Question

The disk in a CD player does not rotate at a constant angular speed, but spins at a rate which is decided by a control unit so that the linear speed of the track being read is constant. The laser beam used to read the data on the disk starts at an inner radius of 5cm and continues to read until reaching an outer radius of 10 cm. If the disk rotates at 600 rev/min at the start, what will be its rotation rate at the end ?

• A. 600 rev/min
• B. 1200 rev/min
• C. 300 rev/min
• D. 150 rev/min

Answer 1

Answer: (C)

300 rev/min

Answer 2

The linear speed ($v$) of the data track being read on the CD is constant. The relationship between linear speed, angular speed ($\omega$), and radius ($r$) is given by:

$$v = r\omega$$

Since the linear speed is constant from the inner radius ($r_i$) to the outer radius ($r_o$), we can set the product of radius and angular speed equal at both points:

$$r_i \omega_i = r_o \omega_o$$

where $\omega_i$ is the initial angular speed at $r_i$, and $\omega_o$ is the final angular speed at $r_o$.

To find the final angular speed ($\omega_o$), we rearrange the equation:

$$\omega_o = \frac{r_i \omega_i}{r_o}$$

Given values are: Inner radius, $r_i = 5$ cm Outer radius, $r_o = 10$ cm Initial angular speed, $\omega_i = 600$ rev/min

Substituting these values into the equation:

$$\omega_o = \frac{(5 \text{ cm}) \times (600 \text{ rev/min})}{10 \text{ cm}}$$

The units of centimeters cancel out, leaving the angular speed in rev/min:

$$\omega_o = \frac{5}{10} \times 600 \text{ rev/min}$$ $$\omega_o = \frac{1}{2} \times 600 \text{ rev/min}$$ $$\omega_o = 300 \text{ rev/min}$$