Question

A ring of radius R performs small oscillations around the pivot point O (Figure 6). Determine the period of oscillation.

                                 

Answer 1

Solution

The ring, suspended at the point o is a physical pendulum. The period of oscillation is determined by the formula

                                     

where I is the moment of inertia of the ring about its center, m is the mass of the ring, a is the distance from the pivot point to the centre of the ring.

The moment of inertia of a ring of mass m is equal to  I₀ =mR ² As the distance from the centre of the ring to the pivot point is equal to  then using the parallel axis theorem (aka Huygens-Steiner theorem), we find the total moment of inertia of the pendulum:

                     

Given that a=R, we obtain the following expression for the oscillation period: