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

Options

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

Explanation

To solve the problem step by step, we need to determine how much the reflected ray has turned after the mirror rotates under uniform angular acceleration.

### Step 1: Identify the given values
- Angular acceleration (α) = 2π rad/s²
- Time (t) = 1/4 s

### Step 2: Use the angular displacement formula
The angular displacement (θ) of the mirror can be calculated using the second equation of motion for rotational motion:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Since the mirror starts from rest, the initial angular velocity (ω₀) is 0. Thus, the equation simplifies to:
\[ \theta = \frac{1}{2} \alpha t^2 \]

### Step 3: Substitute the values into the equation
Substituting the known values:
\[ \theta = \frac{1}{2} \times (2\pi) \times \left(\frac{1}{4}\right)^2 \]
Calculating this:
\[ \theta = \frac{1}{2} \times 2\pi \times \frac{1}{16} = \frac{\pi}{16} \text{ radians} \]

### Step 4: Calculate the angle turned by the reflected ray
When the mirror turns by an angle θ, the reflected ray turns by an angle of 2θ. Therefore, the angle turned by the reflected ray (Δθ) is:
\[ \Delta \theta = 2\theta = 2 \times \frac{\pi}{16} = \frac{\pi}{8} \text{ radians} \]

### Step 5: Convert the angle to degrees
To convert radians to degrees, we use the conversion factor \(180^\circ/\pi\):
\[ \Delta \theta = \frac{\pi}{8} \times \frac{180^\circ}{\pi} = \frac{180^\circ}{8} = 22.5^\circ \]

### Final Answer
The reflected ray must have turned through **22.5 degrees** at the end of \( \frac{1}{4} \) seconds.

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