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# 3.2: Moment of Force

First, let’s look at a familiar two-dimensional situation. In Figure III.1 I draw a force $$\textbf{F}$$ and a point O. The moment of the force with respect to O can be defined as

Force times perpendicular distance from O to the line of action of $$\textbf{F}$$.

Alternatively, (Figure III.2) the moment can be defined equally well by

Transverse component of force times distance from O to the point of application of the force.

Either way, the magnitude of the moment of the force, also known as the torque, is $$rF \sin\theta$$ We can regard it as a vector, $$\boldsymbol\tau$$, perpendicular to the plane of the paper:

$$\ \boldsymbol\tau = \textbf{r} \times \textbf{F}\tag{3.2.1}\label{eq:3.2.1}$$

Now let me ask a question. Is it correct to say the moment of a force with respect to (or “about”) a point or with respect to (or “about”) an axis?

In the above two-dimensional example, it does not matter, but now let me move on to three dimensions, and I shall try to clarify.

In Figure III.3, I draw a set of rectangular axes, and a force $$\textbf{F}$$, whose position vector with respect to the origin is $$\textbf{r}$$.

The moment, or torque, of $$\textbf{F}$$ with respect to the origin is the vector

$$\ \boldsymbol\tau = \textbf{r}\times \textbf{F}\tag{3.2.2}\label{eq:3.2.2}$$

The $$x-, y-$$ and $$z$$ -components of $$\boldsymbol\tau$$ are the moments of $$\textbf{F}$$ with respect to the $$x-, y-$$ and z-axes. You can easily find the components of $$\boldsymbol\tau$$ by expanding the cross product $$\ref{eq:3.2.2}$$:

$\boldsymbol\tau = \hat{\textbf{x}}(yF_{z}-zF_{y})+\hat{\textbf{y}}(yF_{x}-xF_{z})+\hat{\textbf{z}}(xF_{y}-yF_{x}) \tag{3.2.3}\label{eq:3.2.3}$

where $$\bf \hat{x},\hat{y},\hat{z}$$ are the unit vectors along the $$x,y,z$$ axes. In Figure III.4, we are looking down the $$x$$-axis, and I have drawn the components $$F_{y}$$ and $$F_{z}$$ , and you can see that, indeed, $$\tau_{x} =yF_{z}-zF_{y}$$ .

The dimensions of moment of a force, or torque, are ML2T-2, and the SI units are N m. (It is best to leave the units as N m rather than to express torque in joules.)