A ray of light is incident on the plane mirror at rest. The mirror starts turning at a uniform angular acceleration of `pi rad s^(-2)` . The reflected ray at the end of `1/4` s must have turned through

Options

A. `90^@`
B. `45^@`
C. `22.5^@`
D. `11.25^@`

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Understand the Problem
We have a plane mirror that is initially at rest and starts rotating with a uniform angular acceleration of \( \pi \, \text{rad/s}^2 \). We need to find out how much the reflected ray has turned after \( \frac{1}{4} \) seconds.

### Step 2: Determine the Angular Displacement of the Mirror
We can use the second kinematic equation for rotational motion to find the angular displacement \( \theta \) of the mirror. The equation is:

\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]

Where:
- \( \theta \) is the angular displacement,
- \( \omega_0 \) is the initial angular velocity (which is \( 0 \) since the mirror starts from rest),
- \( \alpha \) is the angular acceleration (\( \pi \, \text{rad/s}^2 \)),
- \( t \) is the time (\( \frac{1}{4} \, \text{s} \)).

Since the mirror starts from rest, \( \omega_0 = 0 \), the equation simplifies to:

\[ \theta = \frac{1}{2} \alpha t^2 \]

### Step 3: Substitute the Values
Now, substituting the values into the equation:

\[ \theta = \frac{1}{2} \cdot \pi \cdot \left(\frac{1}{4}\right)^2 \]

Calculating \( \left(\frac{1}{4}\right)^2 \):

\[ \left(\frac{1}{4}\right)^2 = \frac{1}{16} \]

Now substituting this back into the equation:

\[ \theta = \frac{1}{2} \cdot \pi \cdot \frac{1}{16} = \frac{\pi}{32} \, \text{radians} \]

### Step 4: Determine the Angular Displacement of the Reflected Ray
According to the law of reflection, if the mirror rotates by an angle \( \theta \), the reflected ray will turn by \( 2\theta \). Therefore:

\[ \text{Angle turned by reflected ray} = 2\theta = 2 \cdot \frac{\pi}{32} = \frac{\pi}{16} \, \text{radians} \]

### Step 5: Convert to Degrees
To convert \( \frac{\pi}{16} \) radians to degrees, we use the conversion factor \( \frac{180}{\pi} \):

\[ \text{Degrees} = \frac{\pi}{16} \cdot \frac{180}{\pi} = \frac{180}{16} = 11.25^\circ \]

### Final Answer
The reflected ray must have turned through \( 11.25^\circ \).

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