Question

A ray of light $I$ is incident on a plane mirror $M$. The mirror is rotated in the direction as shown in the figure by an arrow at frequency $\frac{18}{\pi }$ cps. The light reflected by the mirror is received on the wall $W$ at a distance $10m$ from the axis of rotation. When the angle of incidence becomes $3$? the speed of the spot (a point) on the wall is

• A. $100m/s$
• B. $1000m/s$
• C. $2000m/s$
• D. $500m/s$

Answer 1

The correct option is C $2000m/s$

assume incidence angle i ${37}^{\circ }$

$\alpha =90-{37}^{\circ }={53}^{\circ }$

take $Q=$ angle of rotationtion of mirror $\alpha =2\theta$

We have to find

speed to point $P$ ie $\frac{dx}{dt}$

change in $x$ wir.t

time.

as image is ratated

twice the angle in which

mirror rotates. $\therefore [\frac{d\alpha }{dt}-\frac{2dQ}{dt}]$

how, $\tan \alpha =\frac{x}{10}\Rightarrow x=40\tan \alpha$

$\begin{array}{c}\frac{dx}{dt}=10{\sec }^2\alpha \frac{d\alpha }{d+},\frac{dx}{dt}=20{\sec }^2\alpha \frac{dQ}{dt}\\ \therefore \frac{dQ}{dt}=\omega =2\pi f\Rightarrow 20{\sec }^2\alpha \omega \omega \Rightarrow 20{\sec }^2\alpha \left(2\pi \left(\frac{18}{\pi }\right)\right)\end{array}$

$\frac{dx}{d+}=40\times {\left(\frac{5}{3}\right)}^2\times 18\Rightarrow \frac{720\times 25}{9}$

$\Rightarrow 2000m/s$

: speed of point $p$ is $2000m/s$

Hence option (c) matenes.