IB Physics rigid body mechanics (HL) explained simply: Topic A.4 studies how extended objects rotate, remain balanced, conserve angular momentum, and roll without slipping. Most exam questions reduce to a small set of ideas: torque, angular kinematics, moment of inertia, rotational dynamics, angular momentum, angular impulse, and rotational kinetic energy.
A rigid body is an idealized object whose shape and dimensions do not change while forces act on it. In the current IB Physics course, A.4 is an HL-only topic within Theme A, Space, time and motion. The official IB Physics subject brief identifies rigid body mechanics as part of the HL syllabus, while the current course has been assessed since 2025.
The rotational equivalents you need to recognize
Rigid body mechanics becomes more manageable when you connect it to linear mechanics. Examiners frequently require students to move between these two descriptions.
| Linear quantity | Rotational equivalent | Main relationship |
|---|---|---|
| Displacement, s | Angular displacement, θ | s = rθ |
| Velocity, v | Angular velocity, ω | v = rω |
| Acceleration, a | Angular acceleration, α | Tangential a = rα |
| Mass, m | Moment of inertia, I | Depends on mass distribution |
| Force, F | Torque, τ | τ = Fr sin θ |
| Newton's second law | Rotational Newton's second law | Στ = Iα |
| Momentum, p = mv | Angular momentum, L = Iω | Conserved if external resultant torque is zero |
| Kinetic energy | Rotational kinetic energy | Eₖ,rot = ½Iω² |
| Impulse, FΔt | Angular impulse, τΔt | τΔt = ΔL |
These are analogies, not interchangeable quantities. For example, torque has units of N m, but it is not energy even though a joule is dimensionally equivalent to a newton metre.
Torque and rotational equilibrium
Torque measures the turning effect of a force about an axis:
τ = Fr sin θ
Here, r is the distance from the axis to the point where the force acts, and θ is the angle between r and F. The same equation can be written as torque = force × perpendicular distance from the axis to the force's line of action.
A reliable torque method
For a beam, wheel, lever, or hinged object:
- Mark the axis or pivot.
- Draw every force and its line of action.
- Find the perpendicular distance from the pivot to each line of action.
- Choose clockwise or anticlockwise as positive.
- Add torques algebraically rather than treating every torque as positive.
A body is in rotational equilibrium when Στ = 0. Complete static equilibrium also requires translational equilibrium, ΣF = 0, so zero resultant torque alone does not prove that an object is stationary.
A couple consists of two equal, opposite, parallel forces acting along different lines. Its resultant force is zero, but it produces a torque τ = Fd, where d is the perpendicular separation of the forces. Exam questions may ask students to explain why a couple rotates an object without accelerating its centre of mass.
Angular kinematics
Angular displacement θ is measured in radians, angular velocity ω in rad s⁻¹, and angular acceleration α in rad s⁻². For uniform angular acceleration, the familiar linear equations of motion have rotational equivalents:
| Linear form | Rotational form |
|---|---|
| v = u + at | ωf = ωi + αt |
| s = ut + ½at² | Δθ = ωi t + ½αt² |
| v² = u² + 2as | ωf² = ωi² + 2αΔθ |
| s = ½(u + v)t | Δθ = ½(ωi + ωf)t |
Use these only when angular acceleration is constant. A common error is substituting angular displacement in revolutions rather than radians, so convert using 1 revolution = 2π rad.
Graphs are also examinable. The gradient of an angular displacement-time graph is angular velocity, the gradient of an angular velocity-time graph is angular acceleration, and the area under an angular velocity-time graph is angular displacement.
Moment of inertia and rotational dynamics
The moment of inertia describes how strongly an object resists angular acceleration about a specified axis. Unlike ordinary mass, it depends on both the amount of mass and its distribution relative to that axis.
For point masses:
I = Σmr²
Because distance is squared, moving mass farther from the axis can increase I substantially. A rotating object therefore has no single moment of inertia unless the axis is specified.
For example, two 2.0 kg point masses each located 0.30 m from an axis have:
I = 2(2.0)(0.30²) = 0.36 kg m²
An unbalanced resultant torque causes angular acceleration according to:
Στ = Iα
Suppose the two-mass system experiences a resultant torque of 0.72 N m. Its angular acceleration is α = 0.72 ÷ 0.36 = 2.0 rad s⁻². In an exam solution, writing Στ rather than τ makes it clear that you have included all relevant turning effects.
Students can review the underlying ideas through the A.4 rigid body mechanics lessons and practise moment-of-inertia applications in the A.4.2 moment of inertia resources.
Angular momentum and angular impulse
For a rigid body rotating about a fixed axis:
L = Iω
Angular momentum remains constant when the resultant external torque is zero. If the body's mass distribution changes, conservation may be written as:
Iiωi = Ifωf
This explains why a rotating person speeds up after drawing their arms inward. Bringing mass closer to the axis decreases I, so ω must increase to keep L constant.
Do not assume rotational kinetic energy is also conserved. If a person changes shape, internal forces can do work, changing the kinetic energy even while angular momentum remains constant.
A resultant torque acting over time produces an angular impulse:
τΔt = ΔL = Δ(Iω)
For a variable torque, the area under a torque-time graph equals the change in angular momentum. Its units are N m s, equivalent to kg m² s⁻¹.
Rotational kinetic energy and rolling
The kinetic energy of a rotating rigid body is:
Eₖ,rot = ½Iω² = L²/(2I)
An object rolling without slipping has both translational and rotational kinetic energy:
Eₖ,total = ½Mv² + ½Iω²
The no-slip condition is v = Rω, where v is the speed of the centre of mass. It allows angular and linear quantities to be connected in the same calculation.
For an object rolling down a slope from vertical height h, conservation of energy gives:
Mgh = ½Mv² + ½Iω²
Substitute ω = v/R and the appropriate expression for I before solving. Objects with a larger moment of inertia relative to MR² place a greater fraction of their energy into rotation, so they generally reach a lower translational speed after descending the same height.
At the instant of pure rolling, the point touching the stationary ground is instantaneously at rest relative to the ground. This does not mean the entire wheel is stationary; its centre of mass continues to move at v.
How IB exam questions test the topic
The official Physics specimen papers show the structure used for the course first assessed in 2025. Rigid body mechanics may appear in multiple-choice, short-response, data-based, or extended problems, often combined with energy, forces, circular motion, or graph interpretation.
Typical instructions include:
- Calculate the resultant torque, requiring distances perpendicular to force lines.
- Determine the angular acceleration, usually through Στ = Iα.
- Show that angular momentum is conserved, requiring a statement about external torque as well as a calculation.
- Explain why angular speed changes, requiring reference to both I and conservation of L.
- Determine the final speed of a rolling object, requiring translational and rotational kinetic energy.
- Sketch or interpret a graph, testing gradients and areas rather than equation substitution alone.
In a calculation, show the governing principle before inserting values. A response such as “angular momentum is conserved because no resultant external torque acts” is stronger than writing Iiωi = Ifωf without stating why it applies.
Common mistakes that lose marks
- Using the full distance r when the force is not perpendicular, instead of r sin θ or the perpendicular lever arm.
- Omitting the torque caused by the object's weight through its centre of mass.
- Assuming Στ = 0 automatically means ΣF = 0.
- Treating moment of inertia as depending only on total mass.
- Applying angular-momentum conservation when an external resultant torque acts.
- Conserving rotational kinetic energy in a situation where work is done.
- Forgetting the translational term ½Mv² for a rolling object.
- Using v = Rω when the object is slipping.
- Mixing degrees, revolutions, and radians in angular kinematics.
- Giving final answers without units or without a consistent rotational sign convention.
An exam-focused revision method
Begin by learning the linear-rotational analogy table, then practise identifying the governing principle before calculating. The A.4 rigid body mechanics Questionbank is useful for targeted questions, while A.4 flashcards can reinforce definitions, conditions, and units.
After attempting each problem independently, compare your method with a worked solution. RevisionDojo's Physics worked video resources and question-by-question solutions show how diagrams, equations, substitutions, and explanations are converted into marks. Mixed practice through the broader IB Physics Questionbank then tests whether you can recognize rotational mechanics when the topic is not announced in advance.
Conclusion
IB Physics rigid body mechanics centres on a compact set of connected principles. Identify the axis, calculate signed torques, use the correct rotational analogue, and state the condition behind any conservation law.
The largest improvement usually comes from watching a correct method and then reproducing it on a fresh question. RevisionDojo's topic Questionbank, Jojo AI feedback, flashcards, and worked video solutions can help you diagnose errors and practise the exact steps required in exam-style problems.
Sources and referenced URLs
- Official IB Physics subject brief
- Official IB Physics specimen papers for first examinations in 2025
- RevisionDojo A.4 rigid body mechanics Questionbank
- RevisionDojo A.4 rigid body mechanics lessons
- RevisionDojo A.4 rigid body mechanics flashcards
- RevisionDojo A.4.2 moment of inertia resources
- RevisionDojo Physics worked video resources
- RevisionDojo IB Physics Questionbank