Question

A plane mirror rotates about a vertical axis in its plane at 35 revs s^-1 and reflects a narrow beam of light to a stationary mirror 200 m away. This mirror reflects the light normally so that it is again reflected from the rotating mirror. The light now makes an angle of 2.0 minutes with the path it would travel if both mirrors were stationary. Calculate the velocity of light.

Please can you explain the solution to this question step by step with a clear diagram! I am really confused on what to picture when about a vertical axis in its plane at 35 revs s^-1 and when it says , light now makes an angle of 2.0 minutes with the path? how can an angle be in seconds?

Answer 1

First of all, the angle given in 2 minutes has nothing to do with time minute.

The minute and second are smaller units of angle, one minute is 1/60 of a degree and 1 second is 1/60 of a minute. This has nothing to do with time in second.

Now, let us consider the problem.

Consider the image shown below.

The light first travels from the rotating mirror to the fixed mirror.

Then it reflects back at an angle of 90 degree and comes back to the first mirror.

By this time, the mirror has rotated an angle 'a'. so that the light gets reflected at an angle of 2 minutes.

According to the laws of reflection, the incident angle and reflected angle are equal.

i + r = 2' (' is the symbol given for minute in the case of angle)

i = r = a (angle of mirror from initial position.

Using these, we get that a = 1' = 1/60 degrees.

Now, the mirror is 200 m away. For a round trip journey, the distance traveled is 400 m

If t is the time taken for this, the speed is given by

v = 400/t

We know that the angular speed of the mirror is 35 rev/s

This means that the mirror rotates an angle of 360*35 = 12600 degrees in one second.

So, time taken for one degree rotation = 1/12600 s

Time taken for 1/60 degree rotation = 1/(60*12600) = 1.323*10^-6 s

So, speed is

v = 400/(1.323*10^-6) = 3.023*10⁸ m/s