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Conservation of Angular Momentum, Explained

Here's how angular momentum works β€” the single principle behind a spinning skater, a diving cat, and a planet's orbit.

What is angular momentum?

Angular momentum is a physical quantity representing the strength of a rotating object's motion β€” it equals the moment of inertia (how far the mass is distributed from the rotation axis) multiplied by the angular velocity (how fast it's spinning). Much like momentum in straight-line motion, it represents how strongly a rotating object tends to keep spinning.

The law of conservation of angular momentum

If no external torque (a force that causes rotation) acts on an object, its angular momentum stays constant throughout the rotation. If the moment of inertia changes, the angular velocity changes inversely to keep the total angular momentum conserved.

Why a figure skater spins faster with arms pulled in

When a spinning figure skater pulls their outstretched arms in toward their body, their moment of inertia decreases, and their rotational speed (angular velocity) increases sharply to conserve angular momentum. Extending the arms back out increases the moment of inertia again, slowing the spin.

Why a planet's orbital speed changes

A planet on an elliptical orbit moves faster when it's closer to the sun and slower when it's farther away, because its angular momentum stays constant along the orbit. This is also described by Kepler's second law of planetary motion (equal areas in equal times), and is a classic astronomical example of angular momentum conservation.

What is moment of inertia?

Moment of inertia is an object's resistance to changing its rotational state β€” for the same mass, the farther it's distributed from the rotation axis, the larger the moment of inertia. That's why extending the arms in figure skating increases the moment of inertia (mass moves farther from the axis), and pulling them in decreases it.

Divers and falling cats spinning in mid-air

A diver tucking their body in mid-air to spin faster, then extending it right before entering the water to slow down, is a technique that uses conservation of angular momentum. A falling cat twisting its body to land feet-first is another example β€” using the rotation of part of the body to change posture while keeping the total angular momentum conserved.

Why conservation of angular momentum matters

The law of conservation of angular momentum is a core principle for understanding rotating objects, explaining everything from athletic moves to the orbits of celestial bodies. It gives you an intuitive way to understand that a rotating object's spin speed changes whenever its shape changes.

Gravity and orbital motion, too

The orbit a planet follows around the sun is itself created by gravity. The two ideas work together: gravity keeps a planet on its orbital path, while angular momentum conservation explains why its speed changes along the way.

Frequently Asked Questions

How does the law of conservation of angular momentum relate to Newton's laws?

The law of conservation of angular momentum is what you get when you apply Newton's laws of motion to rotational motion. Just as momentum is conserved in straight-line motion when there's no external force, angular momentum is conserved in rotational motion when there's no external torque.

Why does spinning speed up when arms are pulled in β€” does that mean energy increases?

As the moment of inertia decreases, the angular velocity increases to keep angular momentum constant, but because the skater is actually using muscle force to pull their arms inward, the rotational kinetic energy actually increases in the process. In other words, the energy spent pulling the arms in goes into increasing the spin speed.