Torque in Physics: Formula τ = Fd and Examples

When you push a door open, you push near the edge — not near the hinge. Instinctively you know that the same force applied further from the pivot produces a greater turning effect. That turning effect is torque. Torque is to rotation what force is to linear motion: it is the quantity that causes angular acceleration, just as force causes linear acceleration. Understanding torque is essential for analysing everything from the balance of see-saws to the design of gears, crankshafts, and robotic arms.

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."

Frequently Asked Questions