Question

A table with smooth horizontal surface is fixed in a cabin that rotates with a uniform angular velocity $\omega$ in a circular path of radius $R$. A smooth groove $AB$ of length $L(<<R)$ is made on the surface of the table.
The groove makes an angle $\theta$ with the radius $OA$ of the circle in which the cabin rotates. A small particle is kept at point A in the groove and is released to move along $AB$. Find the time taken by the particle to reach the point $B$.

Answer 1

Let the cabin rotates with angular velocity $\omega$ and radius $R$

Force experienced by the particle= $mR{\omega }^2$

The component of $mR{\omega }^2$ along the groove which exerts force on the particle to move along AB.

$\therefore mR{\omega }^{2}cos\theta =ma=>a=R{\omega }^{2}cos\theta$

We know that,

$L=ut+\frac{1}{2}a{t}^2=>L=\frac{1}{2}R{\omega }^{2}cos\theta t^2$

$=>t^2=\frac{2L}{R{\omega }^{2}cos\theta }=>t=\sqrt{\frac{2L}{R{\omega }^{2}cos\theta }}$