Angular Momentum Calculator
Calculate angular momentum with L = I omega from moment of inertia and angular velocity, or with L = mvr for a point mass on a circular path.
Angular Momentum Calculator
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Formula: L = Iω or L = mvr
Worked example — 10 kg·m²/s
Formula
L = Iω or L = mvr
Rotational analog of linear momentum. Conserved in the absence of external torques.
Worked Examples
Example 1: Spinning wheel
Problem:I=2 kg·m², ω=5 rad/s
Solution:L=2×5=10 kg·m²/s
Result:10 kg·m²/s
Frequently Asked Questions
What is angular momentum?
Angular momentum (L) is the rotational equivalent of linear momentum. For a rigid body, L = Iω, where I is the moment of inertia (kg·m²) and ω is the angular velocity (rad/s). Its SI unit is kg·m²/s.
How do you calculate angular momentum?
Use L = Iω for rotating bodies, or L = mvr for a point mass moving in a circle. For example, a wheel with I = 2 kg·m² spinning at ω = 5 rad/s has L = 2 × 5 = 10 kg·m²/s.
What units does angular momentum use?
The SI unit of angular momentum is kg·m²/s, equivalent to J·s (joule-seconds). The moment of inertia is in kg·m² and angular velocity in rad/s, giving kg·m²/s when multiplied.
When is angular momentum conserved?
Angular momentum is conserved when no net external torque acts on the system. A classic example: an ice skater spinning at 2 rev/s with arms extended halves their moment of inertia by pulling arms in, instantly doubling their spin rate to 4 rev/s.
What is moment of inertia and what are the values for common shapes?
Moment of inertia is the rotational analogue of mass, defined as the sum of each mass element times the square of its distance from the axis. About a central axis a thin hoop has MR squared, a solid disc or cylinder one half MR squared, a solid sphere two fifths MR squared, and a thin rod about its centre one twelfth ML squared. The axis chosen changes the value, so state it.
What is the difference between the angular momentum forms L = I omega and L = m v r?
Both give the same angular momentum. L = I omega applies to an extended rigid body rotating about a fixed axis, where I is the moment of inertia. L = m v r is the special case of a point mass on a circular path of radius r. They agree, because a point mass has moment of inertia m r squared and tangential speed omega times r. Use the point-mass form for orbiting bodies and the rigid-body form for flywheels.
How is angular momentum related to torque?
Torque is the time rate of change of angular momentum, the rotational counterpart of Newton's second law. A constant torque applied for a given time produces an angular impulse equal to torque times time, and that equals the change in L. This is why angular momentum stays fixed when the net external torque is zero, and why internal forces alone can never change it.
How do I convert between rad/s, revolutions per second and RPM?
One revolution is 2 x pi radians. Angular velocity in radians per second equals 2 x pi times revolutions per second, or RPM times 2 x pi divided by 60, which is about 0.10472 times RPM. A shaft at 3000 RPM turns at 50 revolutions per second, close to 314.2 rad/s. Always convert to rad/s before using L = I x omega, since the formula assumes SI units.
Why does a spinning gyroscope resist being tipped over?
Angular momentum is a vector along the spin axis, so tilting the wheel means changing the direction of L, which requires an external torque. When gravity applies a torque perpendicular to L, the axis does not fall; instead it sweeps sideways at a precession rate equal to the torque divided by L. Larger spin means more L and slower precession, which is why a fast-spinning wheel resists being twisted and why a gyrocompass holds its heading.
Should I size a flywheel by its angular momentum or by its rotational kinetic energy?
Size it by energy. Rotational kinetic energy can be written as KE = L squared divided by 2I, so for a fixed moment of inertia the stored energy grows with the square of L while L itself grows only linearly with omega. A rotor with I = 2 kg·m² at omega = 5 rad/s carries L = 10 kg·m²/s and KE = 25 J, since 100 divided by 4 is 25. Doubling omega doubles L but quadruples the stored energy, which is why flywheels are built light and fast rather than heavy and slow. Angular momentum matters instead when you care about gyroscopic reaction torques on the bearings and mounts.
Background & Theory
Angular Momentum Conservation in Practice
Angular momentum L = Iω is conserved whenever no external torque acts on a system — this is the rotational twin of linear momentum conservation, and it explains several everyday phenomena that look almost magical until you see the math.
| Example | What changes | Why |
|---|---|---|
| Figure skater pulling in arms | Spin rate roughly doubles | I drops as mass moves closer to the axis; L stays fixed so ω rises |
| Diver tucking mid-air | Faster somersault rotation | Same principle — smaller I, larger ω, same L |
| Collapsing star → pulsar | Rotation speeds up to hundreds of times per second | Stellar radius shrinks by orders of magnitude, I falls dramatically |
| Flywheel energy storage | Stores KE = ½Iω² for later release | High ω is favored since energy scales with ω² but L only scales with ω |
Note that rotational kinetic energy and angular momentum scale differently with speed (ω² versus ω), which is why the skater's kinetic energy actually increases when they pull their arms in — that extra energy comes from the muscular work done pulling the arms inward against the outward "centrifugal" tendency.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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