotation rate of the disc in order to keep the part of the disc from which information is being read moving at a constant linear speed of \(1.30 \mathrm{m} / \mathrm{s} .\) Compare the rotation rates of a 12.0 -cm-diameter CD when information is being read (a) from its outer edge and (b) from a point \(3.75 \mathrm{cm}\) from the center. Give your answers in \(\mathrm{rad} / \mathrm{s}\) and \(\mathrm{rpm}\).

Linear speed refers to the distance covered by an object moving along a straight path in a given unit of time. For a rotating object, like a CD in a CD player, linear speed is significant because it represents the speed at which the data under the laser beam is moving relative to the laser itself as the CD spins.

Imagine drawing a line on the surface of a CD and then watching how fast that line moves past a fixed point when the CD is played. That's the linear speed we're talking about. In the exercise provided, the compact disc player keeps this speed constant at 1.30 meters per second (m/s).

To maintain a constant linear speed across portions of the CD with different radii, the CD player must vary the rotational speed, spinning faster when accessing data closer to the center. This is necessary because points closer to the center of a rotating disc have to travel a smaller distance to complete one revolution compared to points near the edge.

Rotational motion is the motion of a body that spins about an axis. In the context of the exercise, this axis would be the central point of the CD through which it's attached to the player. A key parameter that describes rotational motion is angular velocity, denoted as \(\omega\). It represents how fast the angle changes as an object rotates.

To understand this within a CD player, envision that as the CD spins, each point on the disc moves through a certain angle in a specific amount of time, which translates to the angular velocity. However, unlike linear speed, angular velocity is the same for every point on the CD, regardless of how far it is from the center, during a single rotation. This concept underpins the ability of a CD player to read data consistently without the data speed changing as the laser moves across the radius of the CD.

Mathematically, angular velocity (\(\omega\)) and linear speed (\(v\)) are related by the radius (\(r\)) of the circular path with the equation \(v = \omega r\), which was used in the solution of our exercise.

Rotation rates in a CD player are a practical application of the physics concepts of linear speed and rotational motion. CD players must adjust rotation rates to ensure that the data being read by the laser does so at a consistent linear speed, no matter where the laser is positioned on the disc.

In the exercise, we were tasked with comparing rotation rates while reading data from the outer edge and from a point 3.75 cm from the center of a standard 12 cm CD. We see that the rotation rate must be higher when the laser reads from near the center than from the edge to compensate for the shorter circular path around the disc's center.

The step-by-step solution showed how to calculate these rates using the radius of each point and the constant linear speed, reflecting the CD player's sophisticated mechanism that adjusts the rotation speed dynamically. This is crucial for the reliable functioning of the CD player, as it allows for consistent data retrieval at the designated speed regardless of the read position on the CD. The conversion of angular velocity to revolutions per minute (rpm) gives us a more intuitive sense of how fast the CD is spinning, often quoted in the technical specifications of CD players.