Torque and angular acceleration (A-level only) — AQA A-Level Physics

This topic extends the concepts of forces and linear motion into rotational mechanics. You will learn how a force applied at a distance from a pivot creates a torque (or moment), and how this torque causes angular acceleration. The key equation τ = I α is the rotational analogue of Newton's second law (F = ma), where moment of inertia I replaces mass and angular acceleration α replaces linear acceleration. Understanding torque and angular acceleration is essential for analysing rotating systems such as wheels, gears, and planets.

In the wider context of A-level Physics, this topic builds on your knowledge of moments, circular motion, and Newton's laws. It is a core part of the AQA A-level specification and often appears in exam questions involving calculations of torque, moment of inertia, and angular acceleration, as well as in practical contexts like determining the moment of inertia of a flywheel. Mastery of this topic is crucial for further study in engineering and physics.

• →Torque (τ) is the turning effect of a force, defined as τ = F d sin θ, where d is the distance from the pivot to the point of application and θ is the angle between the force and the lever arm. The SI unit is the newton metre (N m).
• →Angular acceleration (α) is the rate of change of angular velocity, α = Δω / Δt, measured in rad s^-2. It is related to linear acceleration by a = r α for a point at distance r from the axis.
• →Moment of inertia (I) is the rotational analogue of mass, representing how mass is distributed relative to the axis of rotation. For a point mass, I = m r^2; for a rigid body, I = Σ m r^2. Standard formulas exist for common shapes.
• →Newton's second law for rotation: The net torque on a body is equal to the product of its moment of inertia and its angular acceleration: τ_net = I α. This is the rotational equivalent of F = ma.
• →Rotational kinetic energy is given by E_k = 1/2 I ω^2, and angular momentum L = I ω. These concepts are often used in conjunction with torque and angular acceleration.

• Torque is defined by T = Fr, where r is the perpendicular distance from the axis to the line of action of the force, not necessarily the length of the object.
• The rotational form of Newton's second law is T = Iα, where I is the moment of inertia about the chosen axis and α is angular acceleration in rad s⁻².
• For several forces, calculate the resultant torque by taking clockwise and anticlockwise contributions with opposite signs before applying T = Iα.
• Moment of inertia depends on mass and its distribution about the axis, so the same torque produces different angular accelerations for different bodies or axes.
• Angular acceleration links to linear acceleration at a point by a = rα, allowing rotational and linear dynamics to be combined.
• The equations apply to a rigid body rotating about a fixed axis, with torque and moment of inertia taken about that same axis.

• 💡Draw a diagram showing the axis, each force and its perpendicular distance before substituting into T = Fr.
• 💡State the direction convention for torques, then calculate the resultant torque before using T = Iα.
• 💡Check units: torque in N m, I in kg m² and α in rad s⁻², and convert degrees to radians where needed.
• 💡Always draw a clear diagram for torque problems, showing the pivot, the line of action of the force, and the perpendicular distance. This helps avoid errors with angles and distances.
• 💡When calculating moment of inertia, check the axis of rotation and use the correct formula for the shape. If the axis is not through the centre of mass, you may need to use the parallel axis theorem (though this is not required for AQA A-level, you should know the standard formulas).
• 💡In explanations, use precise terminology: 'torque' not 'force', 'moment of inertia' not 'mass', and 'angular acceleration' not 'acceleration'. This demonstrates understanding and can gain marks in written answers.

• Using the full length of a rod as r when the force is not perpendicular to it: the correction is to use the perpendicular distance from the axis to the line of action, or resolve the force.
• Adding torques arithmetically without direction: the correction is to assign opposite signs to clockwise and anticlockwise torques and find the resultant.
• Mixing up I and α with linear m and a: the correction is to use T = Iα for rotation and keep the axis consistent throughout the calculation.
• Students often think that torque and force are the same thing. Torque is the turning effect of a force and depends on both the force and the distance from the pivot. A large force applied close to the pivot may produce less torque than a smaller force applied further away.
• Students sometimes believe that moment of inertia is a fixed property of an object. In fact, it depends on the axis of rotation. For example, a rod has a different moment of inertia when rotated about its centre compared to when rotated about one end.
• Students may incorrectly use linear equations for rotational problems, such as using F = ma instead of τ = I α. It is essential to use the rotational analogues: torque instead of force, moment of inertia instead of mass, and angular acceleration instead of linear acceleration.

1. 1Start by reviewing the definitions of torque and moment of inertia. Learn the formula τ = F d sin θ and practice calculating torque for forces applied at various angles. Use diagrams to identify the perpendicular distance.
2. 2Learn the standard formulas for moment of inertia for common shapes: point mass (I = m r^2), solid sphere (I = 2/5 m r^2), hollow sphere (I = 2/3 m r^2), solid cylinder (I = 1/2 m r^2), hollow cylinder (I = m r^2), and rod about centre (I = 1/12 m L^2). Make flashcards to memorise them.
3. 3Apply Newton's second law for rotation (τ = I α) to solve problems. Start with simple cases where torque is constant, then move to problems involving multiple forces or torques. Practice calculating angular acceleration from given torque and moment of inertia.
4. 4Work through past paper questions on this topic. Focus on questions that combine torque with other concepts like rotational kinetic energy or angular momentum. Check your answers against mark schemes to understand the level of detail required.
5. 5Create a summary sheet linking torque, angular acceleration, moment of inertia, and rotational kinetic energy. Use it for quick revision before the exam.

• 📋Calculation of torque: Given a force applied at an angle to a lever, calculate the torque. Advice: Always resolve the force into perpendicular and parallel components, or use τ = F d sin θ. Ensure the angle is between the force and the lever arm.
• 📋Calculation of angular acceleration: Given a torque and the moment of inertia of a body, calculate the angular acceleration using τ = I α. Advice: Check that the moment of inertia is for the correct axis. Rearrange the equation carefully and include units.
• 📋Explanation of moment of inertia: Explain why a hollow cylinder has a larger moment of inertia than a solid cylinder of the same mass and radius. Advice: Refer to the distribution of mass relative to the axis; more mass is concentrated further from the axis in the hollow cylinder.
• 📋Practical context: Describe an experiment to determine the moment of inertia of a flywheel or a rotating system. Advice: Outline the measurements (e.g., time, distance, mass) and how they are used to calculate I, often using energy considerations or τ = I α.