Spinny Skaters

Today's article describes a direct consequence of angular momentum. An example is shown in Figure 1. The ice skater is able to increase her rotational speed without touching the ground. She only moves her arms in closer towards her body. Before understanding how angular momentum works, we need to learn some background. We'll start simple. Rotation always happens around an axis. This is intuitive. For our ice-skater example, the axis she spins around is vertical through her body, from head to toe. Next, we have ways to measure how fast something is rotating around an axis. It is often measured in rotations per minute (rpm) or radians per second (rad/s). This is angular speed. If we add a direction, we can make it angular velocity. For instance, if we define clockwise as positive and counter-clockwise as negative, angular speed can be recognized as angular velocity. If the ice-skater is spinning at 5 rad/s, then we would know that every second, she rotates 5 radians in the clockwise direction.

Another aspect of rotational motion is the moment of inertia. There's a simple way of imagining this. Say we push two boxes, one heavy and one light. The light box will have a greater speed than that of the heavy box. This is because the light box poses less resistance to a change in linear ("normal") motion. Similarly, the moment of inertia measures the amount of resistance an object poses to a change of rotational motion. However, this isn't as simple as the weight of the object, because rotation also depends on an axis. As a general rule, the moment of inertia is calculated as the distribution of the mass of an object away from the axis. If most of the mass of the object is farther from the axis, then the moment of inertia is high. If the mass of the object is close to the axis, then the moment of inertia is lower.

Finally, we can come back to angular momentum. From here it's simple. Angular momentum is calculated as the product of the angular velocity and moment of inertia. Something special about angular momentum is that it is conserved in the absence of a non-zero net external torque. Let's break this down. Many of us are familiar with Newton's first law. This states that an object in motion will stay in the same motion unless acted upon by a non-zero net external force. This means that without any forces, a sliding box would never stop sliding. This is similar to the case with angular momentum. Torque is simply the rotational form of a force. We are essentially stating that angular momentum will always be the same unless if a torque that is not zero is acted on it. When the figure skater is spinning, there is no external torque. This means, as long as she pulls her arms in and no one else pushes her, the angular momentum will be conserved. Now we have a "before" and "after". Angular momentum will be the same in both. In the "after", the arms (mass) are closer to the axis of rotation. Referring to the previous paragraph, this means the moment of inertia is decreased in the "after". Angular momentum is constant and is = angular velocity * moment of inertia. Therefore, if the moment of inertia decreases, in order to keep a constant angular momentum, the angular velocity of the ice skater increases. This is why she spins faster after bringing her arms in.