Question

A spotlight $S$ rotates in a horizontal plane with a constant angular velocity of $0\cdot 1rad/S$. The spot $S$ of light moves along the wall at a distance $3m$. The velocity of the spot $P$ when $\theta ={45}^{\circ }$ is $v\left(m/s\right)$. Find $5v$.

Answer 1

Draw a rough Diagram
$x$ is the distance travelled by spot at any time $t$
Find angular speed of car.
From figure
$\tan \theta =\frac{3}{x}$
L.H.S:
$\frac{d}{dt}\left(\tan \theta \right)=\frac{d}{d\theta }\left(\tan \theta \right)\cdot \frac{d\theta }{dt}$
$\frac{d}{dt}\left(\tan \theta \right)=se{c}^2\theta \cdot w$
R.H.S:
$\frac{d}{dt}\left(\frac{3}{x}\right)=-\frac{3}{x^2}\frac{dx}{dt}=-\frac{3}{x^2}v$
$\therefore se{c}^2\theta \cdot w=\frac{dx}{dt}=-\frac{3}{x^2}v$
ANSWER
Negative sign indicated that as theta increase speed decreases
Given: $x=3m,w=0.1rad/s,\theta ={45}^{\circ }\text{and}sec{45}^{\circ }=\sqrt{2}$
We got
$2\times 0\cdot 1=\frac{3}{3^2}\times v$
$\therefore v=0\cdot 6m/s$
Then $5v=3m/s$
Final Answer : 3