Question

A particle is describing circular motion in a horizontal plane in contact with the smooth inside surface of a fixed right circular cone with its axis vertical and vertex down. The height of the plane of motion above the vertex is $h$ and the semi-vertical angle of the cone is $\alpha .$ The period of revolution of the particle:

• A. increases as $h$ increases
• B. decreases as $h$ increases
• C. increases as $\alpha$ increases
• D. decreases a $\alpha$ increases

Answer 1

Answer: (A), (C)

The correct options are
A increases as $h$ increases
C increases as $\alpha$ increases

As, $N$ $sin\alpha =mg$
$Ncos\alpha =m{\omega }^{2}r$
If $T$ is the time period of revolution, then
$tan\alpha =\frac{g}{{\omega }^{2}r}$
$\therefore T^2\propto tan\alpha$
Since $tan$ is an increasing function
$\therefore$ When $\alpha$ increases $T$ also increases,
Also $T^2\propto rtan\alpha$
But $r=htan\alpha$
$\therefore T^2\propto hta{n}^2\alpha$
For constant $\alpha$
$T^2\propto h$
Thus, when $h$ increases $T$ also increases.