Did you ever find yourself wishing for an angular momentum calculator? This user-friendly tool lets you find angular momentum in two ways, so you can use it with whatever data you have gathered. Below we also explain the conservation of angular momentum and walk through some worked examples. If you would rather compute straight-line motion, try our momentum calculator instead.
What Is Angular Momentum?
Angular momentum, occasionally referred to as the moment of momentum or rotational momentum, is the momentum of an object around its axis. Just as linear momentum is "mass in motion," angular momentum is the rotational correspondent of linear momentum. It is usually denoted by the letter L.
Similar to angular velocity, angular momentum has two distinct types:
- Spin angular momentum — the angular momentum of an object around its own center-of-mass axis; and
- Orbital angular momentum — the momentum of the center of mass about an external origin.
These two contributions sum to give the total angular momentum of an object. The direction of angular momentum is perpendicular to the object's plane of rotation: if you curl the fingers of your right hand in the direction of rotation, your thumb points along the direction of L (the right-hand rule).
Angular Momentum Formula
This calculator supports the two standard ways to find angular momentum:
1. Point Mass / Orbital Motion (L = m · v · r)
Consider the planets revolving around the Sun ☀️ — each is an object moving around a central point. In this situation the angular momentum is the product of the mass m, the velocity v, and the radius r of the circular path:
L = m × v × r
2. Rigid Body Rotating on Its Axis (L = I · ω)
The Earth's rotation on its axis is an example of a rigid body spinning about its axis. Here the angular momentum is the product of the moment of inertia I and the angular velocity ω:
L = I × ω
Our calculator uses both formulas so you can pick whichever matches the data you have. If you need the rotational inertia of a shape first, see our moment of inertia calculator; for the spin rate, see the angular velocity calculator.
Dimensions and Unit of Angular Momentum
Using the formula for angular momentum, we can easily derive its units:
L = m × v × r = kg · m/s · m = kg · m²/s
Hence the SI unit of angular momentum is kg · m²/s, which is equivalent to a newton-metre-second (N · m · s) and to a joule-second (J · s). From the unit we can also derive its dimensions:
L = kg · m²/s = M L² / T
In the US customary system angular momentum is often expressed in slug · ft²/s or lb · ft²/s. This calculator reports the result in metric and imperial units side by side.
Angular Momentum in the Context of Linear Momentum
Linear momentum p = m · v describes motion in a straight line; angular momentum is its rotational analogue. For a point mass moving along a circle of radius r, the two are linked by L = p · r = m · v · r. More generally L = r × p (a vector cross product), so angular momentum depends not only on how much momentum a body has, but also on how far it is from the axis. This is why moving mass closer to or farther from the axis changes the rotation, even when the linear momentum is unchanged.
Conservation of Angular Momentum
In the absence of an external torque, the total angular momentum of a system is conserved: L = I · ω = constant. This is one of the most important conservation laws in physics. Because I · ω stays fixed, decreasing the moment of inertia I must increase the angular velocity ω, and vice versa.
Angular Momentum Examples
- Figure skater: a spinning skater pulls in their arms, reducing I, so ω rises and they spin faster — angular momentum is conserved.
- Earth's spin: with a polar moment of inertia of about 8.0 × 10³⁷ kg · m² and ω ≈ 7.29 × 10⁻⁵ rad/s, the Earth's spin angular momentum is roughly 5.8 × 10³³ kg · m²/s.
- Neutron stars: when a massive star collapses, its radius shrinks dramatically, so its rotation rate skyrockets — pulsars can spin hundreds of times per second.
- Gyroscopes: a large angular momentum makes the spin axis resist reorientation, which is what keeps bicycles, spacecraft, and navigation gyros stable.
Angular Momentum in Other Fields of Physics
Angular momentum is not limited to mechanics. In quantum mechanics, particles carry intrinsic angular momentum called spin, and orbital angular momentum is quantized in units of the reduced Planck constant ℏ. In electromagnetism, light itself carries both spin (polarization) and orbital angular momentum. In astrophysics, the conservation of angular momentum shapes accretion disks, planetary systems, and galaxies. The same simple idea — L = I · ω — echoes across every scale of the universe.
FAQs
What is the SI unit of angular momentum?
Kilogram metre squared per second (kg · m²/s), equivalent to N · m · s and to J · s.
Can angular momentum be negative?
Yes. The sign indicates the sense of rotation (clockwise vs. counter-clockwise) relative to a chosen axis.
What is the difference between the two formulas?
Use L = m · v · r for a small object moving along a circular path (orbital motion), and L = I · ω for an extended rigid body spinning about its own axis. Both give angular momentum in kg · m²/s.
Is angular momentum the same as torque?
No. Torque is the rotational analogue of force and equals the rate of change of angular momentum: τ = dL/dt. A constant angular momentum means zero net torque.