Moment of a Force

Magnitude that measures the turning effect of a force about an axis. Theory, formulas, worked examples and interactive graphs.

Conceptual context

The moment of a force (also called torque) measures the ability of a force to produce rotation about an axis. Knowing the force $F$ is not enough: where it is applied and in which direction also matter. A large force may produce little rotation if its line of action passes close to the axis, while a smaller force may be very effective if it acts far away and almost perpendicularly.

This page studies one individual contribution, not the sum of several torques. The central quantity is $\tau$, which summarises force, geometry and sense of rotation. The perpendicular arm $d_{perp}$ is the effective distance between the axis and the line of action of the force. The radial distance $r$ and the angle $\theta$ allow that arm to be obtained when the force is not perpendicular.

🟢 Essential level

A door opens more easily when pushed far from the hinges than when pushed near them. It also opens more easily when pushed perpendicular to the door than when pushed towards the hinges. That everyday experience summarises the idea of torque: rotation depends on force, distance to the axis and direction.

The relevant arm is not always the direct distance to the point where the force is applied. If the force is oblique, only part of that distance serves to produce rotation. That is why the perpendicular arm is used: it measures the distance from the axis to the line of action.

The result indicates a tendency to rotate about the chosen axis. Its modulus measures intensity; its sign is assigned by a clockwise or anticlockwise convention. This page studies one isolated force to understand its rotational effectiveness before adding several contributions.

Before assigning a sign, state the rotation axis and observe the tendency to rotate that the force would produce by itself. A force whose line of action crosses the axis generates no torque, even if its modulus is large. This geometric test avoids confusing the distance to the point of application with the perpendicular arm.

🔵 Formal level

Basic formulas

If the perpendicular arm is known directly, the most direct definition of torque is:

definition with perpendicular arm

When the radial distance $r$ and the angle $\theta$ between force and radius are known, the effective arm is first identified by means of:

geometric relation of the lever arm

Substituting that relation, the torque can also be written as:

trigonometric definition

The three expressions describe the same physics. The first uses $d_{perp}$ directly. The second explains how that arm arises from the geometry. The third combines force, radial distance and angle in a single relation.

Real student questions

Why can a large force produce no rotation at all? Because if its line of action passes through the axis, the perpendicular arm is zero.

Are $r$ and $d_{perp}$ the same? Only when the force is perpendicular to the radius. In general, $d_{perp}$ is the effective part of $r$.

Why does the torque have a sign? Because the sign indicates whether the force would tend to rotate in the chosen positive or negative sense.

Is N m the same as energy? It has the same dimension, but the torque is interpreted as a tendency to rotate, not as work.

Final synthesis

The torque of a force quantifies the tendency of a force to produce rotation about an axis. It can be computed with $F$ and $d_{perp}$, or with $r$, $F$ and $\theta$. The quantity $\tau$ increases with force, arm and perpendicularity, and vanishes if the line of action passes through the axis. This page measures one individual contribution; summing several contributions belongs to the analysis of rotational equilibrium.