The big idea: Spin on an office chair and pull your arms in — you suddenly whirl faster. That stored spin is angular momentum (L), rotation's version of momentum.
Where straight-line momentum is p = mv, spinning momentum is L = Iω — moment of inertia × angular velocity. A fast, heavy, spread-out spin has lots of it.
- angular momentum (kg m² s⁻¹)
- moment of inertia (kg m²)
- angular velocity (rad s⁻¹)
A disc has a moment of inertia of 0.20 kg m² and spins at 15 rad s⁻¹. Find its angular momentum.
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Spin is conserved: If no external torque acts, angular momentum stays constant:
L = Iω is the same before and after.
So if a spinning body pulls its mass inward (I gets smaller), its spin ω gets faster — this is why an ice skater speeds up when they pull their arms in.
A skater spins at 2.0 rad s⁻¹ with a moment of inertia of 4.0 kg m². They pull their arms in, lowering it to 1.5 kg m². Find their new angular velocity.
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A spinning body also stores kinetic energy — flywheels use this to store energy. It mirrors ½mv², with I in place of m and ω in place of v.
- rotational kinetic energy (J)
- moment of inertia (kg m²)
- angular velocity (rad s⁻¹)
A rolling object has both: Something that rolls (a wheel, a ball) is moving and spinning, so its total KE = ½mv² + ½Iω² — translational plus rotational.
The disc above (I = 0.20 kg m²) spins at 15 rad s⁻¹. Find its rotational kinetic energy.
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How this is tested — angular momentum and rotational energy are HL only (A.4):
Paper 1A
- A quick L = Iω, ½Iω², or 'what happens to ω when I changes?'.
Paper 2
- A conservation problem (something lands on a spinning disc) or comparing the rotational and translational KE of a rolling object.
The classic trap: Using energy conservation for a sticking collision. When objects stick, angular momentum is conserved but kinetic energy is not — always use I₁ω₁ = I₂ω₂.
Three easy marks: (1) Spot the magic words 'no external torque' → use I₁ω₁ = I₂ω₂. (2) Energy is not conserved when objects stick (some is lost), but angular momentum is. (3) For rolling, remember the two KE terms.
A turntable (moment of inertia 0.50 kg m²) spins freely at 12 rad s⁻¹. A lump of clay is dropped onto it, adding 0.10 kg m² to the moment of inertia. Determine the new angular velocity.
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