A CD has a diameter of 12.0 cm. If the CD is rotating at a constant frequency of 3.18 Hz, then the period of the rotational motion is
A) 0.314 s.
B) 0.441 s.
C) 0.582 s.
D) 0.698 s.
E) 0.750 s.

Question

Quizplus · Accepted Answer

The period of a wave or oscillation is the time it takes to complete one full cycle. In this case, one full cycle of the CD's rotation would be a complete revolution, or one circumference of the circle. The circumference of the CD can be found by using the formula: circumference = π × diameter.

circumference = π × 12.0 cm = 37.68 cm

The period can be found by dividing the time for one revolution (which is the circumference divided by the tangential velocity) by the frequency:

period = (circumference / tangential velocity) / frequency
tangential velocity = radius × angular velocity

Since the tangential velocity is constant, we can use the radius to find the angular velocity, which is given by:

angular velocity = tangential velocity / radius

The radius is half of the diameter, or 6.0 cm. The tangential velocity is the same as the linear speed of a point on the edge of the CD, which is the distance traveled (circumference) divided by the time taken (period). So we have:

tangential velocity = circumference / period

Putting everything together, we get:

angular velocity = tangential velocity / radius = (circumference / period) / radius = (2π × 6.0 cm) / period
period = (2π × 6.0 cm) / angular velocity = (2π × 6.0 cm) / (2π × 3.18 Hz) = 0.314 s

Therefore, the period of the CD's rotational motion is 0.314 seconds, which corresponds to choice A.

A CD Has a Diameter of 12