Page History

Also called the "lever arm", the moment arm is the distance of closest approach between the [line of action] of a force and the [axis of rotation].  It is used to compute the [torque] produced by the force about the axis of rotation.

||Page Contents|| 
|{toc:indent=10px}|

h2. Geometrical Definition

h4. Illustration of the Method

The moment arm of a force about a specific [axis of rotation] can be found geometrically by constructing the force's [line of action] and then finding the shortest distance between the line of action and the axis.  The procedure is shown in the figures below.

For vertical forces:

|!vertmarm1.png|width=75%!|!vertmarm2.png|width=75%!|!vertmarm3.png|width=75%!|
|Given forces.|Construct line of action for each.|Find the shortest distance (shortest distance \\ always perpendicular to line of action, which\\ implies horizontal moment arm for vertical force).|

For horizontal forces:

|!hormarm1.png|width=75%!|!hormarm2.png|width=75%!|!hormarm3.png|width=75%!|
|Given forces.|Construct line of action for each.|Find the shortest distance (shortest distance \\ always perpendicular to line of action, which\\ implies vertical moment arm for horizontal force).|

General angles:

|!oddmarm1.png|width=60%!|!oddmarm2.png|width=60%!|!oddmarm3.png|width=60%!|
|Given forces.|Construct line of action for each.|Find the shortest distance.|



h4. Key Points

Some key points to remember:

* The moment arm must be perpendicular to the force. 
* The moment arm is *never* perpendicular to the position vector that gives the point of application of the force with respect to the axis of rotation.
* The moment arm for vertical forces (e.g. gravity) is always perfectly horizontal.
* The moment arm for horizontal forces is always perfectly vertical.
* The moment arm must always be the shortest distance between the line of action and the axis, so it will always be less than or equal to the distance from the force's point of application to the axis of rotation.


h2. Utility

h4. Calculating Torque

The moment arm is often given the symbol:

{latex}\begin{large}\[ \mbox{moment arm = } r_{\perp}\]\end{large}{latex}

When the moment arm for a given force _F_ about a chosen [axis of rotation] is known, the magnitude of the [torque] due to _F_ about the axis is:

{latex}\begin{large}\[ |\tau| = Fr_{\perp}|]\end{large}{latex}

h4. Relevant Examples

Examples where it is useful to employ the moment arm when calculating torque are: