Physics · Classical Mechanics · Dynamics & Forces
Angular Momentum Calculator
Calculates the angular momentum of a rotating object using moment of inertia and angular velocity, or using linear momentum and perpendicular distance.
Calculator
Formula
L is angular momentum in kg·m²/s. In the rotational form, I is the moment of inertia (kg·m²) and ω is the angular velocity (rad/s). In the linear form, m is mass (kg), v is the linear speed (m/s), r is the distance from the pivot point (m), and θ is the angle between the velocity vector and the radius vector.
Source: Halliday, Resnick & Krane — Physics, 5th Edition, Chapter 12; NIST Classical Mechanics Reference Data.
How it works
Angular momentum (L) is the rotational analogue of linear momentum. Just as linear momentum p = mv describes how hard it is to stop a translating object, angular momentum describes how hard it is to stop a rotating one. It is a vector quantity whose direction is given by the right-hand rule along the axis of rotation. The conservation of angular momentum — the fact that L remains constant in the absence of external torques — is one of the most powerful principles in all of physics, explaining phenomena as diverse as the collapse of a spinning neutron star and a diver tucking into a somersault.
There are two equivalent formulations. The rotational form, L = Iω, applies to rigid bodies rotating about a fixed axis, where I is the moment of inertia (a measure of how mass is distributed relative to the rotation axis, in kg·m²) and ω is the angular velocity in rad/s. The linear form, L = mvr sin θ, is used when a point mass moves in a curved path: m is mass (kg), v is linear speed (m/s), r is the straight-line distance from the pivot or reference point (m), and θ is the angle between the velocity vector and the radius vector. When the motion is circular and v is perpendicular to r, sin θ = 1, simplifying the expression to L = mvr.
Practical applications span a wide range of fields. In mechanical engineering, angular momentum analysis governs flywheel energy storage, gyroscopic stabilisation in ships and aircraft, and the design of rotating machinery. In orbital mechanics, conservation of angular momentum explains why planets sweep equal areas in equal times (Kepler's second law) and why satellites speed up as they approach periapsis. In sports science, it informs coaching decisions in gymnastics, diving, and ice skating. In quantum mechanics, quantised angular momentum defines atomic orbitals and particle spin states.
Worked example
Example 1 — Rotational Method (spinning flywheel):
A steel flywheel has a moment of inertia of I = 2.5 kg·m² and spins at ω = 3.0 rad/s.
L = I × ω = 2.5 × 3.0 = 7.5 kg·m²/s
The rotational kinetic energy is KE = ½Iω² = 0.5 × 2.5 × 9.0 = 11.25 J.
Example 2 — Linear Method (orbiting satellite):
A satellite of mass m = 500 kg orbits at a speed of v = 7,800 m/s at an orbital radius of r = 6,800,000 m, with the velocity perpendicular to the radius (θ = 90°), so sin θ = 1.
L = m × v × r × sin θ = 500 × 7,800 × 6,800,000 × 1 = 2.652 × 10¹³ kg·m²/s
This enormous angular momentum remains nearly constant throughout the orbit (conservation of angular momentum), so as the satellite dips lower it speeds up automatically.
Example 3 — Non-perpendicular angle:
A ball of mass m = 2 kg moves at v = 5 m/s at a distance r = 3 m from a pivot, but its velocity is at θ = 60° to the radius.
L = 2 × 5 × 3 × sin(60°) = 30 × 0.8660 = 25.98 kg·m²/s
Limitations & notes
This calculator treats angular momentum as a scalar magnitude, suitable for planar problems with a fixed rotation axis. In three-dimensional problems, angular momentum is a full vector (L = r × p) and requires vector cross-product treatment. The rotational method assumes a rigid body with a constant, well-defined moment of inertia; for deformable or flexible bodies, I changes with time and the calculation becomes significantly more complex. The calculator does not account for external torques — if a net torque acts on the system, angular momentum is not conserved and changes at a rate dL/dt = τ. For relativistic situations (speeds approaching the speed of light), classical angular momentum must be replaced by its relativistic counterpart. Quantum mechanical systems have quantised angular momentum governed by L = ħ√(l(l+1)), which is outside the scope of this classical tool.
Frequently asked questions
What are the SI units of angular momentum?
Angular momentum is measured in kilogram-square-metres per second (kg·m²/s), which is equivalent to joule-seconds (J·s) or newton-metre-seconds (N·m·s). These units reflect that angular momentum equals the product of moment of inertia (kg·m²) and angular velocity (rad/s), where radians are dimensionless.
Why is angular momentum conserved?
Angular momentum is conserved whenever the net external torque acting on a system is zero. This is a direct consequence of Newton's second law for rotation: τ = dL/dt. If τ = 0, then L is constant. This principle explains why a figure skater spins faster when they pull in their arms — reducing their moment of inertia forces their angular velocity to increase to keep L constant.
What is the difference between orbital and spin angular momentum?
Orbital angular momentum arises from an object's motion through space around a central point (like a planet orbiting the Sun), calculated as L = mvr sin θ. Spin angular momentum arises from an object rotating about its own centre of mass, calculated as L = Iω. Both are physically real and additive; the total angular momentum of a system is their vector sum.
How do I find the moment of inertia for common shapes?
Moment of inertia depends on both the total mass and how that mass is distributed around the axis. For a solid cylinder or disk rotating about its central axis, I = ½mr². For a solid sphere, I = (2/5)mr². For a thin rod rotating about one end, I = (1/3)mL². For a point mass or thin ring at radius r, I = mr². Engineering handbooks and physics textbooks provide comprehensive tables for more complex geometries.
How does angular momentum relate to torque?
Torque (τ) is the rotational analogue of force and is related to angular momentum by the equation τ = dL/dt — torque equals the rate of change of angular momentum with respect to time. This is the rotational form of Newton's second law. Applied over a time interval, an impulse of torque (τ × Δt) produces a change in angular momentum, exactly analogous to how a linear impulse (F × Δt) changes linear momentum.
Last updated: 2025-01-15 · Formula verified against primary sources.