Torque & Lever

Introduction

A lever is one of the simplest and most ancient machines ever devised — a rigid beam balanced on a pivot, capable of multiplying force or motion depending on where loads are placed. The physical quantity that governs whether a lever tips or stays balanced is called torque: the rotational equivalent of force. Understanding torque explains why a long spanner loosens a stubborn bolt more easily than a short one, how a see-saw balances children of different weights, and how cranes lift enormous loads. Torque and the conditions for rotational equilibrium are foundational ideas in the study of rotation and rigid-body mechanics.

The Physics Explained

When a force is applied to an object that can rotate about a fixed point — called the pivot or fulcrum — the tendency of that force to cause rotation is called torque. Torque depends on two things: the magnitude of the applied force and the perpendicular distance from the pivot to the line of action of that force. This distance is called the lever arm or moment arm. A larger force or a longer lever arm both produce a greater torque. This is why pushing a door near its hinges requires far more effort than pushing near the handle — the lever arm is much shorter at the hinge end.

Torque is a signed quantity in two dimensions. By convention, a force that would cause anticlockwise rotation produces a positive torque, while one that would cause clockwise rotation produces a negative torque. For a lever carrying weights, the torque due to each weight is simply the weight (mass times gravitational acceleration) multiplied by its horizontal distance from the pivot, with a sign that reflects which way it would make the beam rotate.

A lever — or any rigid body — is in rotational equilibrium when the net torque about any chosen pivot point is zero. This is sometimes stated as the principle of moments: the sum of all clockwise torques equals the sum of all anticlockwise torques. There is also a translational condition for full static equilibrium: the net force on the object must also be zero, meaning the upward reaction force at the pivot must equal the total weight of all masses on the beam. Both conditions must hold simultaneously for the system to remain perfectly still.

The three classical classes of lever — first, second, and third — differ only in the relative positions of the fulcrum, the load, and the effort. A see-saw is a first-class lever with the fulcrum between load and effort. A wheelbarrow is a second-class lever with the load between the fulcrum and effort. Tweezers are a third-class lever with the effort between the fulcrum and the load. All three obey the same torque equations.

Key Equations

Torque τ = F · d

Torque due to a weight τ = m · g · d

Rotational equilibrium (principle of moments) Σ τ = 0 → m₁ · g · d₁ = m₂ · g · d₂

Simplified balance condition (g cancels) m₁ · d₁ = m₂ · d₂

Translational equilibrium (pivot reaction force) R = Σ m · g = (m₁ + m₂ + … + mₙ) · g

Net torque about the pivot (multiple masses) Σ τ = m₁ · g · d₁ − m₂ · g · d₂ + m₃ · g · d₃ − …

Key Variables

Symbol	Name	Unit	Meaning
τ	Torque	N·m	Rotational effect of a force about a pivot; positive anticlockwise
F	Force	N	The applied force that generates the torque
d	Lever arm	m	Perpendicular distance from the pivot to the line of action of the force
m	Mass	kg	Mass of an object placed on the lever
g	Gravitational acceleration	m/s²	Acceleration due to gravity; approximately 9.81 m/s² near Earth's surface
d₁	Distance of mass 1 from pivot	m	Lever arm length for the first mass
d₂	Distance of mass 2 from pivot	m	Lever arm length for the second mass
R	Pivot reaction force	N	Upward force exerted by the fulcrum to maintain translational equilibrium
Σ τ	Net torque	N·m	Algebraic sum of all torques; zero when the lever is in rotational equilibrium

Real World Examples

See-saw: Two children of different weights can balance a see-saw by adjusting their positions relative to the pivot. The heavier child sits closer to the centre so that the product of mass and distance is equal on both sides — a direct application of the principle of moments.
Crowbar: A crowbar is a first-class lever that uses a very long lever arm on the effort side to generate an enormous torque with relatively little force. Even a modest push can prise apart objects held together by thousands of newtons of contact force.
Beam balance (weighing scales): A traditional balance scale achieves equilibrium when the torques on each arm are equal. Because gravity cancels from both sides, it directly compares masses — not weights — making it accurate regardless of local gravitational variation.
Construction cranes: A tower crane balances a heavy load at one end of its jib with a concrete counterweight on the other. Engineers position the counterweight so that the torques on each side of the central pivot are equal, maintaining rotational equilibrium even as the lifted load changes.
Human forearm: The elbow acts as a pivot, the bicep muscle applies the effort very close to the joint, and the hand holds the load at the end of the forearm. This third-class lever arrangement sacrifices force multiplication for a large range of motion and speed at the hand.

How the Simulation Works

The simulation displays a horizontal beam balanced on a central pivot. You can drag masses onto any position along the beam on either side of the fulcrum. Each mass is assigned a value in kilograms using the controls provided, and you can reposition masses by clicking and dragging them left or right. The simulation continuously calculates the torque produced by each mass — mass times gravitational acceleration times its signed distance from the pivot — and sums them to find the net torque on the beam.

If the net torque is non-zero, the beam rotates in the direction of the dominant side, just as a real lever would. When the net torque reaches zero (or falls within a small tolerance), the beam settles into a horizontal, balanced position and a balance indicator lights up. A readout panel shows the individual torque contribution of each mass, the total clockwise torque, the total anticlockwise torque, and whether the system is currently in rotational equilibrium. Experimenting with different mass values and positions lets you discover the principle of moments directly: doubling a mass on one side can be compensated by halving its distance from the pivot, or by adding an equal torque on the other side.