A small plane mirror kept at the centre of a sphere of diameter 3 m, makes 12 revolution per second. A thin light beam is made incident on the mirror. The linear speed of the light spot on the sphere, formed after reflection-from the mirror is:

Options

A. 18`pi` m/s
B. `36pi` m/s
C. 72`pi`m/s
D. `144pi`m/s

Explanation

To find the linear speed of the light spot on the sphere formed after reflection from the mirror, we can follow these steps:

### Step 1: Determine the radius of the sphere
The diameter of the sphere is given as 3 meters. Therefore, the radius \( r \) of the sphere is:
\[ r = \frac{\text{Diameter}}{2} = \frac{3 \, \text{m}}{2} = 1.5 \, \text{m} \]

### Step 2: Calculate the angular speed of the mirror
The mirror makes 12 revolutions per second. To find the angular speed \( \omega \) in radians per second, we use the conversion:
\[ \omega = 2\pi \times \text{Revolutions per second} = 2\pi \times 12 \, \text{rev/s} = 24\pi \, \text{rad/s} \]

### Step 3: Calculate the linear speed of the light spot
The linear speed \( v \) of the light spot on the surface of the sphere can be calculated using the formula:
\[ v = r \cdot \omega \]
Substituting the values we have:
\[ v = 1.5 \, \text{m} \times 24\pi \, \text{rad/s} \]
Calculating this gives:
\[ v = 36\pi \, \text{m/s} \]

### Step 4: Final Result
Thus, the linear speed of the light spot on the sphere is:
\[ v \approx 113.1 \, \text{m/s} \quad (\text{using } \pi \approx 3.14) \]

### Summary
The linear speed of the light spot on the sphere formed after reflection from the mirror is \( 36\pi \, \text{m/s} \) or approximately \( 113.1 \, \text{m/s} \).

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