Problem

A particle $P$ of mass $m$ is moving in a horizontal circle with angular speed $\omega_1$ on the smooth inner surface of a hemispherical shell of radius $r$. The angle between the upward vertical and the normal reaction of the surface on $P$ is $\theta_1$, where $\tan\theta_1=\frac34$.

When the angular speed is increased to $\omega_2$, the angle between the upward vertical and the normal reaction of the surface on $P$ becomes $\theta_2$, where $\tan\theta_2=\frac43$.

Find the ratio $\frac{\omega_1}{\omega_2}$.