Angular Momentum Calculator
Calculate angular momentum L (kg·m²/s) — the rotational analogue of linear momentum. Choose from direct I·ω entry, shape-based moment-of-inertia formulas, or the orbital point-mass formula L = m·v·r.
Input method
kg·m²
rad/s
L = I · ω — conserved when no external torque acts
- 1
Angular momentum L = I × ω
2 × 5 = 10
How does this calculator work?
Angular momentum L = I·ω (kg·m²/s). For a point mass in orbit L = m·v·r. Moment of inertia I depends on shape: solid disk I = ½mr², solid sphere I = ²⁄₅mr², point mass I = mr². L is conserved when no external torque acts — if I decreases, ω increases proportionally.
Formula
How this is calculated
Angular momentum L is the rotational analogue of linear momentum and is conserved whenever no net external torque acts on a system. For a rigid body rotating about a fixed axis, L = I · ω, where I (kg·m²) is the moment of inertia and ω (rad/s) is the angular velocity. Moment of inertia depends on how mass is distributed: a solid disk has I = ½mr², a solid sphere I = ²⁄₅mr², and a point mass at radius r has I = mr².
For a single point mass moving in a circle of radius r with tangential speed v, L = m·v·r = m·r²·ω, consistent with I = mr². This form is widely used in orbital mechanics: the orbital angular momentum of a planet is m·v·r, which is conserved as r and v change along the ellipse (Kepler's second law — equal areas in equal times).
Conservation of angular momentum means that if I decreases (a spinning skater pulls in their arms), ω must increase to keep L constant. The calculator reports L, I, and ω regardless of input method. It does not model precession, gyroscopic effects, or non-fixed axes, which require vector treatment.
Frequently asked questions
When no external torque acts on a system, its total angular momentum L = I·ω stays constant. If a spinning body changes shape (reducing I), it spins faster to compensate. Examples include a figure skater pulling in their arms, a collapsing stellar cloud forming a neutron star, and a planet speeding up as it approaches the Sun.
kg·m²/s (kilogram-metre-squared per second), equivalent to J·s (joule-seconds) or N·m·s. These are the same units as Planck's constant ℏ, which is the quantum of angular momentum.
Torque τ (N·m) is the rate of change of angular momentum: τ = dL/dt. A non-zero net torque changes L; zero net torque means L is conserved. This is the rotational equivalent of force being the rate of change of linear momentum.
TG we-Calculate Editorial Team. (2026). Angular Momentum Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/angular-momentum-calculator
TG we-Calculate Editorial Team. "Angular Momentum Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/angular-momentum-calculator.
TG we-Calculate Editorial Team, "Angular Momentum Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/angular-momentum-calculator
@misc{wecalculate_angular_momentum_calculator, title = {Angular Momentum Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/angular-momentum-calculator}}, year = {2026}, note = {TG we-Calculate} }
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