Spotlight $S$ rotates in a horizontal plane with constant angular velocity of $0.1 \mathrm{rad} / \m

Question

Spotlight $S$ rotates in a horizontal plane with constant angular velocity of $0.1 \mathrm{rad} / \mathrm{s}$. The spot of light $P$ moves along the wall at a distance of 3 m . The velocity of the spot $P$ when $	heta=45^{\circ}$ (see fig.)is $\qquad$ $\mathrm{m} / \mathrm{s}$.\

Professor PC Thomas Classes · Answer

If $x$ is the distance of point $P$ from $O$ then from figure\ $	an \phi=(x / h) \quad$ or $\quad x=h 	an \phi$\ $\frac{d x}{d t}=h\left(\sec ^{2} \phiight) \frac{d \phi}{d t} \quad$ i.e. $\quad v=h \sec ^{2} \phi \omega$\ [as $(d x / d t)=v$ and $(d \phi / d t)=\omega$ ]\ Here $h=3 \mathrm{~m}$,\ $\phi=180^{\circ}-\left(45^{\circ}+90^{\circ}ight)=45$,\ $\omega=0.1 \mathrm{rad} / \mathrm{s}$.\ So $\quad v=3 	imes(\sqrt{2})^{2} 	imes 0.1=0.6 \mathrm{~m} / \mathrm{s}$