Two pendulums consist of a massless rigid rod of equal length attached to a small sphere of equal radius, with one sphere hollow for one pendulum and the other solid. Each pendulum undergoes damped SHM. Which pendulum has the largest time period?

This question requires a qualitative answer seeing as we do not have an expression for the time period of a pendulum undergoing damped SHM. This is what the examiners will be expecting.There are a few different ways we could answer this, all being equally valid. I shall answer this question using the concept of inertia. Inertia is simply a measure of how hard it is to accelerate an object, i.e. the mass of an object. For example, it’s much harder to accelerate a lorry by pushing it than a car to the same speed as the lorry has a larger mass.We can make the reasonable assumption that the hollow sphere has a lighter mass than the solid one (this would be usually stated in the question).Answer: Since the pendulums are undergoing damped harmonic motion, a force must be acting on the pendulums in the opposite direction to their velocity. This force will be equal for both pendulums as they have the same radius. We know from Newton’s second law that F=mawhere F is the force, m is the mass and a is the acceleration of the object. If one pendulum has a smaller mass than the other but has the same force acting on it, then the smaller mass pendulum will have a larger acceleration, or rather deceleration. Therefore, the solid sphere pendulum must have a larger time period as it has a smaller declaration and will take longer to slow down compared to the hollow one.