∑τ = dL/dt

where L is angular momentum. Using the defintion of angular momentum (L = r×p, where r is position and p is the momentum), torque is often written as:

τ = r×F

but this only holds if the product of the sine of the angle between p and r, and the length of r is constant. This happens fairly often whenever a solid body is rotating about an axis, and when forces are always directed along a radius from the origin (like the force due to gravity between the sun and the earth).

Another often used formulation for angular momentum yields yet another equation for torque. Angular momentum is often defined as Iω (where I is a tensor called the Moment of Inertia, and ω is a vector called the angular frequency defined such that ω×r = v), and torque then becomes:

τ = Iα = Idω/dt

This definition only works when I is constant (i.e. a free rotating solid body, or a solid body rotating at the end of a rod), and when this definition for L holds (it holds during fixed axis rotations of rigid bodies).

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measured either in foot-pounds (English) or Newton-meters (metric, SI_units). Its dimensional analysis units are force * distance as distinct from energy with units distance * force.

The measurement of torque is important in automotive engineering, being concerned with the tranmission of power from the drive train to the wheels of a vehicle. It is also used where the tightness of bolts is crucial (see Torque Wrench).

Torque is also the easiest way to explain Mechanical advantage in just about every Simple machine except the Pulley

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