Torque in Physics Explained: Formula & Real Examples

You already apply torque correctly before you know what it is. You push a door near the edge, not near the hinge. You use a long wrench, not a short one, when a bolt is stuck. You know intuitively that the turning effect of a force depends on both the force magnitude and where it's applied. Torque formalises that intuition: τ = rF sinθ, where r is the distance from the pivot and θ is the angle between the force and the lever arm.

Torque is to rotation what force is to linear motion — the cause of angular acceleration. Every gear system, every engine crankshaft, every balanced beam, every joint in the human body involves torque. It's also the quantity that determines whether a structure is in rotational equilibrium, which is the foundation of statics and structural engineering.

The Torque Formula

where τ (tau) is torque (N·m), F is force (N), d is the distance from the axis of rotation to the point where the force is applied (m), and θ is the angle between the force vector and the line from pivot to point of application.

When the force is perpendicular to the moment arm (θ = 90°, sinθ = 1), torque is maximised:

When the force is parallel to the moment arm (θ = 0° or 180°, sinθ = 0), torque is zero — a force directed straight toward or away from the pivot produces no rotation.

The Moment Arm

The moment arm (also called the lever arm) is the perpendicular distance from the axis of rotation to the line of action of the force. It is not necessarily the distance from pivot to point of application — it is the shortest (perpendicular) distance from the pivot to the extended line along which the force acts.

For a force perpendicular to the lever: moment arm = distance from pivot to point of application.

For a force at angle θ: moment arm = d sinθ, so τ = F × (d sinθ) = Fd sinθ.

Direction of Torque: Clockwise and Anticlockwise

Torque is a vector — it has both magnitude and direction. By convention:

• Anticlockwise (counterclockwise) torques are positive.

• Clockwise torques are negative.

The direction is determined by the right-hand rule: curl the fingers of the right hand from the moment arm toward the force direction; the thumb points in the direction of the torque vector (perpendicular to the plane of rotation).

Rotational Equilibrium: The Principle of Moments

An object is in rotational equilibrium when the net torque about any axis is zero:

This is the Principle of Moments: for equilibrium, the sum of clockwise torques about any pivot equals the sum of anticlockwise torques. This is the fundamental principle behind levers, see-saws, balance scales, and structural engineering.

Worked Example: See-saw

A 40 kg child sits 1.5 m from the centre of a see-saw. How far must a 60 kg adult sit on the other side for the see-saw to balance?

The adult must sit 1.0 m from the centre — lighter people sit further from the pivot to balance heavier people sitting closer.

More Worked Examples

Example 2: Tightening a bolt

A mechanic applies 80 N perpendicular to a 0.25 m spanner. What torque acts on the bolt?

Example 3: Force at an angle

A 100 N force acts at 35° to a 0.4 m moment arm.

Newton's Second Law for Rotation

Just as F = ma connects force and linear acceleration, the rotational equivalent connects torque and angular acceleration α:

where I is the moment of inertia (kg·m²) — the rotational analogue of mass — and α is angular acceleration (rad/s²). A larger moment of inertia means the same torque produces less angular acceleration. This is why it is harder to spin a long, heavy flywheel than a small, light disc — even if both have the same mass, the flywheel's mass is concentrated further from the axis, giving it a greater moment of inertia.

Torque and Work in Rotation

Work done by a torque through angle θ (in radians):

Power delivered by a torque at angular velocity ω (rad/s):

This mirrors the linear relationships W = Fd and P = Fv exactly. For a car engine: engine torque × angular velocity = power output. High-revving engines (large ω) can produce high power at moderate torque; diesel engines produce high torque at low revs. Torque and power are related by P = τω — they are not independent quantities.

Real-World Applications of Torque

Door handles: positioned at the edge of the door (maximum moment arm) to minimise the force needed to open it. A handle near the hinge would require enormous force for the same torque.

Wheelie bars on dragsters: prevent the car from rotating (wheelie) by extending the effective wheelbase, increasing the anticlockwise torque of the rear downforce.

Torque wrenches: allow engineers to apply precisely specified torques to bolts, preventing both under-tightening (bolt works loose) and over-tightening (stripping threads or warping flanges).

Seesaws and levers: all governed by the principle of moments — the fundamental application of rotational equilibrium dating back to Archimedes, who reportedly said: "Give me a lever long enough and a fulcrum on which to place it, and I shall move the world."

The Torque Formula

Torque τ (tau) is the rotational equivalent of force — it causes angular acceleration. For a force F applied at distance r from a pivot at angle θ:

SI unit: N·m (not joules — torque and energy have the same units but are different quantities). Maximum torque when θ = 90° (force perpendicular to the lever arm). Zero torque when θ = 0° or 180° (force parallel to lever arm, passes through pivot).

Worked Examples

Example 1: Spanner on a bolt. A 25 N force applied perpendicular (θ = 90°) to a 0.30 m spanner. τ = 0.30 × 25 × sin 90° = 7.5 N·m.

Example 2: Force at an angle. Same spanner, force at 60° to lever arm. τ = 0.30 × 25 × sin 60° = 0.30 × 25 × 0.866 = 6.50 N·m.

Example 3: Equilibrium. A 4 m seesaw pivoted at centre. 60 kg person sits 1.5 m left of pivot. Where must a 45 kg person sit (right side) for balance?

Anticlockwise torque = clockwise torque: 60 × 9.8 × 1.5 = 45 × 9.8 × d → d = 60 × 1.5/45 = 2.0 m from pivot.

Torque and Angular Acceleration

Newton's second law for rotation: τ_net = Iα, where I is moment of inertia and α is angular acceleration. A net torque of 10 N·m on a flywheel (I = 5 kg·m²): α = τ/I = 10/5 = 2 rad/s². This is the rotational analogue of F = ma.

Conditions for Equilibrium

For a rigid body in static equilibrium: (1) ΣF = 0 (no net force — no linear acceleration); (2) Στ = 0 about any pivot point (no net torque — no angular acceleration). Both conditions must hold. Choose the pivot point strategically — placing it at an unknown force eliminates that force from the torque equation, simplifying the algebra.

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