Angular Momentum Calculator
Calculate angular momentum of a rotating object using mass, velocity, and radius, or moment of inertia and angular velocity.
Angular momentum is the rotational equivalent of linear momentum. Just as a moving object has momentum (mass × velocity), a rotating object has angular momentum — and just like linear momentum, it is conserved unless acted upon by an external torque. This principle explains some of the most fascinating phenomena in physics.
Two Forms of the Formula
Form 1 — Point mass moving in a circle: L = m × v × r
Where:
- L = angular momentum (kg·m²/s)
- m = mass (kg)
- v = linear velocity (m/s)
- r = radius of circular path (m)
Form 2 — Rotating rigid body: L = I × ω
Where:
- L = angular momentum (kg·m²/s)
- I = moment of inertia (kg·m²)
- ω = angular velocity (radians per second)
To convert RPM to rad/s: ω = RPM × 2π ÷ 60
Common Moments of Inertia
| Shape | Formula | Example |
|---|---|---|
| Solid sphere | 2/5 × m × r² | Ball bearing, planet |
| Hollow sphere | 2/3 × m × r² | Tennis ball |
| Solid disk / cylinder | 1/2 × m × r² | Flywheel, coin |
| Thin ring | m × r² | Bicycle tire |
| Thin rod (center) | 1/12 × m × L² | Propeller blade |
Conservation of Angular Momentum
When no external torque acts on a system, angular momentum is conserved. This is why:
- An ice skater spins faster when arms are pulled in (radius decreases, ω increases)
- A collapsing star becomes a pulsar spinning hundreds of times per second
- A diver tucking during a somersault rotates faster before extending to slow down for entry
Real-World Applications
Angular momentum concepts appear in gyroscopes (navigation systems), bicycle wheel stability, figure skating, planetary motion, and the angular momentum of electrons in atomic orbitals.
Units
SI unit for angular momentum is kg·m²/s, also expressed as N·m·s or J·s. In CGS units: g·cm²/s.