3 Moments and Couples

Learn how forces create moments about points and axes, how to calculate moments in two and three dimensions, and why couples produce reference-point-independent moments.

of a force about a point

A force creates a rotational effect about a point when its does not pass through that point. The depends on both the force and the position of its relative to the reference point.

The vector and its magnitude

For a force $\mathbf{F}$ acting at a point, the about reference point $O$ is calculated using the $\mathbf{r}$ , drawn from $O$ to any point on the force’s :

\mathbf{M}_O = \mathbf{r} \times \mathbf{F}.

The gives the ’s magnitude and direction. If $\theta$ is the angle between $\mathbf{r}$ and $\mathbf{F}$ , and $d$ is the perpendicular distance from $O$ to the force’s , then

|\mathbf{M}_O| = rF\sin\theta = Fd.

A force whose passes through $O$ has zero about $O$ .

Two-dimensional moments

For two-dimensional calculations, take $x$ to the right and $y$ upward, and use counterclockwise as positive and clockwise as negative. The points in the $z$ direction, with signed value

M_{O,z}=r_xF_y-r_yF_x.

For example, if $\mathbf{F}=(30,-50)\,\text{N}$ acts at $\mathbf{r}=(0.40,0.20)\,\text{m}$ from $O$ , then

M_{O,z}=(0.40)(-50)-(0.20)(30)=-26\,\text{N}\cdot\text{m}.

The negative sign indicates a clockwise . Equivalently, the force’s two components produce moments that add algebraically about $O$ .

Moments in three dimensions

In three dimensions, use the vector form of the and calculate the by components:

\mathbf{M}_O= \begin{vmatrix} \mathbf{i}&\mathbf{j}&\mathbf{k}\\ r_x&r_y&r_z\\ F_x&F_y&F_z \end{vmatrix} = (r_yF_z-r_zF_y)\mathbf{i} +(r_zF_x-r_xF_z)\mathbf{j} +(r_xF_y-r_yF_x)\mathbf{k}.

The direction follows the right-hand rule. The magnitude of the vector is

|\mathbf{M}_O|=\sqrt{M_x^2+M_y^2+M_z^2}.

For example, with $\mathbf{r}=(1,2,0)\,\text{m}$ and $\mathbf{F}=(0,0,10)\,\text{N}$ ,

\mathbf{M}_O=\mathbf{r}\times\mathbf{F}=(20,-10,0)\,\text{N}\cdot\text{m}.

Its magnitude is $\sqrt{20^2+(-10)^2}=22.36\,\text{N}\cdot\text{m}$ . The components show how the force tends to rotate the body about each coordinate axis. To find the about a particular axis with unit direction $\mathbf{u}$ , project the vector onto that axis:

M_{\text{axis}}=\mathbf{u}\cdot\mathbf{M}_O.

Couples

A consists of forces $\mathbf{F}$ and $-\mathbf{F}$ acting along distinct parallel lines. Their net force is zero, but together they produce a . If $\mathbf{r}$ points from the of $-\mathbf{F}$ to the of $\mathbf{F}$ , the is

\mathbf{M}_c=\mathbf{r}\times\mathbf{F}.

Its magnitude is $M_c=Fd$ , where $d$ is the perpendicular separation between the lines of action. Unlike the of a single force about a point, a is independent of the reference point. It is a free vector and can be added directly to other moments.

For example, two opposite forces of $40\,\text{N}$ , separated by a perpendicular distance of $0.30\,\text{m}$ , form a with magnitude

M_c=(40)(0.30)=12\,\text{N}\cdot\text{m}.

The direction is determined by the right-hand rule or, in a planar problem, by whether the pair tends to rotate the body clockwise or counterclockwise.

Combining moments and checking calculations

When calculating or combining moments, draw $\mathbf{r}$ from the center to the force’s and use $\mathbf{r}\times\mathbf{F}$ , not $\mathbf{F}\times\mathbf{r}$ . Keep coordinate signs consistent with the chosen axes and use a right-handed coordinate system.

Include both force moments and any applied couples when summing moments. By , a force’s can be calculated from its components, with their moments added algebraically.

Moments and couples have units of force times distance, such as $\text{N}\cdot\text{m}$ . Although these units are dimensionally the same as a joule, a is not energy.

Moment of a force about a point

The moment vector and its magnitude