Torque

Torque is a quantity in classical mechanics that describes the turning effect of a force about a specified point or axis. Commonly denoted by $\boldsymbol{\tau}$, it depends on the force’s magnitude, direction, and point of application. Torque is central to the analysis of rotating bodies: an unbalanced torque changes angular momentum, whereas opposing torques can balance without producing rotation. Its SI unit is the newton metre, written $\mathrm{N\,m}$. (openstax.org)

Definition and geometry

For a force $\mathbf F$ applied at a position described by $\mathbf r$, measured from a chosen origin, torque is defined using the cross product:

Its magnitude is

where $\theta$ is the angle between the position and force vectors. The distance $d=r\sin\theta$, called the moment arm or lever arm, is the perpendicular distance from the origin to the force’s line of action. A force whose line of action passes through the origin produces zero torque about that origin. (openstax.org)

The torque vector is perpendicular to the plane containing $\mathbf r$ and $\mathbf F$, with its direction determined by the right-hand rule. For planar motion, torque can instead be represented by a signed scalar: counterclockwise turning is conventionally positive and clockwise turning negative. Torque about a particular axis is the component of the torque vector along that axis. (openstax.org)

The geometry explains why a door is easier to turn when pushed near its outer edge than near its hinges. Increasing the perpendicular distance increases torque without increasing force. Likewise, applying force perpendicular to a wrench handle maximizes torque for a given force and handle length; an oblique force has a smaller turning effect. (openstax.org)

Units and distinction from energy

In the International System of Units, one newton metre is the torque produced by a force of one newton acting with a perpendicular moment arm of one metre. In base units,

Torque therefore has the same physical dimensions as energy and work, but it is a different kind of quantity. Torque describes a turning effect, while work describes energy transferred through displacement. The unit name joule is consequently not used for torque, even though a joule also equals a newton metre algebraically. (nist.gov)

Rotational dynamics

For a particle, angular momentum about an origin is $\mathbf L=\mathbf r\times\mathbf p$, where $\mathbf p$ is its linear momentum. For a system of particles, the corresponding quantities are summed. About a fixed origin in an inertial reference frame, the rotational equation of motion is

Thus, net external torque equals the time derivative of total angular momentum. Zero external torque implies constant angular momentum; it does not imply that the body is stationary. (openstax.org)

For a rigid body rotating about a fixed axis with constant moment of inertia $I$, the axial equation becomes

where $\alpha$ is angular acceleration. This is the rotational counterpart of Newton’s second law. Moment of inertia depends both on mass and on how that mass is distributed relative to the axis: mass farther from the axis contributes more strongly. The same net torque therefore produces less angular acceleration when the moment of inertia is larger. (openstax.org)

The scalar equation should not be interpreted as a universal vector equation for arbitrary three-dimensional motion. The more general angular-momentum equation permits torque to change either the magnitude or the direction of angular momentum. In gyroscopes, a torque can chiefly redirect angular momentum, producing precession rather than simply increasing or decreasing the spin rate. (openstax.org)

Balance and static equilibrium

A body in static equilibrium is at rest and has both zero net external force and zero net external torque:

The two conditions are independent. Balanced forces alone do not guarantee rotational balance, and balanced torques alone do not guarantee translational balance. When net force is zero, changing the origin does not change the net torque, so an equilibrium calculation may use whichever origin simplifies the geometry. (openstax.org)

In a balance beam, for example, opposing torques are equal when each applied force multiplied by its perpendicular distance from the pivot has the same magnitude. Consequently, a larger load can be balanced by a smaller load placed farther from the pivot. Support forces must also satisfy the force-balance condition. Such calculations underpin the treatment of beams, ladders, and other supported structures. (openstax.org)

Work and power in rotation

When a body rotates about a fixed axis, torque transfers energy through angular displacement. The work done by an axial torque is the integral

For constant torque, this reduces to $W=\tau\Delta\theta$, with the angle expressed in radians. Net rotational work changes the body’s kinetic energy, whose fixed-axis rotational expression is $K=\tfrac12 I\omega^2$. (openstax.org)

Instantaneous power is

where $\omega$ is angular velocity and the torque is its signed axial component. Torque and power are therefore not interchangeable: a stationary shaft can sustain torque while transferring no rotational mechanical power. At a specified torque, increasing rotational speed increases the associated power. (openstax.org)