# Torque

Torque is defined as the tendency to rotate an object when it is subjected to a force. It is a function of the magnitude of the force, the direction of the force, and the "arm". See figure below.

Define the following variables:

*τ* is the torque (a vector)

*r* is the arm (a vector) from the point of rotation

*O* to the contact point

*P* (where the force acts)

*F* is the force (a vector)

*φ* is the angle between

*r* and

*F*, as shown

*u* is a unit vector acting in the direction of the tοrque

*τ*
Tοrque can be expressed mathematically, as follows

In the last two equations,

*τ*,

*r*, and

*F* do not have the vector "hat" because here they represent the magnitude of vectors

*τ*,

*r*, and

*F* (respectively).

The rotation point

*O* (and therefore

*r*) can be chosen arbitrarily when solving a dynamics problem, since this choice will not affect the solution. However, it may be easier and mathematically more convenient to choose a point which "intuitively" makes sense, such as the hinges of a door. Also,

*O* does not need to be a fixed point. It can be a moving point on a body (e.g. the center of mass). This becomes clearer when solving actual dynamics problems, which many times involve calculating tοrque about the center of mass.

The direction of

*τ* can be visualized using the right-hand rule. If one imagines which direction the object will tend to rotate when acted upon by the force

*F*, and then curling the fingers of your right hand in that direction, your thumb will give the direction of the

*τ* vector. For the figure above, the

*τ* vector is pointing out of the page.

Note that tοrque is sometimes called the "moment" of a force, which specifically means the "moment arm

*r*sin

*φ*" of the force

*F*. The moment arm

*r*sin

*φ* is the component of

*r* that is perpendicular to

*F*. This is simply a different naming convention.

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