Question

A particle of mass $m$ is allowed to oscillate on a smooth parabola: $x^2=4ay,a>1$ as shown in the figure. The angular frequency $\left(\omega \right)$ of small oscillations is

• A. $\omega =\sqrt{\frac{g}{4a}}$
• B. $\omega =\sqrt{\frac{g}{2a}}$
• C. $\omega =\sqrt{\frac{2g}{a}}$
• D. $\omega =\sqrt{\frac{g}{a}}$

Answer 1

The correct option is B $\omega =\sqrt{\frac{g}{2a}}$

Ref.Image.

$x^2=4ay$

$2x=4a\frac{dy}{dx}$

$tan\theta =\frac{dy}{dx}=\frac{2x}{4a}$

fresing = -mg $sin\theta$

$tan\theta -sin\theta \simeq \theta .tan\theta$ is very small

$tan\theta =\frac{x}{20}$

$F=-mg\frac{X}{20}$

$F=-k/l$

$K=\frac{mg}{2a}K=m{w}^2$

$\frac{mg}{2a}=m{w}^2$

$w2\sqrt{\frac{y}{2a}}$