Pendulum similarity with a ball attached to a uniform disk with constant [closed]

Question

A particle of mass $m$ is supported by a frictionless horizontal disk which rotates about a vertical axis through its center with a constant angular velocity $\omega$ . The particle is connected by a massless string of length $\ell$ to a point located a distance $a$ from the center of the disk. Show that the motion of the particle with respect to the disk is similar to that of a simple pendulum, and find the frequency of small oscillations of the system. Set up the equations in this problem by means of Lagrange's equations.

No graphics, no examples, the answer is $ \ddot{\Theta}+ a \, \omega^{2} \sin(\Theta )/\ell=0 $

How do you understand this problem because, I really can't see the pendulum similarity.

Answer 1

with respect to the disk

This point is key. When you transform into the reference frame of the disk, the (pseudo) centrifugal force appears, which points towards the outside of the disk.

Place your rotating reference frame (remember it is rotating with the disk) such that the anchor point is exactly below the center of the disk. This means the outward pointing centrifugal force is pointing down. Now you can see that this is a mass on a string with a force pulling down, i.e. a pendulum.