Calculators
Angular Momentum Calculator
Calculate angular momentum from rotational inertia or particle motion.
Calculate scalar angular momentum for a rigid body using L = Iω or the magnitude for a point particle using L = mvr sin θ.
Angular momentum, L
10 kg·m²/s
I from L and ω: 2.5; ω from L and I: 4
The rigid-body form assumes rotation about the axis used to define I. The particle form reports magnitude relative to the chosen origin. Angular momentum is fundamentally a vector; signs and magnitudes here are simplified one-axis/scalar representations.
About This Tool
Angular momentum measures rotational motion relative to an axis or origin. This calculator covers two common introductory-mechanics models: a rigid body rotating about a fixed axis, where L = Iω, and a point particle moving relative to an origin, where the angular-momentum magnitude is L = mvr sin θ. Keeping the models separate helps make the required geometry and assumptions visible rather than treating angular momentum as a single plug-in formula.
How To Use It
- Choose Rigid body when you know the moment of inertia about the rotation axis and angular velocity.
- Choose Point particle when you know mass, speed, distance from the chosen origin, and the angle between the position vector and velocity.
- Use SI inputs shown beside each field to obtain angular momentum in kg·m²/s.
- For the particle model, enter 90° when velocity is perpendicular to the radius, 0° or 180° for purely radial motion, or the actual angle for other geometry.
- Interpret sign in the rigid one-axis model using your chosen positive rotation direction; remember that full angular momentum is a vector.
Examples
Rigid body
For I = 2.5 kg·m² and ω = 4 rad/s, L = Iω = 10 kg·m²/s.
Opposite rotation direction
If I = 3 kg·m² and the chosen sign convention gives ω = −2 rad/s, the scalar one-axis result is L = −6 kg·m²/s.
Perpendicular particle motion
For m = 2 kg, v = 3 m/s, r = 4 m, and θ = 90°, L = mvr = 24 kg·m²/s.
Non-perpendicular particle motion
With the same m, v, and r but θ = 30°, sin 30° = 0.5, so the angular-momentum magnitude is 12 kg·m²/s.
Useful Notes
Rigid-body formula: L = Iω
For rotation about a fixed principal axis in the scalar introductory model, angular momentum is moment of inertia I multiplied by angular velocity ω. Moment of inertia must be defined about the same axis. In general three-dimensional rigid-body motion, angular momentum and angular velocity need not point in the same direction, so the full vector/tensor relationship is more general than this calculator.
Point-particle formula: |L| = mvr sin θ
For a point particle, angular momentum about an origin is the cross product L = r × p, with p = mv. Its magnitude is mvr sin θ, where θ is the angle between the position vector r and velocity v. The perpendicular component of velocity is what contributes to angular momentum about that origin.
Why radial motion gives zero
When velocity points directly toward or away from the chosen origin, θ is 0° or 180° and sin θ is zero. The particle can still have linear momentum, but its angular momentum about that particular origin is zero at that instant.
Units and dimensions
In SI units, moment of inertia is kg·m² and angular velocity is rad/s, so angular momentum is kg·m²/s. The radian is dimensionless in SI dimensional analysis. The particle formula gives the same units: kg × m/s × m.
Conservation and scope
Angular momentum is conserved for a system when the net external torque about the chosen origin is zero. This calculator only evaluates simple instantaneous/scalar formulas; it does not determine whether a system is isolated, calculate external torque, model collisions, or solve general vector dynamics.
FAQ
What is angular momentum?
Angular momentum is a vector quantity describing rotational motion relative to an origin or axis. For simple fixed-axis rigid rotation it can be represented by L = Iω; for a point particle it is L = r × p.
Is angular momentum the same as moment of inertia?
No. Moment of inertia describes how mass is distributed relative to an axis. Angular momentum also depends on motion; in the simple fixed-axis model, L equals I multiplied by angular velocity.
Why does the particle formula include sin θ?
Angular momentum comes from a cross product. Only the component of motion perpendicular to the radius contributes, which introduces the sin θ factor.
Can angular momentum be negative?
A vector itself is not simply positive or negative, but in a one-axis scalar treatment a sign can represent direction relative to a chosen positive axis. Reversing rotation reverses that signed component.
When is angular momentum conserved?
Total angular momentum about an origin is conserved when the net external torque about that origin is zero. Whether that condition holds depends on the physical system and is not inferred by this calculator.
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