Angular Momentum Calculator – Point Mass and Rigid Body
Calculate angular momentum L using L = mvr for a point mass or L = Iω for a rigid body rotating about a fixed axis.
Select the object type, enter the required values, and instantly compute angular momentum in kg·m²/s.
Angular Momentum Calculator – Point Mass and Rigid Body
Calculate angular momentum L using L = mvr for a point mass or L = Iω for a rigid body rotating about a fixed axis.
About the Angular Momentum Calculator
Angular momentum is a fundamental conserved quantity in physics that describes the rotational inertia of a system. Just as linear momentum p = mv quantifies the inertia of an object moving in a straight line, angular momentum L quantifies the tendency of a rotating or orbiting object to continue its rotation. The SI unit is kg·m²/s, which is equivalent to J·s (joule-seconds).
For a point mass moving in a circular path, angular momentum is L = m × v × r, where m is the mass in kilograms, v is the tangential velocity in metres per second, and r is the perpendicular distance from the axis of rotation to the line of motion (the moment arm) in metres. This formula applies to planets orbiting the sun, electrons in atomic orbitals (treated classically), and any small object moving in a curved path.
For a rigid body rotating about a fixed axis, angular momentum is L = I × ω, where I is the moment of inertia in kg·m² and ω is the angular velocity in rad/s. This is the direct rotational analogue of linear momentum, with I playing the role of mass and ω playing the role of velocity. The moment of inertia depends on both the mass and its distribution relative to the rotation axis.
One of the most important properties of angular momentum is its conservation: in the absence of external torques, the total angular momentum of an isolated system remains constant. This principle explains why a spinning ice skater speeds up when pulling in her arms (reducing I forces ω to increase), why gyroscopes maintain their orientation, why the Earth keeps rotating, and why spiral galaxies maintain their structure over billions of years.
Angular momentum also plays a central role in quantum mechanics, where it is quantised in units of ħ (the reduced Planck constant). The orbital and spin angular momenta of electrons determine atomic structure, chemical bonding, and the selection rules for spectroscopic transitions.
This calculator handles two common scenarios: a point mass with mass, velocity, and orbital radius (relevant for orbital mechanics, circular motion, and lever-arm problems), and a rigid body with moment of inertia and angular velocity (relevant for flywheels, spinning discs, rotors, and any extended rotating object).
Angular Momentum Examples
Four worked examples ranging from planetary orbits to laboratory-scale rotators.
| Input | Result | Notes |
|---|---|---|
| Planet in orbit: m = 1×10²⁴ kg, v = 2.98×10⁴ m/s, r = 1.5×10¹¹ m | L ≈ 4.47×10³⁹ kg·m²/s | Point mass model. L = 1e24 × 2.98e4 × 1.5e11. |
| Ball on string: m = 0.5 kg, v = 3 m/s, r = 1.2 m | L = 1.8 kg·m²/s | Point mass. L = 0.5 × 3 × 1.2 = 1.8 kg·m²/s. |
| Flywheel: I = 2.5 kg·m², ω = 10 rad/s | L = 25 kg·m²/s | Rigid body. L = I × ω = 2.5 × 10 = 25 kg·m²/s. |
| Earth: I = 8.04×10³⁷ kg·m², ω = 7.27×10⁻⁵ rad/s | L ≈ 5.845×10³³ kg·m²/s | Rigid body model for Earth's spin angular momentum. |
How to use the Angular Momentum Calculator
- Select 'Point Mass' if you have a mass moving in a circular path at a given radius, or 'Rigid Body' if you have a rotating object with a known moment of inertia.
- For point mass, enter the mass m (kg), tangential velocity v (m/s), and perpendicular radius r (m). The result is L = m × v × r.
- For rigid body, enter the moment of inertia I (kg·m²) and angular velocity ω (rad/s). The result is L = I × ω.
- Click Calculate to display the angular momentum L in kg·m²/s. Click Reset to clear all inputs.
Angular Momentum FAQ
What is angular momentum and why is it important?
Angular momentum L is the rotational equivalent of linear momentum. It measures how much rotational motion an object has and in what direction. It is important because it is conserved in systems with no external torque — this conservation law explains gyroscopes, planetary motion, and figure skaters speeding up when they tuck their arms.
What is the difference between the two calculation methods?
The point mass formula L = mvr applies to objects treated as particles moving along a curved path — orbiting planets, swinging pendulums, or balls on strings. The rigid body formula L = Iω applies to extended objects rotating about a fixed axis — flywheels, spinning discs, turbines, and planets (as self-rotating bodies).
How do I find the moment of inertia I?
Common values: solid disc I = ½mr²; solid sphere I = ⅖mr²; thin ring I = mr²; thin rod about centre I = (1/12)mL². For complex shapes, use the parallel-axis theorem or look up the formula for your geometry. I has units of kg·m².
What are the units of angular momentum?
Angular momentum is measured in kg·m²/s. This is equivalent to N·m·s (newton-metre-seconds) and J·s (joule-seconds). In quantum mechanics, angular momentum is quantised in integer or half-integer multiples of ħ ≈ 1.055×10⁻³⁴ J·s.
How is angular momentum conserved in practice?
When no external torque acts on a system, its total angular momentum is constant. A spinning ice skater brings her arms in (reducing I) so ω must increase to keep L = Iω constant. A planet moves faster when close to the sun (smaller r) so v must increase to keep mvr constant.
Can angular momentum be zero?
Yes. A stationary object has zero angular momentum. An object moving directly towards or away from the reference point has zero angular momentum because the perpendicular component of velocity is zero (r × v_perp = 0). In quantum mechanics, electrons in s-orbitals have zero orbital angular momentum.