Torque

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

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

where denotes the angle from to . We can interpret as the length of the projection of onto the tangent direction (the line orthogonal to in the direction of the force), so that we can write

where , thus getting back to Eq.(B.26).

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University