How much work is done by the motor in a CD player to make a CD spin, starting from rest? The CD has a diameter of 12.8 cm and a mass of 15.7 g. The laser scans at a constant tangential velocity of 1.23 m/s. Assume that the music is first detected at a radius of 19.1 mm from the center of the disk. Ignore the small circular hole at the CD's center.

Question

How much work is done by the motor in a CD player to make a CD spin, starting from rest? The CD has a diameter of 12.8 cm and a mass of 15.7 g. The laser scans at a constant tangential velocity of 1.23 m/s. Assume that the music is first detected at a radius of 19.1 mm from the center of the disk. Ignore the small circular hole at the CD's center.

Answer 1

To calculate the work done by the motor in a CD player to make a CD spin, starting from rest, we need to consider the following steps:

Step 1: Determine the initial and final rotational kinetic energies of the CD.
Step 2: Calculate the work done by the motor to increase the rotational kinetic energy.

Let's begin.

Step 1: Determine the initial and final rotational kinetic energies of the CD.

The rotational kinetic energy of an object can be calculated using the formula:

Rotational Kinetic Energy (KE_rot) = (1/2) * moment of inertia * (angular velocity)^2

In this case, the moment of inertia of the CD can be approximated as that of a thin disc:

Moment of Inertia (I) = (1/2) * m * r^2

Where:
- m is the mass of the CD
- r is the radius of the CD

Since the CD is starting from rest, the initial rotational kinetic energy is zero.

For the final rotational kinetic energy, we'll use the given tangential velocity at which the laser scans (1.23 m/s). The tangential velocity of a point on the CD can be calculated using the formula:

Tangential Velocity (v_t) = r * angular velocity

Rearranging the formula, we find that the angular velocity (ω) can be calculated as:

Angular Velocity (ω) = v_t / r

Now, we can calculate the final rotational kinetic energy using the formula for rotational kinetic energy:

KE_rot(final) = (1/2) * I * ω^2

Step 2: Calculate the work done by the motor to increase the rotational kinetic energy.

The work done by the motor is equal to the change in rotational kinetic energy:

Work (W) = KE_rot(final) - KE_rot(initial)

By substituting the calculated values into the equation, we can find the work done by the motor.

Note: To perform these calculations, we need the values of the CD's mass, diameter, and the radius at which the music is first detected.