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. 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 |
📐 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).
✏️ 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.
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.
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.
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
- 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.