About

Angular momentum L is the rotational analogue of linear momentum. For a point particle, L = m × v × r × sin(θ). For a rigid body rotating about a fixed axis, L = I × ω, where I is the moment of inertia. Miscalculating I by choosing the wrong geometric model leads to engineering errors in flywheel design, satellite attitude control, and gyroscopic stability analysis. This calculator implements exact formulas for 12 standard rigid-body shapes per classical mechanics references and converts results to SI units (kg⋅m²/s), CGS units (g⋅cm²/s), and reduced Planck constant ℏ for quantum-scale work.

Note: all formulas assume uniform mass distribution and rigid-body constraints. Real-world objects with non-uniform density require numerical integration of the inertia tensor. The angle θ in orbital mode is measured between the position vector r and the velocity vector v. At θ = 0° the particle moves radially and L vanishes. Pro tip: for compound shapes, compute each component's I separately and sum via the parallel axis theorem.

Formulas

Two primary computation modes are supported: orbital angular momentum for point particles and spin angular momentum for rigid bodies rotating about a fixed axis.

L_orbital = m ⋅ v ⋅ r ⋅ sin(θ)

Where m is mass (kg), v is linear velocity (m/s), r is the distance from the rotation axis (m), and θ is the angle between r and v.

L_spin = I ⋅ ω

Where I is the moment of inertia (kg⋅m²) determined by shape geometry and ω is the angular velocity (rad/s). Angular velocity relates to rotational speed n (RPM) by:

ω = 2πn60

The reduced Planck constant for quantum-scale conversion: ℏ = 1.054571817 × 10⁻³⁴ J⋅s. The CGS conversion factor: 1 kg⋅m²/s = 10⁷ g⋅cm²/s.

Reference Data

Shape	Axis	Moment of Inertia I	Notes
Point mass	Distance r	mr²	Orbital only
Solid sphere	Through center	25 mr²	Uniform density
Hollow sphere (thin shell)	Through center	23 mr²	Shell thickness → 0
Solid cylinder / disc	Central axis	12 mr²	Flywheels, wheels
Hollow cylinder (thin wall)	Central axis	mr²	Pipe, hoop
Thick-walled cylinder	Central axis	12 m(r₁² + r₂²)	r₁ inner, r₂ outer
Thin rod	Center, perpendicular	112 mL²	L = rod length
Thin rod	End, perpendicular	13 mL²	Pendulum approximation
Rectangular plate	Center, perpendicular	112 m(a² + b²)	a, b = side lengths
Solid cone	Central axis	310 mr²	Base radius r
Ellipsoid	Through center, along c	15 m(a² + b²)	Semi-axes a, b perpendicular to axis
Torus (ring)	Central axis	m(R² + 34r²)	R = major, r = minor radius
Annular disc	Central axis	12 m(r₁² + r₂²)	Washer, brake disc

Frequently Asked Questions

At θ = 0° or 180°, sin(θ) = 0, which makes the orbital angular momentum L = 0. Physically this means the particle moves purely radially toward or away from the axis, producing no rotational effect. Maximum L occurs at θ = 90°.

The parallel axis theorem states I = I_cm + md², where d is the distance between the center-of-mass axis and the parallel axis. This calculator computes I_cm only. If your rotation axis is offset, add the md² term manually to the displayed I value before computing L.

A thick-walled cylinder has mass distributed between inner radius r₁ and outer radius r₂. Its moment of inertia is 12m(r₁² + r₂²). Setting r₁ = 0 reduces it to a solid cylinder. Setting r₁ ≈ r₂ gives a thin-walled hoop.

This calculator computes classical angular momentum. Quantum spin is an intrinsic property with eigenvalues S = ħ√s(s + 1), which does not derive from mass or radius. However, the ħ unit output lets you compare classical results against quantum scales.

The calculator handles this automatically. Enter angular velocity in either rad/s or RPM and select the corresponding unit. Internally, RPM is converted via ω = 2π ⋅ n60. For example, 3000 RPM = 314.16 rad/s.

Yes. All shapes in this calculator assume uniform (homogeneous) density. For objects with variable density ρ(r), you must integrate I = ∫ r² dm numerically. The error from using uniform formulas on non-uniform bodies can exceed 30% for strongly graded materials.