Question

A spotlight is placed at a distance of $4\text{m}$ from a vertical wall and is rotating at a rate of $1{\text{rad s}}^{-1}$. The beam moves on the wall. Find the speed of the beam on the wall when the light makes an angle of ${45}^{\circ }$ with the horizontal.

• A. $4{\text{ms}}^{-1}$
• B. $6{\text{ms}}^{-1}$
• C. $8{\text{ms}}^{-1}$
• D. None of these

Answer 1

The correct option is C $8{\text{ms}}^{-1}$

$\text{s}$ is the light shown in the figure and $x$ is the distance moved by the beam in time $t$.

From the figure, $\frac{x}{4}=\tan \varphi$
$\Rightarrow x=4\tan \varphi$
Speed of the beam is given by
$v=\frac{dx}{dt}=4{\sec }^2\varphi \frac{d\varphi }{dt}$
and given $\frac{d\varphi }{dt}=\omega =1\text{rad/s}$
$\therefore v=4{\sec }^2{45}^{\circ }\times 1$ (when $\theta ={45}^{\circ },\varphi ={45}^{\circ }$)
$v=8{\text{ms}}^{-1}$