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 (figure 7-E3). A smooth groove AB of length L(<<R) is made on the surface of the table. The groove makes an angle $\omega$ with the radius OA of the circle in which the cabin rotates. A small particle is kept at the 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

The cabin rotates with angular velocity $\omega$ and radius R.

$\therefore$ The particle experiences a force $mR{\omega }^2.$

The component of $mR{\omega }^2$ along the provided force to the particle to move along AB.

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

$\Rightarrow a=R{\omega }^{2}cos\theta$

Length of groove L,

$L=ut+\frac{1}{2}a{t}^2$

$\Rightarrow L=\frac{1}{2}R{\omega }^{2}cos\theta t^2$

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

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