Angular Momentum and Conservation Laws

Angular momentum is the rotational analog of linear momentum, L = Iω for a rigid body (or L = mvr for a particle moving in a circle), and the law of conservation of angular momentum states that if the net external torque on a system is zero, its total angular momentum remains constant. In brief: the net torque equals the rate of change of angular momentum, τ = ΔL ÷ Δt. When the moment of inertia of a spinning system decreases, its angular velocity increases (a skater pulling in the arms), so that Iω stays constant. The unit of angular momentum is kg·m²/s. Planetary orbits obey Kepler’s second law because of the conservation of angular momentum.

Figure skaters, divers, planets, neutron stars and spinning tops all demonstrate conservation of angular momentum. This guide defines angular momentum, states the conservation law and its consequences, discusses gyroscopes, compares linear and angular momentum in a table, works through numerical examples and corrects common mistakes.

What is angular momentum?

For a rigid body rotating with angular velocity ω about a fixed axis, the angular momentum is L = Iω. For a point mass m moving with velocity v at a perpendicular distance r from the axis, L = mvr (more generally, L = r × p). Angular momentum is a vector along the axis of rotation, given by the right-hand rule. Newton’s second law for rotation can be written as τ_net = dL/dt: the net torque is the rate of change of angular momentum, just as force is the rate of change of linear momentum. The torque that changes the angular momentum can come from the outside (friction, gravity, a push).

What is conserved?

If the net external torque on a system is zero, then dL/dt = 0, and the total angular momentum is constant: L_initial = L_final, or I₁ω₁ = I₂ω₂ for a rotating system whose shape changes. A decrease of I must be accompanied by an increase of ω. The kinetic energy ½Iω² = L² ÷ 2I increases when I decreases at constant L: the extra energy comes from the work done by the skater’s muscles in pulling the arms in against the “centrifugal” effect. When two rotating objects are joined (a spinning disc dropped on another), the total angular momentum is conserved but kinetic energy is lost, similar to an inelastic collision. Conservation of angular momentum is a fundamental law, linked to the symmetry of space under rotations.

What are some examples?

• Figure skater: spinning slowly with the arms out, then pulling them in, increases the spin rate. Divers tuck to spin faster in the air and extend to slow before entering the water.
• Planetary orbits: the gravitational force on a planet acts along the line to the Sun, so there is no torque about the Sun, and L = mvr is constant: a planet moves faster when closer (perihelion). This is Kepler’s second law.
• Neutron stars and pulsars: a star with a radius of about 10⁶ km collapsing to a radius of 10 km spins up by a factor of 10¹⁰, as I ∝ R², producing rotation periods of milliseconds to seconds.
• Gyroscopes and spinning tops: a spinning body resists changes in the direction of its axis. A torque perpendicular to L (such as gravity on a tilted top) changes the direction of L rather than its magnitude, causing precession. Gyroscopes are used in navigation systems and for stabilizing ships, spacecraft and cameras. A rotating bicycle wheel, a football thrown with a spiral and a bullet with rifling all gain stability from angular momentum.
• Helicopters: the tail rotor counters the torque that would make the body spin in the opposite direction to the main rotor.

Linear and angular momentum compared

Worked example: skater, merry-go-round and planet

1. Skater. A skater with arms out has I₁ = 3.0 kg·m² and spins at ω₁ = 2.0 rad/s. Pulling in the arms gives I₂ = 1.0 kg·m². ω₂ = I₁ω₁ ÷ I₂ = 3.0 × 2.0 ÷ 1.0 = 6.0 rad/s (3 times faster). Kinetic energy before: ½ × 3.0 × 4.0 = 6.0 J; after: ½ × 1.0 × 36 = 18 J. The increase of 12 J is supplied by the skater’s muscles.

2. Merry-go-round. A disc with I = 500 kg·m² rotates at ω = 1.0 rad/s. A child of mass 30 kg jumps on at the rim, radius 2.0 m, with no initial rotation. The system’s I = 500 + 30 × 2.0² = 620 kg·m². L is conserved: ω’ = 500 × 1.0 ÷ 620 = 0.81 rad/s.
3. Planet. A comet at its closest approach (perihelion) to the Sun is 0.60 AU away and moves at 60 km/s. At the farthest point, 30 AU away, its speed v: from mv₁r₁ = mv₂r₂, v₂ = 60 × 0.60 ÷ 30 = 1.2 km/s.
4. Torque and angular momentum. A wheel with I = 0.50 kg·m² is spun from rest to 20 rad/s. L = 0.50 × 20 = 10 kg·m²/s. A constant torque of 2.0 N·m would take t = ΔL ÷ τ = 10 ÷ 2.0 = 5.0 s.
5. Neutron star. A star of radius 7.0 × 10⁵ km rotates once in 30 days and collapses to a radius of 10 km (treating it as a uniform sphere of the same mass). The period shrinks by (10 ÷ 7.0 × 10⁵)² = 2.0 × 10⁻¹⁰, giving T = 30 × 86,400 × 2.0 × 10⁻¹⁰ = 5.3 × 10⁻⁴ s (about half a millisecond; in reality, mass loss and other effects change the result).

How was the law established?

Johannes Kepler’s second law (1609) is a consequence of the conservation of angular momentum, as Newton showed in the Principia (1687). Leonhard Euler and Daniel Bernoulli stated the principle for rigid bodies in the 1740s to 1750s. Foucault demonstrated the rotation of the Earth with a gyroscope and a pendulum in 1851 to 1852. The discovery of pulsars by Jocelyn Bell Burnell and Antony Hewish in 1967 was explained as rapidly rotating neutron stars.

Which mistakes are common?

Mistake 1: Saying that kinetic energy is conserved when angular momentum is

When I changes, kinetic energy changes. Only L is conserved.

Mistake 2: Forgetting to include the added mass in I

When objects join the rotating system, add their mr² to the moment of inertia.

Mistake 3: Applying conservation when an external torque acts

Check that the net external torque is zero (for example, no friction at the axle).

Mistake 4: Treating angular momentum as a scalar

It is a vector. Its direction matters, for example, in the precession of a gyroscope.