A small sized mass $m$ is attached by a massless string (of length $L$) to the top of a fixed fricti

Question

A small sized mass $m$ is attached by a massless string (of length $L$) to the top of a fixed frictionless solid cone whose axis is vertical. The half angle at the vertex of the cone is $	heta$. If the mass $m$ moves around in a horizontal circle at speed $v$, what is the maximum value of $v$ for which mass stays in contact with the cone? ($g$ is acceleration due to gravity.)

Professor PC Thomas Classes · Answer

Let $N$ be the normal reaction and $T$ be the tension. Vertical equilibrium: $T\cos	heta + N\sin	heta = mg$. Radial equation: $T\sin	heta - N\cos	heta = \frac{mv^2}{L\sin	heta}$. For maximum speed $v$ while maintaining contact, normal reaction $N 	o 0$. Thus $T\cos	heta = mg \implies T = mg/\cos	heta$. Substituting into radial equation gives $(mg/\cos	heta)\sin	heta = \frac{mv^2}{L\sin	heta} \implies g	an	heta = \frac{v^2}{L\sin	heta} \implies v = \sqrt{gL\sin	heta	an	heta}$.