Law of Conservation of Angular Momentum
Simple Explanation
When no external torque acts on a rotating system, its total angular momentum stays exactly constant over time β if the system's moment of inertia changes (by redistributing its mass), its angular velocity must change to compensate, keeping L = IΟ constant.
Why Do We Need It?
This law explains a wide range of rotational phenomena, from figure skating spins to planetary orbits, and is one of the most fundamental conservation laws in physics, alongside conservation of energy and linear momentum.
Formula
Conservation of Angular Momentum
IβΟβ = IβΟβ
When no external torque acts on a system, its total angular momentum stays constant β so if its moment of inertia changes, its angular velocity must change to compensate.
- Iβ, Οβ
- β Moment of inertia and angular velocity at an initial time
- Iβ, Οβ
- β Moment of inertia and angular velocity at a later time
When to use it: Use whenever a rotating system changes shape (redistributing its mass) with no external torque acting on it, such as a spinning figure skater pulling in their arms.
Worked Example
A figure skater's spin
A skater spinning at 2 rad/s with arms out (Iβ = 4 kgΒ·mΒ²) pulls their arms in, reducing their moment of inertia to 1 kgΒ·mΒ². Find their new angular velocity.
Why Does This Work?
With no external torque, there is nothing to change the system's total angular momentum β so if I decreases (mass pulled closer to the axis), Ο must increase proportionally to keep the product IΟ constant, and vice versa.
Real-Life Example
A collapsing star spinning faster
When a massive star collapses into a much smaller, denser neutron star, its rotation speeds up dramatically.
As the star's radius shrinks dramatically, its moment of inertia decreases sharply β with no significant external torque during the collapse, conservation of angular momentum forces its angular velocity to increase enormously, producing the extremely fast-spinning neutron stars (pulsars) observed in space.
Practice
A rotating platform with Iβ = 6 kgΒ·mΒ² spins at Οβ = 3 rad/s. Its moment of inertia is increased to Iβ = 9 kgΒ·mΒ² (no external torque). Find the new angular velocity.
HardCommon mistake
Applying conservation of angular momentum when an external torque IS present β this law only holds when the net external torque on the system is zero; friction or an external push/twist would violate the conditions needed for L to stay constant.
Quick Review
- With zero external torque, total angular momentum (L = IΟ) stays constant.
- IβΟβ = IβΟβ lets you find a new angular velocity after a moment-of-inertia change.
- Reducing I (pulling mass inward) increases Ο, and vice versa.