Question

A spotlight is fixed $4\text{m}$ from the vertical wall and is rotating at a rate ${\text{1 rads}}^{-1}$. The spot moves vertically on the wall. Find the speed ( in ${\text{ms}}^{-1}$) of the spot on the wall when the spotlight makes an angle of ${45}^{\circ }$ with the wall.

Answer 1

s is the spot light shown in Fig, and x is the distance moved by spot in time t.

From Fig
$\frac{x}{4}=\tan \varphi$
$x=4\tan \varphi$
$v=\frac{dx}{dt}=4{\sec }^2\varphi \frac{d\varphi }{dt}$
Given $\because \frac{d\varphi }{dt}=\omega =1$
$v=8{\text{ms}}^{-1}$