How does the law of conservation of angular momentum control a figure-skater's rate of spin?

Moment of inertia is a measure of how much resistance an object has to a change in its rotation. Think of it as the rotational equivalent of mass in linear motion. The moment of inertia depends not only on the mass of an object but also on how that mass is distributed relative to the axis of rotation.
For instance, a figure skater with arms extended has a larger moment of inertia because their mass is spread out further from the center of their body. Conversely, when they pull their arms in close, the moment of inertia decreases as the mass is now nearer to the axis of rotation.
Mathematically, the moment of inertia can be represented as:
\[ I = \frac{1}{12} M (L^2 + W^2) \] where:

• \(M\) is the mass
• \(L\) is the length of the object
• \(W\) is the width of the object

Although these variables can change with different shapes, the idea is that the further the mass is spread from the center, the larger the moment of inertia.

Angular velocity describes how fast something is rotating. It measures the rate of rotation and is represented as \(\text{\textbackslash omega}\) in the equation. You can think of it as the speed of rotation and it is typically measured in radians per second.
For instance, if a figure skater speeds up their spin, their angular velocity increases. If they slow down, their angular velocity decreases.
The relationship between angular velocity and angular momentum is pivotal in understanding figure skating dynamics. The angular momentum formula is:
\[ L = I \text {\textbackslash omega} \]

• \(L\) is the angular momentum
• \(I\) is the moment of inertia
• \(\text{\textbackslash omega}\) is the angular velocity

From this, it's clear that if the moment of inertia decreases (like when a skater pulls in their arms), the angular velocity must increase to keep the angular momentum constant.

The dynamics of a figure skater's spin provide a perfect real-world example of the conservation of angular momentum. When a skater begins to spin with arms extended, they have a high moment of inertia and a relatively lower angular velocity.
If the skater pulls in their arms, they reduce their moment of inertia considerably. Since the angular momentum must stay the same (as there’s no external torque acting), their angular velocity increases, and they spin much faster.
On the opposite end, if a skater extends their arms, the moment of inertia increases. Consequently, their angular velocity decreases to conserve angular momentum, causing them to spin slower.
This concept can be observed in many skating performances, where skaters pull their limbs in to accelerate spins and extend them to decelerate. Understanding these dynamics is critical for anyone studying rotational motion and is a beautiful demonstration of the conservation laws in physics.