Angular Momentum Study Pack

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Last updated May 27, 2026

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Angular Momentum Study Guide

Master the rotational analog of linear momentum by working through L = Iω, net torque, and conservation of angular momentum — including how changing moment of inertia affects spin rate and how the right-hand rule determines direction.

Key Takeaways

  • •Angular momentum (L) is defined as the product of a rotating object's moment of inertia (I) and its angular velocity (ω), expressed as L = Iω, making it the rotational analog of linear momentum.
  • •The net external torque acting on a system equals the rate of change of that system's angular momentum, mirroring Newton's second law for rotation.
  • •When the net external torque on a system is zero, the total angular momentum of that system remains constant — a principle called the conservation of angular momentum.
  • •Because angular momentum is conserved, a spinning object that decreases its moment of inertia (pulls mass closer to the rotation axis) must increase its angular velocity proportionally, and vice versa.
  • •Angular momentum is a vector quantity: its direction is determined by the right-hand rule and lies along the axis of rotation, meaning both magnitude and direction are conserved when net torque is absent.
  • •For a single point mass moving in a straight or curved path, angular momentum relative to a reference point is calculated as L = r × p = mvr sinθ, where r is the position vector and p is the linear momentum.

Defining Angular Momentum

Angular momentum is the rotational counterpart to linear momentum and quantifies how much rotational motion an object or system possesses about a chosen axis.

Angular Momentum of a Rigid Rotating Body

  • •For an extended object rotating about a fixed axis, angular momentum L equals the product of the object's moment of inertia I and its angular velocity ω: L = Iω.
  • •The moment of inertia I depends on both the total mass of the object and how that mass is distributed relative to the rotation axis — objects with mass concentrated far from the axis have larger I values.
  • •Angular velocity ω is measured in radians per second and specifies how fast and in which direction (clockwise vs. counterclockwise) the object spins.

Angular Momentum of a Point Particle

  • •For a single particle with mass m moving with velocity v, angular momentum about a reference point is L = r × p, where r is the displacement vector from the reference point to the particle and p = mv is its linear momentum.
  • •The magnitude of this cross product is L = mvr sinθ, where θ is the angle between the position vector r and the velocity vector v.
  • •For circular motion, θ = 90°, so sinθ = 1, and the expression simplifies to L = mvr.

Angular Momentum as a Vector

  • •Because L is a vector, it carries directional information: the direction of L points along the rotation axis and is found using the right-hand rule — curl the fingers of the right hand in the direction of rotation and the extended thumb points in the direction of L.
  • •SI units of angular momentum are kilogram-meters squared per second (kg·m²/s).

Torque as the Driver of Changes in Angular Momentum

Just as a net force changes an object's linear momentum, a net external torque changes an object's angular momentum — this relationship is the rotational form of Newton's second law.

The Rotational Form of Newton's Second Law

  • •The net external torque τ_net acting on a system equals the time rate of change of its angular momentum: τ_net = ΔL/Δt (or in calculus terms, τ_net = dL/dt).
  • •This equation is more general than τ = Iα, because it applies even when the moment of inertia I is changing over time, not just when ω is changing at constant I.
  • •When τ and α both point in the same direction as the existing angular momentum vector, the object speeds up rotationally; when they oppose L, the object slows down.

Torque Defined in Terms of Cross Products

  • •Torque itself is a vector cross product: τ = r × F, where r is the position vector from the pivot to the point of force application and F is the applied force.
  • •The magnitude of torque is τ = rF sinφ, where φ is the angle between r and F; maximum torque occurs when the force is applied perpendicular to the moment arm (φ = 90°).
  • •Only the component of torque directed along the rotation axis actually changes the angular speed; torques perpendicular to the spin axis can tilt the axis itself, producing precession in gyroscopes.

Conservation of Angular Momentum

Conservation of angular momentum is one of the fundamental conservation laws in physics, arising whenever a system experiences no net external torque.

The Conservation Principle

  • •If τ_net = 0 for a system, then ΔL = 0, meaning the total angular momentum of the system stays constant in both magnitude and direction.
  • •This condition holds whether the system is an isolated spinning object, a pair of colliding rotating discs, or a planet orbiting the Sun — as long as no outside torque acts on the system.
  • •Internal torques between parts of a system (e.g., friction between two meshed gears in a closed box) cannot change the total angular momentum of the system; they only redistribute angular momentum among the parts.

The Moment of Inertia–Angular Velocity Trade-off

  • •Because L = Iω = constant when τ_net = 0, any decrease in I must be accompanied by a proportional increase in ω, and any increase in I must cause a proportional decrease in ω.
  • •A figure skater pulling their arms inward reduces their moment of inertia, causing them to spin faster; extending their arms increases I and slows the spin — total L remains the same throughout.
  • •Divers tuck their bodies to increase angular speed during a flip, then extend to reduce angular speed before entering the water, all without any external torque changing their angular momentum.

Astrophysical and Planetary Applications

  • •A protostellar cloud that begins to collapse under gravity conserves angular momentum; as its radius shrinks dramatically, its rotation rate increases dramatically, explaining why neutron stars (pulsars) can spin hundreds of times per second.
  • •Kepler's second law — that a planet sweeps out equal areas in equal times — is a direct consequence of conservation of angular momentum, since gravity from the Sun acts along the planet's position vector and produces zero torque about the Sun.

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