13.3.5. On the optical axis $AB$ of the collecting lens there is a flat mirror rotating with an angular velocity $\omega$ around the axis passing through point $A$ and perpendicular to the drawing plane. A parallel beam of rays falls on the mirror, which after reflection focuses on the screen. The focal length of the lens is $f$. Find the speed of the light spot on the screen at the moment when it crosses the optical axis. The screen plane is perpendicular to the optical axis.
Solution
The reflected beam incident on the lens rotates with double the angular velocity $2\omega$ relative to the optical axis, because when the mirror rotates by some angle $\varphi$, both the angle of incidence and the angle of reflection change by the same amount. This can also be seen from the drawing:
$$\gamma=\pi-2\varphi\quad\to\quad\frac{d\gamma}{dt}=2\omega \ - direction \ is \ not \ so \ important$$$$y=f\tan\gamma$$$$v(\gamma)=\frac{dy}{dt}=f\frac{d(\tan\gamma)}{d\gamma}\cdot\frac{d\gamma}{dt}=2\omega f\frac{1}{\cos^2(\gamma)}$$$$v(0)= 2\omega f$$
