An insect of mass \(100 \mathrm{mg}\) is resting on the edge of a flat uniform disc of mass \(3 \mathrm{~g}\) and radius \(50 \mathrm{~mm}\), which is rotating at 60 r.p.m. about a smooth pivot. The insect crawls in towards the centre of the disc. Find the angular velocity when it reaches it, and the gain in kinetic energy. Where does this kinetic energy come from, and what happens to it when the insect crawls back out to the edge?

Think of the moment of inertia as an object’s resistance to change in its rotation. It is the rotational equivalent to mass in linear motion. The more an object’s mass is distributed far from the axis of rotation, the higher its moment of inertia will be, making it harder to spin or stop once it starts spinning.

For our disc and insect system, we have two different moments of inertia to consider. For the uniform disc that spins around a pivot at its center, we use the formula: \[I_{disc} = \frac{1}{2}MR^2\], where \(M\) is the mass of the disc and \(R\) is its radius. The moment of inertia of the insect, when at the edge of the disc, is calculated like a point mass at a distance \(R\) from the pivot: \[I_{insect} = mR^2\]. When the insect moves, the total moment of inertia of the system changes because the distribution of mass relative to the axis of rotation changes. This plays a crucial role in the conservation of angular momentum, which we’ll explore in depth later in the article.

In a rotational system, kinetic energy is associated with the rotation of an object. The formula to determine kinetic energy for a rotating object is \[KE = \frac{1}{2}I\omega^2\], where \(I\) represents the moment of inertia and \(\omega\) represents the angular velocity. This tells us that the kinetic energy of an object depends not just on how fast it's spinning \(\omega\), but also on how its mass is distributed \(I\).

Initially, the disc-insect system has a certain kinetic energy based on their joint moment of inertia and the speed of rotation. When the insect moves inward, towards the axis, it affects both the system's total moment of inertia and kinetic energy. Contrary to what one might think, even though no external forces are doing work on the system, the kinetic energy changes! This is due to the energy that the insect itself expends crawling inward, which brings us to the law of conservation of energy.

Conservation of energy is a fundamental principle that states that energy cannot be created or destroyed, only transformed from one form to another. In the case of our rotating system, this concept assures that as the insect moves, the energy within the system changes form but the total energy remains constant.

The insect crawling toward the center of the disc does work against the centripetal force, and this work gets transformed into increased kinetic energy as the system speeds up. This is where the gain in kinetic energy originates. When the insect moves back out, the process reverses: kinetic energy is converted back to potential energy, specifically the potential energy associated with the insect being farther from the axis of rotation. There’s a clear exchange between kinetic and potential energy while the overall energy within the system remains unchanged, showcasing the conservation of energy in action.