Animated Solution for Physics - Rotational Motion: A child is standing with folded hands at the centre of a platform rotating about its central axis. The kinetic energy of the system is $K$. The child now stretches his arms so that the moment of inertia of the system doubles. The kinetic energy of the system now is

A Socratic Thought Experiment

Imagine you are standing at the exact center of a frictionless, rotating merry-go-round. Your arms are folded tightly across your chest, and you are spinning at a comfortable, constant speed. Suddenly, you throw your arms wide open. What happens? You immediately feel yourself slowing down. But why? And more importantly, what happens to the energy of your rotation?

This classic physics problem is a beautiful demonstration of the interplay between mass distribution, rotational speed, and energy. Let's break it down step by step.

Conservation of Angular Momentum

The most crucial observation here is that there are no external twisting forces—no external torques—acting on the child-platform system. The friction at the axle is negligible, and the child's movement is entirely internal.

According to Newton's laws applied to rotation, when the net external torque is zero, the total angular momentum ($L$) of the system must remain absolutely constant.

Mathematically, this is written as:

Here, $I$ represents the moment of inertia (how hard it is to change the spin), and $\omega$ represents the angular velocity (how fast it is spinning).

When the child stretches his arms outward, he is moving mass further away from the axis of rotation. This increases the system's moment of inertia. The problem states that the moment of inertia exactly doubles:

Because the product of $I$ and $\omega$ must remain constant, if $I$ goes up by a factor of 2, $\omega$ must go down by a factor of 2 to compensate.

Let's see the math:

Canceling $I_1$ from both sides, we find:

The platform now spins at exactly half its original speed.

Now comes the real question: what happens to the kinetic energy? The rotational kinetic energy ($K$) is given by the formula:

Let's calculate the new kinetic energy, $K_2$, using our new values for inertia and angular velocity:

Substitute $I_2=2I_1$ and ${\omega }_2=\frac{{\omega }_1}{2}$:

When we square the angular velocity term, the factor of $\frac{1}{2}$ becomes $\frac{1}{4}$:

Notice that the term inside the parenthesis is exactly our initial kinetic energy, $K_1$ (which the problem calls $K$). Therefore:

Where Did It Go?

This result often confuses students. If energy is conserved in the universe, how can the kinetic energy of the platform just disappear?

The answer lies in the biology of the child. When the child is spinning, his arms want to fly outward due to inertia (often felt as a "centrifugal force"). To keep his arms folded, his muscles must exert an inward force. When he stretches his arms out, he is allowing them to move in the direction of that outward pull, but he is still controlling the motion.

In physics terms, the child's muscles do negative internal work to control the outward movement of his arms. This internal work consumes exactly half of the rotational kinetic energy, converting it into internal biochemical energy (or heat) within the child's body.

This is a profound lesson in physics: Angular momentum is strictly conserved in the absence of external torques, but mechanical kinetic energy is NOT conserved if internal non-conservative forces (like muscles) are doing work.