Question

A particle of mass $5 \times 10^{-5}\, kg$ is placed at the lowest point of a smooth parabola having the equation $20 x^{2}=y(x, y$ in $m ) .$ Here $y$ is the vertical height. If it is displaced slightly and it is constrained to move along the parabola, find the angular frequency (in $rad / s$ ) of small oscillations.

Answer 1

Potential energy of particle $=m g y$
$\therefore F_{x}=\frac{-\partial U}{\partial x}=\frac{-\partial\left(20 m g x^{2}\right)}{\partial x}=-40\, m g x$
$F \propto-x$, thus $\omega^{2}=\frac{40 \times 5 \times 10^{-5} \times 10}{5 \times 10^{-5}}$
$\Rightarrow \omega=20$