Moment of a Force About a Point and an Axis

The moment of a force describes the rotational effect produced by a force. It depends on both the force magnitude and the position of its line of action relative to the point or axis about which rotation is considered.

Moment of a force about a point

Let a force $\vec F$ act at point $A$, and let $O$ be the point about which the moment is required. The moment vector is defined by the cross product

$$\vec M_O=\vec r\times\vec F,$$

where $\vec r=\overrightarrow{OA}$. Its direction follows the right-hand rule. Its magnitude can be written as $M_O=rF\sin\alpha$, where $\alpha$ is the angle between $\vec r$ and $\vec F$. Since $d=r\sin\alpha$ is the perpendicular distance from $O$ to the line of action, the principal practical formula follows.

The magnitude of the moment of a force about a point equals the force magnitude multiplied by the moment arm, the shortest distance from the point to the force's line of action:

$$M_O = Fd$$

where $M_O$ is the moment about point $O$, $F$ is the force magnitude, and $d$ is the moment arm. In a planar problem, the sign represents the sense of rotation: counterclockwise is commonly taken as positive and clockwise as negative.

Sign convention in planar statics

For a planar force system, moments are conveniently treated as algebraic quantities. Counterclockwise rotation is commonly taken as positive and clockwise rotation as negative. Choose a convention once and use it consistently throughout the equilibrium equations.

When the moment is zero

If the force's line of action passes through point $O$, the moment arm is $d=0$ and the moment about that point is zero. Consequently, moving a force anywhere along its line of action does not change its moment about an arbitrary point.

Coordinate calculation

In the $xy$ plane, if $\vec r=(x,y)$ and $\vec F=(F_x,F_y)$, the moment about the origin is

$$M_O=xF_y-yF_x.$$

This form is especially useful when a problem gives the coordinates of the point of application and the Cartesian components of the force.

Moment of a force about an axis

The moment about a specified axis is the projection of the moment vector about any point on that axis onto the axis direction. If $\vec e$ is a unit vector along the axis,

$$M_{axis}=\vec e\cdot(\vec r\times\vec F).$$

In three-dimensional statics, this measures the tendency of the force to rotate the body specifically about the selected axis.

Example

A force $F=200$ N acts on a lever with a perpendicular distance $d=0.30$ m from the pivot to the force's line of action. The moment magnitude is $M=200\cdot0.30=60$ N·m. Its sign depends on the resulting sense of rotation and the adopted sign convention.

Skills for solving problems

Before calculating a moment, identify the reference point or axis, locate the force's line of action, determine the perpendicular moment arm or resolve the force into components, and check the sign. For several forces, their moments are summed algebraically. This procedure forms the basis of equilibrium equations in statics.