Principle

Conservation of angular momentum

L_{i} = L_{f} (τ_{net,ext} = 0)

Statement

With no net external torque, total angular momentum stays constant, even if the moment of inertia itself changes.

Symbols and units

L_{i}, L_{f}

initial, final angular momentum

kg·m²/s

When it applies

Zero net external torque about the axis in question.

Derivation

Since net torque equals the rate of change of angular momentum, zero net torque means angular momentum is unchanging:

τ_{net} = \frac{d L}{d t} = 0 \Rightarrow L_{i} = L_{f}

Worked examples

Figure skater pulling in their arms

I drops from 4.0 to 1.0 kg·m² while spinning at 2.0 rad/s:

ω_{f} = \frac{I _{i} ω _{i}}{I _{f}} = \frac{( 4.0 ) ( 2.0 )}{1.0} = 8.0 rad/s

Four times faster — no torque was applied, only the mass distribution changed.

Watch out for

If moment of inertia decreases (mass pulled toward the axis), angular velocity must increase to keep L = Iω constant — a spinning skater speeds up by pulling in their arms.

Connects to

Conservation of momentum