Question

Consider the figure here. A particle of mass $m$ is constrained to move inside a smooth vertical groove of radius $R$ and is connected to a light spring of spring constant $K$ in equilibrium. $O$ is centre of the groove. $x$ and $y$ are horizontal and vertical axes respectively. Different physical parameters are related as $2KR=7mg$ . Angular frequency of the oscillations, if the particle is slightly displaced from the shown equilibrium position is

• A. $\sqrt{\frac{g}{R}}$
• B. $\sqrt{\frac{g}{2R}}$
• C. $2\sqrt{\frac{g}{R}}$
• D. $\sqrt{\frac{2g}{R}}$

Answer 1

Answer: (C)

Restoring torque on small displacement $d\theta$
$\tau =K{R}^{2}d\theta +\frac{mg}{2}Rd\theta$
$\Rightarrow \alpha =\frac{\tau }{l}=\frac{\left(K{R}^2+\frac{mg}{2}R\right)d\theta }{m{R}^2}$
$=\left(\frac{K}{m}+\frac{g}{2R}\right)d\theta ={\left(\omega \right)}^{2}d\theta$
$\Rightarrow \omega =\sqrt{\frac{K}{m}+\frac{g}{2R}}=2\sqrt{\frac{g}{R}}$