Rotation of a plane mirror

The figure above shows the experiment used by Henry Cavendish to determine the constant of universal gravitation (G). We must remember that during his experiment it was necessary to measure the rotation suffered by the set of two spheres A and B, represented in the figure above. Because the rotation is very small and difficult to measure, Cavendish adapted a plane mirror to the vertical wire and then focused a thin ray of light on it. In this way, when the system rotated, the reflected ray also rotated.

Our intention now is to verify what happens when a mirror undergoes a certain rotation. In the figure below, we represent a ray RA that strikes a plane mirror E, with an angle of incidence α, so AD is the reflected ray.

Assuming that we rotate the mirror causing an angle Δ to appear around its own plane and also supposing that the incident ray hits the mirror, B being the new point of incidence, and the reflected ray BF appearing, the angle we call Δ is formed between the two prolongations of the rays AD and BF, and to this angle we call the angular deviation suffered by the reflected ray.

We know that the inner sum of any triangle is equal to 180º. Thus, in the triangle CAB in the figure we have:

Ө + (90º- α) + 2α + β = 180º

Isolating Ө we will have:

Ө = 90º – α – β

For triangle GAB we have:

Δ + 2α + 2β = 180º

Isolating Δ we have:

Δ = 180º – 2α – 2β

Substituting the first equation into the second we have:

Δ = 2Ө

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