3.2: Varignon's Theorem
- Page ID
- 102528
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Varignon's Theorem, also often called the principle of moments, is a very useful tool in scalar moment calculations. In cases where the perpendicular distance is hard to determine, Varignon's Theorem offers an alternative to finding that distance.
In its basic form, Varignon's Theorem states that if we have two or more concurrent forces, the sum of the moments that each force creates about a single point will be equal to the moment created by the sum of those forces about the same point.
![A lever with one end attached to the wall, with two different forces applied to its free end. A third vector, the sum of those two forces, is also shown acting upon the free end.](https://eng.libretexts.org/@api/deki/files/41708/varignonstheorem.png?revision=1)
On its surface this doesn't seem that useful, but in practice we will often use this theorem in reverse by breaking down a force into components (the components being a set of concurrent forces). We can solve for the moment exerted by each component (where perpendicular distance \(d\) is easier to find) and then simply add together the moments from each component to find the moment from the original force.
![A rectangular shape has a force applied upwards and to the right at the bottom left corner, causing rotation of the shape about its center point. In addition to the vector of the applied force, the x- and y-components of said force are shown, as well as the vertical and horizontal distances from the force's point of application to the center point.](https://eng.libretexts.org/@api/deki/files/41711/varignons_theorem_reverse.png?revision=1)
Use Varignon's Theorem to find the moment that the forces in the diagram below exert about point A.
![A lever 0.5 meters long is attached to a wall at one end; the center of this region of contact is point A. Three forces are simaltaneously applied to the free end: one of magnitude 70 N upwards, one of magnitude 150 N to the right, and one of magnitude 300 N downwards and to the left, at a 30-degree angle from the vertical.](https://eng.libretexts.org/@api/deki/files/41714/P1_(1).png?revision=1)
- Solution
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Video \(\PageIndex{2}\): Worked solution to example problem \(\PageIndex{1}\), provided by Dr. Jacob Moore. YouTube source: https://youtu.be/JIpgZGxJWu4.
Use Varignon's Theorem to find the moment that the force in the diagram below exerts about point B.
![A 24-by-36-inch rectangle rests with its longer side horizontal. A 200-lb-magnitude force is applied up and to the left, making a 20-degree angle with the horizontal, on the rectangle's right upper corner, point A. The rectagnle's lower left corner is marked as point B.](https://eng.libretexts.org/@api/deki/files/41715/P2_(1).png?revision=1)
- Solution
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Video \(\PageIndex{3}\): Worked solution to example problem \(\PageIndex{2}\), provided by Dr. Jacob Moore. YouTube source: https://youtu.be/jpaLEprFndA.