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 $\theta$ 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)

• A. $\sqrt{gLcos\theta }$
• B. $\sqrt{gLsin\theta }$
• C. $\sqrt{gLsin\theta tan\theta }$
• D. $\sqrt{gLtan\theta }$

Answer 1

Answer: (C)

At maximum velocity the mass will just loose contact with cone and will behave like free conical pendulum with time period

$T=2\pi \sqrt{\frac{Lcos\theta }{g}}$

$\Rightarrow \omega =\frac{2\pi }{T}\sqrt{\frac{g}{Lcos\theta }}$

Hence $V_{\text{max}}=\left(Lsin\theta \right)\omega$
$=\sqrt{gLsin\theta tan\theta }$