A plane mirror rotating at an angular velocity of 3 radian/s reflects a light beam. The angular velocity of the reflected beam is -

Options

A. `3 "rad"//s`
B. `6 "rad"//s`
C. `9 "rad"//s`
D. `12 "rad"//s`

Answer

To find the angular velocity of the reflected beam when a plane mirror rotates, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between the mirror's rotation and the reflected beam**:
- When a plane mirror rotates by an angle θ, the angle of the reflected beam changes by an angle of 2θ. This is due to the law of reflection, which states that the angle of incidence is equal to the angle of reflection.

2. **Define the angular velocity of the mirror**:
- Given that the angular velocity of the mirror (ω_m) is 3 radian/s, we can express this as:
\[ \omega_m = \frac{d\theta}{dt} = 3 \text{ rad/s} \]

3. **Relate the angular velocity of the reflected beam to that of the mirror**:
- Since the reflected beam rotates by an angle of 2θ when the mirror rotates by θ, the angular velocity of the reflected beam (ω_r) can be expressed as:
\[ \omega_r = \frac{d(2\theta)}{dt} = 2 \frac{d\theta}{dt} = 2\omega_m \]

4. **Substitute the value of the mirror's angular velocity**:
- Now, substituting the value of ω_m into the equation for ω_r:
\[ \omega_r = 2 \times 3 \text{ rad/s} = 6 \text{ rad/s} \]

5. **Determine the direction of the angular velocity**:
- The reflected beam's angular velocity is in the opposite direction to the mirror's rotation. Therefore, we express this as:
\[ \omega_r = -6 \text{ rad/s} \]

### Final Answer:
The angular velocity of the reflected beam is **-6 rad/s**.

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