A.4
Rigid Body Mechanics
Torque, rotational equilibrium, moment of inertia, Newton's second law for rotation, angular momentum and rotational kinetic energy. HL only.
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A.4 Rigid Body Mechanics
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A.4 Rigid Body Mechanics
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Key Concepts, Rigid Body Mechanics
Torque
Torque (τ) is the rotational equivalent of force. It measures how effectively a force causes rotation about an axis. The equation is τ = Fr sin θ, where F is the applied force in Newtons, r is the perpendicular distance from the axis of rotation to the line of action of the force (the moment arm), and θ is the angle between the force vector and the line joining the pivot to the point of application. Torque is measured in Newton-metres (N m). Maximum torque occurs when the force is perpendicular to r (sin 90° = 1). If the force is directed straight through the pivot, sin θ = 0 and no torque is produced.
Rotational Equilibrium
A body is in rotational equilibrium when the resultant torque acting on it is zero. This means all clockwise torques exactly balance all anticlockwise torques. This is the rotational equivalent of translational equilibrium (where the resultant force is zero). For complete static equilibrium, both conditions must be satisfied simultaneously: resultant force = 0 and resultant torque = 0. When analysing a problem, choose a convenient pivot point (often where an unknown force acts) to simplify your torque equation.
Angular Quantities: Displacement, Velocity and Acceleration
Angular displacement (θ) is the angle rotated, measured in radians. Angular velocity (ω) is the rate of change of angular displacement, measured in rad/s: ω = Δθ/Δt. Angular acceleration (α) is the rate of change of angular velocity, measured in rad/s²: α = Δω/Δt. These are the rotational equivalents of linear displacement, velocity and acceleration. An unbalanced (resultant) torque causes angular acceleration, just as an unbalanced force causes linear acceleration.
Rotational SUVAT Equations
When angular acceleration is constant, the rotational equations of motion are direct parallels of the linear SUVAT equations, with the substitutions s → θ, u → ω₀, v → ω, a → α: ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ, and θ = ½(ω₀ + ω)t. Use these exactly as you would linear SUVAT: identify your known and unknown rotational quantities, then select the appropriate equation.
Moment of Inertia
The moment of inertia (I) is the rotational equivalent of mass. It measures how difficult it is to change an object's rotational motion and depends on both the total mass and how that mass is distributed relative to the axis of rotation. The further the mass is from the axis, the larger I is, and the harder it is to angularly accelerate. For a point mass: I = mr². For extended bodies, the formula depends on the shape and axis (these are given in the IB data booklet where needed). Units of I are kg m².
Newton's Second Law for Rotation
Just as F = ma for linear motion, the rotational equivalent is τ = Iα, where τ is the resultant torque in N m, I is the moment of inertia in kg m², and α is the angular acceleration in rad/s². This equation is Newton's Second Law applied to rotation. A larger torque produces a larger angular acceleration. For a given torque, an object with a larger moment of inertia will accelerate less quickly rotationally.
Angular Momentum and Its Conservation
Angular momentum (L) is the rotational equivalent of linear momentum. For a rigid body: L = Iω, measured in kg m²/s. The law of conservation of angular momentum states that L remains constant (ΔL = 0) unless a resultant external torque acts on the system. This explains why a spinning ice skater speeds up when pulling their arms in (I decreases, so ω must increase to keep L constant). If a resultant torque does act, it produces an angular impulse equal to the change in angular momentum: τΔt = ΔL.
Rotational Kinetic Energy
A rotating body possesses kinetic energy due to its rotation. The rotational kinetic energy is given by Ek(rot) = ½Iω², which is the direct rotational equivalent of the linear ½mv². For an object that is both translating and rotating (such as a rolling wheel), the total kinetic energy is the sum of translational and rotational kinetic energy: Ek(total) = ½mv² + ½Iω². Both terms must be included in energy conservation problems involving rolling objects.
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Frequently Asked Questions, IB Physics Rigid Body Mechanics
What is Rigid Body Mechanics in IB Physics? ↓
Torque, rotational equilibrium, moment of inertia, Newton's second law for rotation, angular momentum and rotational kinetic energy. HL only. This topic is part of Theme A (Space, Time & Motion) in the current IB Physics syllabus.
Is Rigid Body Mechanics SL or HL in IB Physics? ↓
Rigid Body Mechanics is an HL-only topic. It is not assessed in the SL IB Physics exam.
What equations do I need for IB Physics Rigid Body Mechanics? ↓
The key equations for Rigid Body Mechanics are covered in the concept tutorial above. For a structured set of notes with all equations, conditions and worked examples, the GradePod Exam Pack includes a revision note template for every topic.
What are common exam mistakes in IB Physics Rigid Body Mechanics? ↓
Common mistakes are covered in detail in the exam technique video above. The GradePod Exam Pack also includes exam-style questions with mark schemes so you can see exactly how marks are awarded and where students typically drop them.
How do I revise Rigid Body Mechanics for the IB Physics exam? ↓
Follow the GradePod three-step method. First, watch the concept tutorial and tick off each learning objective on the checklist above as you go. Second, watch the exam technique video to see how IB-style questions are answered under exam conditions. Third, use the Exam Pack to practise independently with knowledge questions, exam questions and mark schemes. That's it. It works. I promise.