Question

A particle of mass $m$ is allowed to oscillate near the point of minima of a vertical parabolic path having the equation $x^{2}=4ay$ , where $x-axis$ is along the horizontal direction and $y-axis$ is along the vertical direction. The angular frequency of small oscillations of the particle is

• A. $\sqrt{\frac{8 g}{a}}$
• B. $\sqrt{\frac{2 g}{a}}$
• C. $\sqrt{\frac{g}{a}}$
• D. $\sqrt{\frac{g}{2 a}}$

Answer 1

Answer: (D)

$\text{ma} = - \text{mg} sin \theta $
$\text{a} = - g sin \theta $ or $\text{a} = - g tan \theta $ ...(i)
(as $\theta $ is small)
Now, $\text{x}^{2} = 4 \text{ay}$
$∴ \, \, \, \frac{\text{dy}}{\text{dx}} = \frac{\text{x}}{2 \text{a}}$
$∴ \, \, \, \text{a} = - \text{g} \frac{\text{x}}{2 \text{a}}$
$- \omega ^{2} \text{x} = - \frac{\text{gx}}{2 \text{a}}$
$\omega = \sqrt{\frac{\text{g}}{2 \text{a}}}$