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.
A few things worth knowing. The point-mass form is really L = m·v·r·sin(θ), where θ is the angle between the velocity and the radius. When the motion is perpendicular to the radius (the usual case), sin(θ) = 1 and it collapses to the simple m·v·r. Angular momentum is also a vector, not just a size: its direction follows the right-hand rule and points along the rotation axis, which is exactly why a spinning gyroscope fights being tilted. And for a planet orbiting the Sun, the conserved L = mvr is Kepler’s second law in disguise — equal areas swept in equal times, because nothing applies an external torque to the orbit.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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