Key Idea: Watch an ice skater pull in their arms and whirl faster, or a spanner turn a stubborn bolt — that is rigid body mechanics, the rotational twin of straight-line motion. Every idea you know for motion in a line has a spinning version — swap distance for angle, mass for moment of inertia, force for torque, momentum for angular momentum.
Turn it
- Torque τ = Fr sin θ about the axis
- Balanced: Στ = 0 and ΣF = 0
- τ = Iα; angular suvat with θ, ω, α
Resist it
- Moment of inertia I = Σmr²
- Mass far from the axis: bigger I
- Shape formulas (½MR², MR²) are given
Keep it
- L = Iω, conserved with no external torque
- Angular impulse τΔt = ΔL
- Eₖ = ½Iω² = L²/2I; rolling shares mgh
Learn the dictionary once and the whole topic falls into place. It is HL only and appears on Paper 1A (quick one-liners) and Paper 2 (a multi-step rotational problem).
The rotation–translation dictionary
| Straight-line motion | Rotational motion |
|---|---|
| displacement s | angle turned θ (rad) |
| velocity v | angular velocity ω |
| acceleration a | angular acceleration α |
| mass m | moment of inertia I = Σmr² |
| force F | torque τ = Fr sinθ |
| F = ma | τ = Iα |
| momentum p = mv | angular momentum L = Iω |
| kinetic energy ½mv² | rotational KE ½Iω² |
| impulse FΔt = Δp | angular impulse τΔt = ΔL |
| speed of a point v | v = ωr at distance r from the axis |
The formulas you're given
- torque (N m) — the turning effect of a force
- moment of inertia (kg m²) — rotation's mass
- angular acceleration (rad s⁻²)
- angular momentum (kg m² s⁻¹) — conserved if no external torque
- rotational kinetic energy (J)
- angular velocity (rad s⁻¹)
1. A body is balanced (rotational equilibrium) when the clockwise torques equal the anticlockwise torques about any point: Στ = 0. 2. Spin is conserved. With no external torque, I₁ω₁ = I₂ω₂ — pull mass inward (smaller I) and the spin speeds up (the ice-skater). 3. Rolling shares the energy. A body rolling without slipping has ½mv² + ½Iω² with v = ωr, so a bigger I means a slower finish down a slope.
IB-style worked examples (one per micro)
A uniform beam of length 3.0 m and weight 200 N is hinged to a wall and held horizontal by a vertical cable at its far end. Determine the tension in the cable.
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Next:
A solid disc (I = ½MR²) has mass 4.0 kg and radius 0.30 m. A net torque of 1.8 N m acts on it. Determine its angular acceleration.
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Next:
A turntable (I = 0.50 kg m²) spins freely at 12 rad s⁻¹. A lump of clay dropped on it adds 0.10 kg m² to the moment of inertia. Determine the new angular velocity.
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Important: 1. Work in radians, not degrees — and convert revolutions with × 2π. 2. In I = Σmr² the distance is squared — never forget to square r. 3. When two objects stick (clay on a disc), angular momentum is conserved but kinetic energy is NOT — use I₁ω₁ = I₂ω₂, never energy conservation.
Quick self-check
Tap each card to reveal the answer.
What is the rotational version of mass? Moment of inertia, I = Σmr² — it measures how hard a body is to spin up, and depends on how far the mass sits from the axis.
When is a rigid body balanced? In rotational equilibrium: the total torque about any point is zero (Στ = 0), clockwise = anticlockwise.
A skater pulls their arms in. What happens to their spin? It speeds up. I decreases, so ω increases to keep L = Iω constant (no external torque).
Torque of a 12 N force 0.40 m from the pivot, at 90°? 4.8 N m — τ = Fr sinθ = 12 × 0.40 × sin90°.
A hoop and a disc have the same M and R — which has more I? The hoop (I = MR²); the disc is ½MR², because its mass is spread inward.
Double a flywheel's angular velocity — what happens to its rotational KE? It quadruples — Eₖ = ½Iω², and ω is squared.
Exam tips
Exam tips
- Build the rotation–translation dictionary in your head; then every rotational question is one you already know.
- Take torques about an unknown force's line so it drops out of the equation.
- Spot 'no external torque' → conserve angular momentum (I₁ω₁ = I₂ω₂), not energy.
- The exam gives you the shape's moment-of-inertia formula — recognise it and substitute.
- Carry units throughout: N m for torque, kg m² for I, kg m² s⁻¹ for L.