Conservation of angular momentum is a foundational concept in rotational dynamics. It tells us that if no external torque acts on a system, its total angular momentum remains constant over time. This principle has broad applications in mechanics, astrophysics, and engineering.
This is the rotational analog of Newton’s second law. If the net external torque is zero, then the derivative is zero, and angular momentum is constant:
Just like conservation of linear momentum, this law implies that without an external influence (torque), the rotational motion of a system cannot spontaneously change. The moment of inertia and angular velocity may vary within a system, but their product remains constant if $I\omega = \text{constant}$.
A figure skater spinning with arms extended decreases her moment of inertia by pulling her arms inward. Since no external torque acts, her spin rate increases to conserve angular momentum:
This demonstrates how internal changes can affect rotational speed without violating conservation laws.
As a star collapses into a neutron star, its radius shrinks drastically, which reduces its moment of inertia. To conserve angular momentum, its angular velocity increases. This is why pulsars rotate so rapidly:
Even though its mass stays roughly the same, the reduction in radius makes $I$ much smaller, so $\omega$ becomes very large.
Setup: A person sits on a frictionless rotating platform (or swivel chair) while holding a spinning bicycle wheel by its axle. The wheel’s axis is initially horizontal, and the wheel is spinning rapidly.
Observation: When the person tries to flip the wheel so the axis is vertical, they experience resistance and the platform begins to rotate in the opposite direction.
Explanation: The spinning wheel has angular momentum $\vec{L}$ directed along its axis of rotation. When the person attempts to change the wheel’s orientation, they are trying to change the direction of $\vec{L}$. But since no external torque acts on the system (assuming negligible friction), the total angular momentum of the system must be conserved.
Therefore, to counter the change in direction of the wheel’s angular momentum, the platform and person begin to rotate in the opposite direction to maintain the same net angular momentum vector.
This example demonstrates that angular momentum is a vector and emphasizes the conservation of its direction and magnitude in isolated systems.
If no external torque acts on a system, its angular momentum remains conserved. This powerful law helps us predict motion in everything from rotating toys to stars and galaxies. When moment of inertia changes due to internal rearrangements, angular velocity changes inversely to keep $L = I\omega$ constant.