A particle moving in a circle of radius $R$ with a uniform speed takes a time $T$ to complete one re

Question

A particle moving in a circle of radius $R$ with a uniform speed takes a time $T$ to complete one revolution. If this particle were projected with the same speed at an angle $	heta$ to the horizontal, the maximum height attained by it equals $2 R$. The angle of projection $	heta$, is then given by

Professor PC Thomas Classes · Answer

(a): We know, $T=\frac{2 \pi R}{u} \Rightarrow u=\frac{2 \pi R}{T}$. \ Here $H_{	ext{max}} = 2 R$. \ $H_{	ext{max}} = \frac{u^{2} \sin ^{2} 	heta}{2 g} \Rightarrow 2 R = \left(\frac{2 \pi R}{T}ight)^{2} 	imes \frac{\sin ^{2} 	heta}{2 g} \Rightarrow \sin ^{2} 	heta = \frac{4 g R \cdot T^2}{4 \pi^2 R^2} = \frac{g T^2}{\pi^2 R} \Rightarrow 	heta=\sin ^{-1}\left(\frac{g T^{2}}{\pi^{2} R}ight)^{1 / 2}$.