Question

A horizontal plane mirror and a vertical wall are placed in such a way that the centre of the horizontal plane mirror is at a distance $d$ from the vertical wall. The mirror is rotated with $\omega$ in clockwise direction.


Incident light falls at centre $C$ as shown in the figure, then linear velocity of spot $\left(S\right)$ formed on wall is

• A. $\omega d$
• B. $2\omega d\tan \theta$
• C. $\omega d{\sin }^2\theta$
• D. $2\omega d{\sec }^2\theta$

Answer 1

The correct option is D $2\omega d{\sec }^2\theta$

When the mirror is rotated by angle $\theta$, the reflected ray will be rotated by $2\theta$.
i.e., the rate of angular displacement of the reflected ray about $C$ is twice the rate of angular displacement of the mirror.

$\Rightarrow \frac{d\theta }{dt}=2\omega ...\left(i\right)$

Let the height of the spot be $x$ above the horizontal mirror.
$\therefore \tan \theta =\frac{x}{d}$
$\Rightarrow x=d\tan \theta$

differentiating w. r. t. time, we get,

$\Rightarrow \frac{dx}{dt}=d{\sec }^2\theta \frac{d\theta }{dt}$

$\therefore v=d{\sec }^2\theta \left(2\omega \right)=2\omega d{\sec }^2\theta$



Hence, $\left(D\right)$ is the correct answer.