Moments

A moment is a force applied at a distance from a point, which causes rotation about that point. The distance that the force is applied from is called the moment arm.

If the bar is pinned at point $ A $ , the force $ F $ , applied at a distance $ d $ from point A, will cause a moment $ M_A $ about point A and make the bar rotate clockwise. The only force that produces a rotation about a point is a force perpendicular to the moment arm (in this case, the moment arm is $ d $).

Moments as scalars

Moments can be written as scalar quantities, where

Scalar Moment

$$ {M_A} = F \ d\ $$

Note that d is the distance from the applied force to the point of interest and that the force F is the force perpendicular to the moment arm.

The moment about point A is dependent on how much of the force $ F $ is perpendicular to the moment arm $ d $ .

You can determine the direction of moments using the right hand rule and wrapping your fingers around where the moment is turning. The direction your thumb is pointing is the direction of the moment. Moments are assumed to be negative when they point in the $ -\hat{k} $ direction (clockwise rotation) and positive when they point in the $ +\hat{k} $ direction (counterclockwise rotation).

Positive and negative moment conventions.

Moments as vectors

Moments can also be written as vector quantities, where the moment about point A is the cross product of the moment arm, $ r $ and the force $ F $.

Vector Moment

$$ \vec{M_A} = \vec{r} \times \vec{F}\ $$

Using this formula, moments could have an $ \hat{i} $, $ \hat{j} $, and $ \hat{k} $ component. The magnitude of the moment can be written as

Moment Magnitude

$$ \vert\vec{M_A}\vert = \vert\vec{r}\vert \vert\vec{F}\vert sin\theta\ $$

Couple moment

A couple moment is a set of two equal and opposite forces with parallel lines of action that produce a moment. The resulting moment from a force couple can be written as

Moment of a force couple

$$ M_A = F \ d_{\perp}\ $$

where $ F $ is the magnitude of one of the force vectors and d is the perpendicular distance between the two vectors. Note: Couple moments are independent of location when applied to a rigid body, which means the resulting moment from a force couple can be "placed" anywhere on the rigid body with the same resulting rotation from the couple moment.

A couple moment

Alternatively, the resulting moment from a force couple can be written with the following equation.

Vector moment of a force couple.

$$ \vec{M}_A = \vec{r}_{AB} \times \vec{F}_B $$

We consider that $ \vec{F}_A $ and $ \vec{F}_B $ are forces forming a couple moment and $ \vec{r}_{AB} $ is any vector pointing from the line of action of $ \vec{F}_A $ to the line of action of $ \vec{F}_B. $

Vector couple moment