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:
is the torque (a vector)
is the arm (a vector) from the point of rotation O
to the contact point P
(where the force acts)
is the force (a vector)
is the angle between r
, as shown
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
, and F
do not have the vector "hat" because here they represent the magnitude of vectors τ
, and F
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
" of the force F
. The moment arm r
is the component of r
that is perpendicular to F
. This is simply a different naming convention.
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