Question

A particle of mass $m$ is released from point $A$ on smooth fixed circular track as shown. If the particle is released from rest at $t=0,$ then variation of normal reaction $N$ with $\left(\right.\theta \left.\right)$ angular displacement from initial position is -

• A.
• B.
• C.
• D.

Answer 1

Answer: (A)

When the block reaches a point where the angular displacement $\left(\theta \right)$ from the initial position is as shown.

The normal reaction force on the mass is
$N=mgsin\left(\theta \right)+\frac{m v^{2}}{R}$ .
Here, $\frac{m v^{2}}{R}$ is the centrifugal force on the mass.
By conservation of mechanical energy,
$\frac{1}{2}mv^{2}=mgh$
$\Rightarrow v^{2}=2gh=2gRsin\left(\theta \right)$
The Normal reaction,
$N=mgsin\left(\theta \right)+\frac{m \left(2 g R sin \left(\theta \right)}{R}$
$\Rightarrow N=3mgsin\left(\theta \right)$ .
Therefore, the normal reaction $N$ varies sinusoidally with $\theta $ as