Theme A · Space, Time and Motion · HL only · 7 hours
A.4 Rigid Body Mechanics
How does the distribution of mass within a body affect its rotational motion?
Higher Level only — no Standard Level content in this topic
Everything in A.1–A.3 was about point masses — objects treated as having no size, so rotation doesn't arise. Real bodies have extent, and when a force acts at a distance from a pivot, it produces rotation. Torque is the rotational equivalent of force; moment of inertia is the rotational equivalent of mass; angular momentum is the rotational equivalent of linear momentum. Every linear equation has a rotational analogue, and the elegance of that correspondence is one of the things that makes this topic satisfying to teach.
Angular momentum is conserved when no net torque acts — this is why a spinning ice skater speeds up when they pull their arms in (decreasing I forces ω to increase to keep L = Iω constant). The same principle governs the formation of solar systems, the spin-up of neutron stars, and the control of spacecraft attitude.
A solid disc of mass 2 kg and radius 0.4 m spins at 6 rad s−1. A student then drops a ring of mass 1 kg and the same radius onto the spinning disc. What is the new angular velocity? (Disc: I = ½MR²; ring: I = MR²)
Key equations
Torque: τ = Frsinθ
Moment of inertia (point masses): I = Σmr²
Newton's second law (rotation): τ = Iα
Angular momentum: L = Iω
Angular impulse: ΔL = τΔt = Δ(Iω)
Rotational kinetic energy: E_k = ½Iω² = L²/2I
What students must understand
Torque τ = Frsinθ — the turning effect of a force about an axis
Rotational equilibrium: resultant torque is zero
Angular displacement, velocity, and acceleration; the four rotational equations of motion analogous to A.1
Moment of inertia depends on the distribution of mass — I = Σmr² for point masses; formula given for extended bodies
Newton's second law for rotation: τ = Iα
Angular momentum L = Iω is conserved unless an external torque acts
Rolling without slipping: simultaneous rotational and translational motion
Linking questions
How does rotation apply to charged particles or satellites in orbit? → D.1, D.3
How does conservation of angular momentum relate to the Bohr radius? → E.1
How can a torque lead to simple harmonic motion? → C.1
How can rotation generate an electric current? → D.4 Induction
Angular motion variables | Moments, torque, and angular momentumRelating angular and regular motion variablesRotational kinematic formulas | Moments, torque, and angular momentumTorqueFinding torque for angled forcesAngular momentum | Moments, torque, and angular momentumRotational version of Newton's second lawAngular momentum of an extended objectConservation of angular momentum | Torque and angular momentumBall hits rod angular momentum exampleRotational kinetic energy of rigid systemsConservation of angular momentumAngular momentum of satellites
Flipping Physics
Torque IntroductionConservation of Angular Momentum Introduction and Demonstrations
The Organic Chemistry Tutor
Torque, Basic Introduction, Lever Arm, Moment of Force, Simple Machines & Mechanical AdvantageAngular Momentum - Basic Introduction, Torque, Inertia, Conservation of Angular MomentumTorque, Moment of Inertia, Rotational Kinetic Energy, Pulley, Incline, Angular Acceleration, Physics
Michel van Biezen
Physics 15 Torque Fundamentals (1 of 13) What is Torque?Physics 13.5 Angular Momentum (1 of 11) What is angular momentum? Basics
WNY Tutor — worked problems
A uniform plank of length 2.00 mThe puck in the figure has a mass of 0.120 kg - angular momentum
Physics with Professor Matt Anderson — full course modules
Module 12 | Torque and Rotation | Physics with Professor Matt Anderson
WNY Tutor — worked-problem sets
Rolling, Torque, and Angular MomentumRotational Equilibrium and Rotational Dynamics