3.2: Moment of Force
- Page ID
- 6940
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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:
\begin{equation} \ \boldsymbol\tau = \textbf{r} \times \textbf{F}\tag{3.2.1}\label{eq:3.2.1} \end{equation}
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
\begin{equation} \ \boldsymbol\tau = \textbf{r}\times \textbf{F}\tag{3.2.2}\label{eq:3.2.2} \end{equation}
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.)