When twisting things, the rotational force we apply about the center is called a torque (or moment, or moment of force). Informally, we think of the torque as the tangential applied force times the moment arm (length of the lever arm)
as depicted in Fig.B.7. The moment arm is the distance from the applied force to the point being twisted. For example, in the case of a wrench turning a bolt, is the force applied at the end of the wrench by one's hand, orthogonal to the wrench, while the moment arm is the length of the wrench. Doubling the length of the wrench doubles the torque. This is an example of leverage. When is increased, a given twisting angle is spread out over a larger arc length , thereby reducing the tangential force required to assert a given torque .
For more general applied forces , we may compute the tangential component by projecting onto the tangent direction. More precisely, the torque about the origin applied at a point may be defined by
where is the applied force (at ) and denotes the cross product, introduced above in §B.4.12.
Note that the torque vector is orthogonal to both the lever arm and the tangential-force direction. It thus points in the direction of the angular velocity vector (along the axis of rotation).
The torque magnitude is
Newton's Second Law for Rotations
Rotational Kinetic Energy Revisited