Question

A spotlight is fixed $4m$ from the vertical wall and is rotating at a rate 1 ${rads}^{-1}$ . The spot moves horizontally on the wall. Find the speed of the spot on the wall when the spotlight makes an angle of ${45}^o$ with the wall

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

Answer 1

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

S is the spot light shown in fig and x is distance moved by spot in time t.

from fig$\frac{x}{y}=\tan \varphi$

$x=y\tan \varphi$

$v=\frac{dx}{dt}=y{sec}^2\varphi \frac{d\varphi }{dt}$ and $\frac{d\varphi }{dt}=\omega =1$

v=4(2)1(when $\theta ={45}^{\circ },\varphi ={45}^{\circ }$ = 8 ${ms}^{-1}$