Question

Spotlight $S$ rotates in a horizontal plane with constant angular velocity of 0.1 rad$s^{-1}$ The spot of light $P$ moves along the wall at a distance of 3 m The velocity of the spot $P$ when $\theta$ = 45 is $m{s}^{-1}$

Answer 1

Linear velocity of rotating object is given by

$v=\frac{\omega r}{{sin}^2\theta }$ , { $\omega$ =angular velocity, r=distance from centre }

if d is the distance from spotlight to wall then distance from the spot of the light from the center at an angle45 is

$r=\frac{d}{cos\theta }=\frac{3}{cos45}=3\sqrt{2}v=\frac{0.1(3\sqrt{2)}}{sin45}=0.6m/sec$