IB Physics HLRigid Body MechanicsHL onlyPaper 1 & 2~12 min read
Angular Momentum
A spinning skater who pulls their arms in speeds up dramatically. A collapsing star spins faster and faster as it shrinks. Both are the same piece of physics: angular momentum — the rotational version of momentum — is conserved. Once you know that a spinning system keeps its angular momentum unless a torque acts on it, a whole class of striking effects suddenly makes sense. This is one of the most elegant conservation laws in physics, and it’s the rotational partner of the linear momentum you already know.
📘 What you need to know
Angular momentum is the rotational equivalent of linear momentum (p = mv)
For a rotating body: L = Iω
L is measured in kg m2 rad s−1
For a point mass: L = Iω = (mr2)(v/r) = mvr
Conservation of angular momentum: total angular momentum is constant unless a resultant torque acts
If I decreases, ω increases (and vice versa), keeping Iω constant
Even objects moving in straight lines can have angular momentum, depending on the axis chosen
What is angular momentum?
Linear momentum measures “how much motion” an object has in a straight line: p = mv. Angular momentum does the same for rotation, using the rotational versions of mass and velocity — moment of inertia and angular velocity:
Angular momentumL = Iω
where L is the angular momentum in kg m2 rad s−1, I is the moment of inertia in kg m2, and ω is the angular velocity in rad s−1. Just swap each linear quantity for its rotational partner and linear momentum becomes angular momentum.
Angular momentum mirrors linear momentum: mass becomes moment of inertia, velocity becomes angular velocity, so p = mv becomes L = Iω.
Angular momentum of a point mass
For a single point mass m moving at speed v a distance r from the axis, its moment of inertia is mr2 and its angular velocity is v/r. Putting these into L = Iω gives a neat result:
Angular momentum of a point massL = Iω = (mr2) × (v/r)L = mvr
So even a mass travelling in a straight line has angular momentum about a point, as long as it isn’t heading straight for that point. This is why an object moving past a pivot can set it spinning on impact.
WE 1
A flywheel has a moment of inertia of 0.80 kg m2 and spins at an angular velocity of 12 rad s−1. Calculate its angular momentum.
Step 1 — use L = Iω
L = I × ω
Step 2 — substituteL = 0.80 × 12L = 9.6 kg m² rad s⁻¹Same structure as p = mv — just multiply the rotational “mass” by the rotational “velocity”.
WE 2
A 0.50 kg point mass moves at 2.0 m s−1 at right angles to a line joining it to an axis, at a distance of 0.30 m. Calculate its angular momentum about that axis.
Step 1 — for a point mass, L = mvr
L = m × v × r
Step 2 — substituteL = 0.50 × 2.0 × 0.30L = 0.30 kg m² s⁻¹A straight-line mover still has angular momentum about a point it isn’t heading toward.
Conservation of angular momentum
Just like linear momentum, angular momentum is conserved. The principle states:
Conservation of angular momentum
Total angular momentum stays constant unless a resultant torque acts
Because L = Iω must stay the same, if the moment of inertia decreases then the angular velocity must increase to compensate — and vice versa. This is the secret behind spinning skaters, divers, gymnasts, and even planets:
Conservation equationIiωi = Ifωf
A skater pulling their arms in cuts their moment of inertia, so their angular velocity rises to keep Iω constant — conservation of angular momentum in action.
WE 3
A spinning skater has a moment of inertia of 4.0 kg m2 and an angular velocity of 2.0 rad s−1. They pull their arms in, reducing their moment of inertia to 1.5 kg m2. Find their new angular velocity.
Step 1 — angular momentum is conserved: Iiωi = Ifωf
Iiωi = IfωfStep 2 — rearrange for ωf
ωf = Iiωi ÷ IfStep 3 — substituteωf = (4.0 × 2.0) ÷ 1.5 = 8.0 ÷ 1.5ωf = 5.3 rad s⁻¹ (2 s.f.)Cutting I to under half more than doubles ω — that’s why arms-in spins are so fast.
Real-world examples
Conservation of angular momentum shows up all over nature and sport. In each case, changing the moment of inertia changes the spin rate to keep Iω fixed:
Situation
What changes
Result
Spinning skater
Pulls arms in (I ↓)
Spins faster (ω ↑)
Diver / gymnast
Tucks into a ball (I ↓)
Rotates faster (ω ↑)
Planet in elliptical orbit
Moves closer to star (r ↓)
Orbits faster
Collapsing star
Radius shrinks (I ↓↓)
Spins up enormously
Tornado
Radius decreases (I ↓)
Winds speed up
The key move in any conservation-of-angular-momentum problem is always the same: write Iiωi = Ifωf, then work out how the moment of inertia changed. For a collision that sticks together, add the moments of inertia of everything that’s now rotating. For a shrinking object, put in the new radius. The angular momentum before always equals the angular momentum after — that’s your anchor equation.
Before Iᵢωᵢ
conserved =
After Iᶠωᶠ
so if I ↓
ω must ↑
🛠️ Solving an angular momentum problem
Find the angular momentum with L = Iω (or L = mvr for a point mass).
Is there a resultant torque? If not, angular momentum is conserved.
Write Iiωi = Ifωf for a conservation problem.
Work out the new moment of inertia — add up all rotating parts after a collision, or use the new radius.
Rearrange and solve for the unknown angular velocity or moment of inertia.
💡 Top tips
No torque, no change. Angular momentum is conserved only when there’s no resultant external torque.
Straight lines count. A mass moving in a straight line has angular momentum mvr about a point it’s not heading for.
Add moments after a collision. When things join and rotate together, sum their moments of inertia.
I and ω trade off. Halve the moment of inertia and the angular velocity doubles.
Quick recap: Angular momentum is the rotational form of momentum, L = Iω (or mvr for a point mass), measured in kg m2 rad s−1. It’s conserved when no resultant torque acts, so if the moment of inertia falls, the angular velocity rises to keep Iω constant — the physics behind spinning skaters and collapsing stars.
⚠ Common mistakes
Thinking angular momentum is conserved even when a resultant torque acts
Forgetting that a mass in a straight line can still carry angular momentum
Adding angular velocities instead of moments of inertia after a collision
Confusing angular momentum L = Iω with angular impulse or torque
Assuming kinetic energy is conserved too — in a “sticking” collision, it usually isn’t, but angular momentum still is
You now know that angular momentum is conserved when no torque acts — but what happens when a torque does act for a while? Just as a force acting over time gives linear impulse, a torque acting over time gives angular impulse, which changes the angular momentum. That’s exactly where we head next.
Want this to actually click before the exam?
Book a free meeting and let’s work through the tricky bits together.
