Angular Momentum Formula

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

The Formula

L=Iω=mvrL = 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

L⃗\vec{L} is angular momentum in kg·m²/s, II is the moment of inertia in kg·m², ω\omega is angular velocity in rad/s, r⃗\vec{r} is the position vector, and τ⃗\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 L⃗=Iω⃗\vec{L} = I\vec{\omega}, where II is the moment of inertia and ω⃗\vec{\omega} is the angular velocity. For a point particle: L⃗=r⃗×mv⃗\vec{L} = \vec{r} \times m\vec{v}. Conservation: dL⃗dt=τ⃗net\frac{d\vec{L}}{dt} = \vec{\tau}_{\text{net}}; if τ⃗net=0\vec{\tau}_{\text{net}} = 0, then L⃗\vec{L} is constant.

Worked Examples

Example 1

medium
A skater has I1=8 kg\cdotpm2I_1 = 8\,\text{kg·m}^2, ω1=1.5 rad/s\omega_1 = 1.5\,\text{rad/s}. They tuck in to I2=3 kg\cdotpm2I_2 = 3\,\text{kg·m}^2. Find ω2\omega_2.

Answer

ω2=4 rad/s\omega_2 = 4 \text{ rad/s}

First step

1
Conserve LL: I1ω1=I2ω2I_1 \omega_1 = I_2 \omega_2.

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Example 2

hard
Show that if KE is conserved when a skater pulls in their arms (II halves), then LL cannot be conserved. (Reach a contradiction.)

Common Mistakes

  • Confusing angular momentum with linear momentum — angular momentum involves rotation about an axis and uses moment of inertia, not just mass. - Fix this by naming the system, checking "Is the interaction short, collision-like, or rotational, and have I checked whether external forces or torques can be ignored?", and attaching units or direction to the final statement.
  • 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. - Fix this by naming the system, checking "Is the interaction short, collision-like, or rotational, and have I checked whether external forces or torques can be ignored?", and attaching units or direction to the final statement.
  • Assuming angular velocity stays constant when the mass distribution changes — when a skater pulls arms in, II decreases and ω\omega must increase to conserve LL. - Fix this by naming the system, checking "Is the interaction short, collision-like, or rotational, and have I checked whether external forces or torques can be ignored?", and attaching units or direction to the final statement.
  • Using angular momentum from a keyword alone - Signal words like momentum, impulse, collision only point to a possible model; the system must match too.

Why This Formula Matters

Angular Momentum is central because forces explain changes in motion and balance. Students who can isolate a system and draw the interactions can avoid treating every force word as the same kind of cause.

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?

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

Why is the Angular Momentum formula important in Physics?

Angular Momentum is central because forces explain changes in motion and balance. Students who can isolate a system and draw the interactions can avoid treating every force word as the same kind of cause.

What do students get wrong about Angular Momentum?

Students often know a formula related to angular momentum but skip the recognition step: Is the interaction short, collision-like, or rotational, and have I checked whether external forces or torques can be ignored? That leads to a correct-looking substitution attached to the wrong physical model.

What should I learn before the Angular Momentum formula?

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

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TorqueMomentum