Angular Momentum Calculator

Calculate angular momentum for a point mass, rigid body, or rotating system.

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Angular Momentum as Rotational Momentum

Angular momentum measures rotational motion relative to a chosen origin or axis, and its direction matters as much as its magnitude. For a particle, the vector definition is L=r×p. Its magnitude is L=mvr sinθ, where θ is the angle between position and velocity. The familiar L=mvr applies only when the motion is perpendicular to the radius.

For a rigid body spinning about a fixed principal axis, angular momentum can be written L=Iω. The moment of inertia I contains the mass distribution: moving mass farther from the axis increases I strongly because distances enter squared. That is why a skater can increase angular speed by bringing mass inward while approximately conserving angular momentum.

L=|r×p|=mvr sinθ,   L=Iω,   τext=dL/dt
SymbolMeaningWhy it appears / units
LAngular momentum magnitudekg·m2/s; rotational analogue of linear momentum.
rDistance from origin or axism; only the perpendicular lever-arm component contributes.
p=mvLinear momentumkg·m/s.
IMoment of inertiakg·m2; depends on geometry and rotation axis.
ωAngular velocityrad/s; rotation rate about the axis.
τextExternal torqueN·m; changes angular momentum.

If net external torque about the chosen origin is zero, total angular momentum is conserved. Internal forces can redistribute angular momentum among parts of a system, but they do not change the isolated system's total. For general three-dimensional rigid-body motion, L and ω need not point in the same direction, so the scalar L=Iω form has limits.

Worked Examples

Example 1: Figure skater: m=60kg, v=2m/s, r=0.8m
L = 60×2×0.8
Result: 96 kg·m²/s
Pulling arms in decreases r, increases ω
Example 2: Flywheel: I=2 kg·m², ω=100 rad/s
L = Iω = 2×100
Result: 200 kg·m²/s
Used in gyroscopes and energy storage
Example 3: Velocity not perpendicular to radius
m=2kg, r=0.40m, v=5m/s, θ=30° → L=mvr sinθ
Result: L=2.00kg·m2/s
Using mvr without sinθ would double the answer because only transverse momentum contributes.
Example 4: Uniform solid disk
M=4kg, R=0.30m, ω=20rad/s → I=(1/2)MR2=0.18kg·m2
Result: L=3.60kg·m2/s
The correct geometry-dependent moment of inertia must be found before applying L=Iω.

Common Mistakes

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Dropping the sinθ factor for particle motion

L=mvr assumes tangential motion. Radial motion has θ=0 and therefore contributes zero angular momentum about that origin.

⚠️
Using the wrong moment of inertia

I depends on shape and axis. A disk, ring, sphere, rod, and off-axis body with the same mass and size generally have different moments of inertia.

⚠️
Assuming angular momentum is always conserved

Conservation requires zero net external torque about the reference point. An applied external torque changes angular momentum according to τext=dL/dt.

Frequently Asked Questions

Why do skaters spin faster pulling in arms?
Conservation of angular momentum: L=Iω = constant. Decreasing I (pulling arms in, reducing r) must increase ω proportionally.
What is moment of inertia?
I depends on how mass is distributed relative to the rotation axis. Solid sphere: I=2/5 mr². Thin ring: I=mr². Hollow sphere: I=2/3 mr².
Is angular momentum a vector?
Yes. Its direction follows the right-hand rule for the cross product r×p. In many introductory fixed-axis problems only one component matters, so a signed scalar is used, but the underlying physical quantity is a vector.
Can an object moving in a straight line have angular momentum?
Yes, relative to an origin that is not on its line of motion. Angular momentum depends on the chosen origin. A particle moving along a line that passes directly through the origin has zero angular momentum about that origin.
What is the difference between torque and angular momentum?
Angular momentum describes the rotational state of motion. Net external torque describes how quickly that angular momentum changes. The relationship τext=dL/dt is analogous to force changing linear momentum.
Why does moving mass outward increase moment of inertia?
Moment of inertia contains terms proportional to mass times distance squared from the axis. Moving the same mass farther out therefore makes it harder to change the rotation rate. With angular momentum conserved, increasing I causes ω to decrease.

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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