Question

Light incident on a rotating mirror M is returned to a fixed mirror N placed $22.5km$ away from M. The fixed mirror reflects it back to M (along the same path) which in turn reflects the light again along a direction that makes an angle of ${27}^{\circ }$ with the incident direction. The speed of rotation of the mirror is:

• A. $250rps$
• B. $500rps$
• C. $1000rps$
• D. $125rps$

Answer 1

The correct option is A $250rps$

the number of revolutions are n.

The angle between the two points of the rotating mirror $\frac{1}{2}\times 27={13.5}^0$.

The time taken by the mirror in rotating through an angle $\theta$ is given by

$d=22.5km=22500m;c=3\times {10}^{8}m/s$

$t=\frac{\theta }{2\pi n}=\frac{{13.5}^0}{2\times 180\times n}$sec ......(1)

This is also the time taken by the light to travel from original point to the fixed mirror and back,thus

$t=\frac{2d}{c}=\frac{2\times 22500}{3\times {10}^8}$ ...............(2)

From (1) and (2)

$\frac{13.5}{2\times 180\times n}=\frac{2\times 22500}{3\times {10}^8}$

or $n=\frac{13.5\times 3\times {10}^8}{2\times 180\times 2\times 22500}=\frac{40.5\times {10}^8}{16200000}=\frac{40500\times {10}^5}{162\times {10}^5}=250$ revolution/s

hence,option A is correct.