Plane Mirrors — Question 8

Question 8

A rotating mirror makes a fixed points image move around a circle

In a two-dimensional cross section, a mirror line passes through the origin at angle beta to the x axis. An object point has fixed polar coordinates (r,alpha). The reflecting geometry remains accessible during a small rotation.

1. Derive the polar angle of the reflected image.

2. If the mirror rotates by delta, find the exact image displacement magnitude and its small-angle limit.

Question 8 - Solution

A rotating mirror makes a fixed points image move around a circle

(a) Method and calculation

$$\boxed{\alpha_I=2\beta-\alpha,\quad r_I=r}.$$$$\Delta\alpha_I=2\delta.$$

(b) Interpretation and further calculation

$$|\Delta\mathbf r_I|=2r|\sin(\Delta\alpha_I/2)|=\boxed{2r|\sin\delta|}.$$$$|\delta|\ll1\Rightarrow\boxed{|\Delta\mathbf r_I|\simeq2r|\delta|}.$$

Physical check

The image rotates at twice the mirror angular rate, while remaining the same distance from the pivot.