Angular Momentum
Reference for angular momentum L = Iw or L = mvr.
Includes moment of inertia for spheres, disks, rods, and cylinders with worked examples.
Need to calculate, not just reference? Use the interactive version.
Open Angular Momentum Calculator →
The Formulas
Point particle: L = m × v × r × sin(θ)
Rotating body: L = I × ω
Rotating body: L = I × ω
Angular momentum measures how much rotational motion an object has. Like linear momentum, angular momentum is conserved when no external torques act on a system.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| L | Angular momentum | kg·m²/s |
| m | Mass of the particle | kg |
| v | Velocity | m/s |
| r | Distance from the axis of rotation | m |
| θ | Angle between r and v vectors | degrees or radians |
| I | Moment of inertia | kg·m² |
| ω | Angular velocity | rad/s |
Example 1
A 2 kg ball moves at 5 m/s in a circle of radius 3 m
L = m × v × r (θ = 90°, sin(90°) = 1)
L = 2 × 5 × 3
= 30 kg·m²/s
Example 2
A solid disk (I = 0.5 kg·m²) spins at 10 rad/s
L = I × ω = 0.5 × 10
= 5 kg·m²/s
When to Use It
Use angular momentum when:
- Analyzing spinning objects (wheels, planets, figure skaters)
- Applying conservation of angular momentum (e.g., a skater pulling arms in spins faster)
- Studying orbital mechanics and satellite motion
- Solving rotational dynamics problems in physics
Key Notes
- Conservation of angular momentum: when no external torques act, L is constant — a figure skater spins faster pulling arms in because reducing r decreases I, so ω must increase to keep L = Iω unchanged
- L is a vector, not just a magnitude; its direction is given by the right-hand rule and lies along the rotation axis — this is why a spinning gyroscope resists tilting (changing the direction of L requires a torque)
- Moment of inertia I depends on how mass is distributed: solid sphere I = 2/5mr², hollow sphere I = 2/3mr², solid disk I = 1/2mr² — a hollow cylinder is harder to spin up than a solid one of the same mass because more mass sits at the rim
- For a planet in orbit, L = mvr (perpendicular velocity) remains constant — this is Kepler's second law rewritten: the planet sweeps equal areas in equal times because angular momentum is conserved in the absence of external torques