Angular Momentum Formula

The Formula

L = I\omega = mvr

When to use: A spinning skater pulling their arms in spins faster — they're conserving angular momentum.

Quick Example

A planet orbiting the sun has angular momentum; it moves faster when closer to the sun.

Notation

\vec{L} is angular momentum in kg·m²/s, I is the moment of inertia in kg·m², \omega is angular velocity in rad/s, \vec{r} is the position vector, and \vec{\tau} is torque in N·m.

What This Formula Means

The rotational equivalent of linear momentum, measuring the quantity of rotational motion in a spinning or orbiting object.

A spinning skater pulling their arms in spins faster — they're conserving angular momentum.

Formal View

Angular momentum of a rigid body is \vec{L} = I\vec{\omega}, where I is the moment of inertia and \vec{\omega} is the angular velocity. For a point particle: \vec{L} = \vec{r} \times m\vec{v}. Conservation: \frac{d\vec{L}}{dt} = \vec{\tau}_{\text{net}}; if \vec{\tau}_{\text{net}} = 0, then \vec{L} is constant.

Common Mistakes

  • Confusing angular momentum with linear momentum — angular momentum involves rotation about an axis and uses moment of inertia, not just mass.
  • Forgetting that angular momentum is a vector — its direction is along the axis of rotation (right-hand rule), and it can point up or down.
  • Assuming angular velocity stays constant when the mass distribution changes — when a skater pulls arms in, I decreases and \omega must increase to conserve L.

Why This Formula Matters

Angular momentum conservation explains why ice skaters spin faster when pulling arms in, how gyroscopes maintain orientation, why planets speed up near the sun, and how helicopters use tail rotors to prevent spinning.

Frequently Asked Questions

What is the Angular Momentum formula?

The rotational equivalent of linear momentum, measuring the quantity of rotational motion in a spinning or orbiting object.

How do you use the Angular Momentum formula?

A spinning skater pulling their arms in spins faster — they're conserving angular momentum.

What do the symbols mean in the Angular Momentum formula?

\vec{L} is angular momentum in kg·m²/s, I is the moment of inertia in kg·m², \omega is angular velocity in rad/s, \vec{r} is the position vector, and \vec{\tau} is torque in N·m.

Why is the Angular Momentum formula important in Physics?

Angular momentum conservation explains why ice skaters spin faster when pulling arms in, how gyroscopes maintain orientation, why planets speed up near the sun, and how helicopters use tail rotors to prevent spinning.

What do students get wrong about Angular Momentum?

Angular momentum depends on both the mass distribution and the rotation rate.

What should I learn before the Angular Momentum formula?

Before studying the Angular Momentum formula, you should understand: torque, momentum.

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Related Concepts

TorqueMomentum