Concept of Torque

When you push open a door, tighten a bolt with a wrench, or pedal a bicycle, you are producing a rotational effect. That rotational effect is called torque (symbol τ, the Greek letter tau). Torque is the rotational analogue of force: just as a net force causes a linear acceleration, a net torque causes an angular acceleration. Understanding torque is essential for analysing any system that rotates or tends to rotate, from car engines and turbines to the human musculoskeletal system. A crucial insight is that torque depends not only on how large a force you apply, but also on where and in which direction you apply it relative to the axis of rotation. This is why a long wrench makes it easier to loosen a stubborn nut than a short one — the greater distance from the pivot amplifies the rotational effect of the same force. This distance, measured perpendicularly from the axis of rotation to the line of action of the force, is called the moment arm (or lever arm). Torque is a vector quantity. Its magnitude is determined by the product of force, moment arm, and the sine of the angle between them, while its direction is given by the right-hand rule, pointing along the axis about which rotation would occur. Mastering torque unlocks the study of rotational equilibrium, angular momentum, and the full dynamics of rigid bodies — topics that form the backbone of classical mechanics beyond simple linear motion.

Key takeaways

From Force to Rotational Effect
Defining Torque as a Cross Product
The Moment Arm Interpretation
Direction: The Right-Hand Rule
Newton's Second Law for Rotation

From Force to Rotational Effect

Consider a rigid body free to rotate about a fixed axis. A force $\vec{F}$ applied at a point whose position vector from the axis is $\vec{r}$ does not necessarily cause rotation. Only the component of $\vec{F}$ perpendicular to $\vec{r}$ produces a rotational effect; the component along $\vec{r}$ (toward or away from the axis) merely pushes on the axis itself. This motivates defining a quantity that captures the 'turning power' of a force.

Defining Torque as a Cross Product

Torque is defined as the cross product $\vec{\tau} = \vec{r} \times \vec{F}$ . The magnitude is $|\tau| = rF\sin\theta$ , where $\theta$ is the angle between $\vec{r}$ and $\vec{F}$ . When $\theta = 90°$ , the force is entirely perpendicular to $\vec{r}$ , giving maximum torque $\tau = rF$ . When $\theta = 0°$ or $180°$ , the force acts along $\vec{r}$ , giving zero torque — no rotation is produced.

The Moment Arm Interpretation

An equivalent and often more intuitive way to compute torque is $\tau = F \cdot d$ , where $d = r\sin\theta$ is the perpendicular distance from the axis of rotation to the line of action of the force. This distance $d$ is called the moment arm or lever arm. Increasing either the applied force or the moment arm increases the torque proportionally, which is why levers and long wrenches are so effective.

Direction: The Right-Hand Rule

Because $\vec{\tau} = \vec{r} \times \vec{F}$ is a cross product, its direction is perpendicular to both $\vec{r}$ and $\vec{F}$ . To find it, point the fingers of your right hand along $\vec{r}$ , then curl them toward $\vec{F}$ ; your thumb points in the direction of $\vec{\tau}$ . For a door hinge axis pointing vertically upward, a force pushing the door open produces a torque pointing upward (counterclockwise rotation when viewed from above). A force closing the door produces a downward torque (clockwise).

Newton's Second Law for Rotation

Just as Newton's second law for linear motion states $\vec{F}_{\text{net}} = m\vec{a}$ , its rotational analogue is $\vec{\tau}_{\text{net}} = I\vec{\alpha}$ , where $I$ is the moment of inertia (rotational analogue of mass) and $\vec{\alpha}$ is the angular acceleration. This equation is the foundation of rotational dynamics: a net torque is required to change the angular velocity of a rigid body, and a larger moment of inertia means a larger torque is needed to produce the same angular acceleration.

The Key Equation

$$\vec{\tau} = \vec{r} \times \vec{F} \implies |\tau| = rF\sin\theta

- **\vec{\tau}** — Torque (N\cdot m): The rotational effect of a force about a chosen axis; a vector quantity. - **\vec{r}** — Position vector (m): Vector from the axis of rotation to the point where the force is applied. - **\vec{F}** — Applied force (N): The force whose rotational effect is being evaluated. - **\theta** — Angle between r and F (degrees or radians): The angle between the position vector and the force vector; determines what fraction of F is perpendicular to r. - **r\sin\theta** — Moment arm (lever arm) (m): The perpendicular distance from the axis of rotation to the line of action of the force. ## Worked Example **Problem.** A mechanic applies a force of 80 N at the end of a wrench that is 0.35 m long. The force is directed at 60° to the wrench handle. Calculate the torque exerted on the bolt.

\tau = rF\sin\theta = (0.35)(80)(\sin 60°) \approx 24.2 \text{ N·m}

**Answer.** The torque on the bolt is approximately 24.2 N\cdot m.

Figure

Diagram

Origins · 1 minArchimedes of Syracuse and how this was discovered

Archimedes of Syracuse

-287–-212 · Greek

Archimedes was the first to formulate the law of the lever in a rigorous mathematical way in his treatise 'On the Equilibrium of Planes'. He showed that a lever is in balance when the products of force and distance are equal on both sides — the earliest statement of what we now call torque balance. His principles of levers and centres of gravity laid the groundwork for all of statics and, centuries later, the formal concept of torque. Archimedes reportedly boasted 'Give me a place to stand and a long enough lever, and I shall move the Earth' — a vivid illustration of how increasing the moment arm can amplify torque to any desired magnitude.

Moment of Inertia — Moment of inertia is the rotational analogue of mass; it appears in the rotational Newton's second law τ = Iα alongside torque.
Angular Acceleration — A net torque produces angular acceleration according to τ_net = Iα, directly linking torque to changes in rotational motion.
Rotational Equilibrium — A rigid body is in rotational equilibrium when the net torque about any axis is zero — the direct application of torque concepts to statics.
Angular Momentum — Torque is the rate of change of angular momentum: τ = dL/dt, making it central to understanding conservation of angular momentum.
Newton's Second Law (Linear) — Torque and angular acceleration mirror the linear relationship F = ma, so understanding the linear law deepens comprehension of its rotational analogue.