A flat mirror revolves at a constant angular velocity making `n=0.4` revolutions per second. With what velocity `("in m s"^(-1))` will a light spot move along a spherical screen with a radius of 15 m, if the mirror is at the centre of curvature of the screen ?

Options

A. 37.7
B. 60.3
C. 68.7
D. 75.4

Answer

To solve the problem, we need to determine the velocity of a light spot moving along a spherical screen when a flat mirror revolves at a constant angular velocity. Here are the steps to find the solution:

### Step-by-Step Solution:

1. **Identify Given Values**:
- Angular velocity in revolutions per second, \( n = 0.4 \, \text{revolutions/second} \)
- Radius of the spherical screen, \( r = 15 \, \text{m} \)

2. **Convert Angular Velocity to Radians**:
- Since \( 1 \, \text{revolution} = 2\pi \, \text{radians} \), we convert the angular velocity from revolutions per second to radians per second:
\[ \omega = n \times 2\pi = 0.4 \times 2\pi = 0.8\pi \, \text{radians/second} \]

3. **Determine Angular Velocity of Reflected Light**:
- The angular velocity of the reflected rays is twice the angular velocity of the mirror:
\[ \omega_{\text{reflected}} = 2 \times \omega = 2 \times 0.8\pi = 1.6\pi \, \text{radians/second} \]

4. **Calculate the Velocity of the Light Spot**:
- The velocity \( v \) of the light spot on the spherical screen can be calculated using the formula:
\[ v = r \times \omega_{\text{reflected}} \]
- Substituting the values:
\[ v = 15 \, \text{m} \times 1.6\pi \, \text{radians/second} \]
- Calculating this gives:
\[ v = 15 \times 1.6 \times 3.14 \approx 75.4 \, \text{m/s} \]

5. **Final Answer**:
- The velocity of the light spot moving along the spherical screen is approximately \( 75.4 \, \text{m/s} \).

### Summary:
The light spot moves along the spherical screen with a velocity of approximately \( 75.4 \, \text{m/s} \).