6 Angular Momentum

Defining angular momentum

Angular momentum describes rotational motion and is a vector quantity, like linear momentum. For a particle with position vector $\mathbf r$ measured from a chosen origin and linear momentum $\mathbf p=m\mathbf v$, it is defined as

The cross product means that $\mathbf L$ points perpendicular to the plane containing $\mathbf r$ and $\mathbf p$. Its direction follows the right-hand rule, and its value depends on the origin used to measure position. The SI unit is $\mathrm{kg\,m^2/s}$.

For a rigid body rotating about a fixed axis, the angular momentum component along that axis is

where $I$ is the moment of inertia about the axis and $\omega$ is angular velocity. In general three-dimensional rotation, angular momentum need not point in the same direction as angular velocity. The scalar relation applies to fixed-axis rotation, or when rotation is about a principal axis.

Takeaway: Angular momentum depends on both motion and the chosen reference point or axis.

Torque and changing motion

Torque describes how a force changes angular momentum about the same origin. For a force $\mathbf F$ applied at position $\mathbf r$,

The second relation says that net torque changes the magnitude, direction, or both, of angular momentum. For a system of particles, total angular momentum is the vector sum of the particles’ angular momenta, and its rate of change is determined by the net external torque on the system. Under the usual Newtonian assumptions, internal torques cancel for an isolated system.

To apply the relation, specify the system and the reference point or axis, then identify the external torques about it. A force may produce no torque about a chosen point if its line of action passes through that point.

Takeaway: Angular momentum changes only when there is net torque about the reference being used.

Conservation of angular momentum

Conservation of angular momentum applies when the net external torque on a system is zero, or negligible over the interval considered:

For rotation about a fixed axis, this becomes

Thus angular speed can change even though angular momentum remains constant: a change in moment of inertia must be accompanied by a corresponding change in angular velocity.

Example: a spinning skater. Suppose the skater’s moment of inertia decreases from $4.0\,\mathrm{kg\,m^2}$ to $2.0\,\mathrm{kg\,m^2}$ while initially spinning at $2.0\,\mathrm{rad/s}$. With negligible external torque,

Pulling the arms inward reduces the moment of inertia, so the skater spins faster. Rotational kinetic energy can increase because the skater’s muscles do work; conserving angular momentum does not mean rotational energy is conserved.

Takeaway: A changing mass distribution can change spin rate without changing total angular momentum, provided external torque is negligible.

Applications in rotating systems

Angular momentum conservation helps explain several kinds of rotational motion:

• Spinning objects: A wheel or gyroscope has angular momentum that tends to maintain its direction. When external torque acts, it changes angular momentum; the spin axis can consequently change direction in a motion called precession.

• Orbital motion: A central force points toward or away from a fixed center, so its torque about that center is zero. A particle’s angular momentum about the center is therefore conserved, even as its distance from the center and speed vary.

• Changing body shape: Divers and skaters can redistribute mass and change their spin rate. With negligible external torque, the product $I\omega$ remains constant.

For a conservation calculation, define the system and reference axis first. Then check whether the net external torque about that reference is zero or negligible. If it is, equate the initial and final angular momentum.

Takeaway: The same principle links changing spin rates, motion under central forces, and the response of gyroscopes to torque.