Vibrations

The damped natural frequency is related to the undamped natural frequency and the damping factor by the following formula:

Given that and , we can calculate the damped natural frequency as:

The natural frequency of a system with rotational motion can be found using the formula for the torsional pendulum. For small angular displacement, the restoring force is proportional to the angular displacement, and the equivalent moment of inertia for the system is considered. The effective spring constant for rotational motion is adjusted by considering the location of the spring above the centre of mass of the disc.

Using energy principles and angular dynamics, the natural frequency is given by:

where is the moment of inertia of the disc. Simplifying the expression:

Thus, the correct answer is Option (4).