Torque is the rotational effect of a force about a specified point or axis. It measures the tendency of a force to cause angular acceleration or rotation and is defined vectorially as tau = r cross F, where r is the position vector from the reference point to the point of application of the force and F is the force vector. The magnitude is tau = rF sin theta, where theta is the angle between r and F.
The SI unit is the newton metre, N m. Although the dimensions of torque are the same as those of energy, torque and energy are physically different quantities and should not be treated as interchangeable. Torque is associated with rotational action, while work and energy involve scalar energy transfer. The amount of torque depends not only on the force magnitude but also on where and how the force is applied relative to the chosen pivot or axis. A force applied farther from a pivot can produce greater torque than the same force applied closer to it.
The perpendicular lever arm provides an especially useful interpretation of torque magnitude. The torque can be written as tau = Fd, where d is the perpendicular distance from the pivot to the force's line of action. This means that a force produces maximum torque when it acts perpendicular to the radius or lever direction, because sin theta equals one. A force directed through the pivot produces zero torque because its line of action has zero perpendicular distance from the pivot.
This explains the operation of tools such as spanners and wrenches. Applying a force at the end of a long handle provides a larger lever arm than applying the same force close to the bolt, so the resulting torque is greater. The same principle explains why door handles are normally positioned away from hinges and why a long bicycle crank can produce greater torque at the axle for a given applied force.
Torque is closely connected to angular acceleration through rotational Newtonian mechanics. For a rigid body rotating about a fixed axis, the net torque is related to angular acceleration by Sigma tau = I alpha, where I is the moment of inertia about the axis and alpha is angular acceleration. Moment of inertia is the rotational analogue of mass and depends on how mass is distributed relative to the axis.
A given torque therefore produces different angular accelerations for objects with different moments of inertia. Concentrating mass farther from the axis increases moment of inertia and makes angular acceleration more difficult for a given torque. This relationship is analogous to F = ma in translational motion, but the rotational quantities are torque, moment of inertia, and angular acceleration. For more complicated systems, the rotational equations may involve tensors and coupled motions rather than a single scalar moment of inertia.
Torque plays a central role in levers and mechanical advantage. A lever balances when the clockwise and anticlockwise torques about a pivot are equal, provided other forces and torques are accounted for. This gives relations such as F1 d1 = F2 d2 for idealised two force lever arrangements. Mechanical advantage can therefore be produced by choosing different distances from the pivot.
A small input force applied over a long lever arm can balance or overcome a larger output force acting over a shorter lever arm. The trade off is that the point farther from the pivot generally moves through a greater distance for the same angular rotation. Ideal machines conserve work, so increased force advantage is accompanied by a corresponding movement disadvantage. Real machines have friction, deformation, and other losses, so their actual mechanical advantage and efficiency differ from ideal values.
Torque has a direction associated with the axis and is therefore properly treated as a vector or, more generally, as an axial vector in three dimensional mechanics. The right hand rule is commonly used to determine its direction. If the fingers of the right hand curl in the direction of rotation produced by the force, the thumb points in the torque direction according to the conventional vector representation.
In planar problems, torque is often assigned a positive sign for anticlockwise rotation and a negative sign for clockwise rotation, or the reverse convention if stated explicitly. The important requirement is consistency. In three dimensions, torques about different axes cannot simply be added as ordinary scalar quantities because their directions matter. The cross product formulation provides the correct vector relationship and captures both magnitude and orientation.
Torque and angular momentum are related but distinct. Angular momentum for a particle is L = r cross p, where p is linear momentum, while torque satisfies tau = dL divided by dt. This relationship generalises the rotational version of Newton's second law. If the net external torque on a system is zero, its total angular momentum is conserved. This principle explains phenomena such as a rotating skater changing angular speed by moving their arms inward or outward.
Pulling mass closer to the rotation axis decreases moment of inertia, and if external torque is negligible, angular velocity increases so that angular momentum remains approximately constant. Torque therefore describes how external interactions change angular momentum, whereas angular momentum describes the rotational state of a system.
Torque is essential in engineering systems involving motors, engines, turbines, shafts, gears, wheels, robotic joints, and rotating machinery. A motor produces torque through electromagnetic interactions, and the combination of torque and angular speed determines mechanical power according to P = tau omega when the torque and angular velocity are aligned appropriately. Gears can trade rotational speed for torque while approximately conserving mechanical power apart from losses. A gear reduction can increase output torque while reducing output angular speed.
Shafts must be designed to withstand twisting stresses, and excessive torque can cause torsional deformation or failure. In vehicles, wheel torque influences the force available at the tyre contact patch, while the actual acceleration also depends on gearing, power, traction, mass, aerodynamic resistance, rolling resistance, and other factors. Torque is therefore a key quantity for describing the mechanical capability of rotating systems but does not by itself determine performance.
Torque depends on the selected reference point or axis, whereas a force itself does not have this same dependence. The same force can have different torques about different points because the lever arm changes. For a rigid body in equilibrium, the net torque is zero about every point if the net force is also zero, although selecting a particular point can greatly simplify calculations.
If the net force is zero but the net torque is nonzero, the system can undergo pure rotational acceleration through a couple. Two equal and opposite parallel forces separated by a distance produce zero net force but nonzero torque. This configuration is called a couple and can rotate a body without producing translational acceleration of its centre of mass. The distinction between force balance and torque balance is therefore fundamental to analysing rigid body motion and equilibrium.
