Definitions

In w:physics, torque (τ) is also called moment), and is a vector that measures the tendency of a force to rotate an object about some axis ^{[1]} (center). The magnitude of a torque is defined as force times the length of the w:lever arm ^{[2]} (radius). Just as a force is a push or a pull, a torque can be thought of as a twist.
 The force applied to a lever, multiplied by its distance from the lever's fulcrum, is its torque. ^{[3]}
 ${\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F}$
where r is the particle's w:position vector relative to the fulcrum, and F is the force acting on the particles.

Units

As with any concept defined by a formula, the units Torque (force times distance ) can be determined by the formula (e.g., newton meter inSI units)^{[4]} Even though the order of "newton" and "meter" are mathematically interchangeable, the BIPM (Bureau International des Poids et Mesures) specifies that the order should be N m not m N. N·m is also acceptable.^{[5]}
The joule, which is the SI unit for energy or work, is also defined as 1 N m, but this unit is not used for torque. While both torque and energy have the same units, one is a scalar and the other is a vector (technically (pseudo) vector)

Relation between torque and energy

 $E=\tau \theta \$
where
 E is the energy
 τ is torque
 θ is the angle moved, in radians.
Other nonSI units of torque include "poundforcefeet" or "footpoundsforce" or "ounceforceinches" or "meterkilogramsforce".

Moment arm

A very useful special case, often given as the definition of torque in fields other than physics, is as follows:
 $\tau =({\textrm {moment\ arm}})\cdot {\textrm {force}}$
The construction of the "moment arm" is shown in the figure below, along with the vectors r and F mentioned above. The problem with this definition is that it does not give the direction of the torque but only the magnitude, and hence it is difficult to use in threedimensional cases. If the force is perpendicular to the displacement vector r, the moment arm will be equal to the distance to the centre, and torque will be a maximum for the given force. The equation for the magnitude of a torque arising from a perpendicular force:
 $\tau =({\textrm {distance\ to\ center}})\cdot {\textrm {force}}$
For example, if a person places a force of 10 N on a spanner which is 0.5 m long, the torque will be 5 N m, assuming that the person pulls the spanner by applying force perpendicular to the spanner.

Force at an angle

If a force of magnitude F is at an angle θ from the displacement arm of length r (and within the plane perpendicular to the rotation axis), then from the definition of cross product, the magnitude of the torque arising is:
 $\tau =rF\sin \theta$

Static equilibrium

For an object to be in static equilibrium, not only must the sum of the forces be zero, but also the sum of the torques (moments) about any point. For a twodimensional situation with horizontal and vertical forces, the sum of the forces requirement is two equations: ΣH = 0 and ΣV = 0, and the torque a third equation: Στ = 0. That is, to solve statically determinate equilibrium problems in twodimensions, we use three equations.

Torque and power

If a force is allowed to act through a distance, it is doing mechanical work. Similarly, if torque is allowed to act through a rotational distance, it is doing work. Power is the work per unit time. However, time and rotational distance are related by the angular speed where each revolution results in the circumference of the circle being travelled by the force that is generating the torque. The power injected by the applied torque may be calculated as:
 ${\mbox{Power}}={\mbox{torque}}\cdot {\mbox{angular speed}}\,$

