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centre of mass of polygon (Topic)

Let $A_1A_2{\ldots}A_n$ be an $n$-gon which is supposed to have a constant surface-density in all of its points, $M$ the centre of mass of the polygon and $O$ the origin. Then the position vector of $M$ with respect to $O$ is

$\displaystyle \overrightarrow{OM} = \frac{1}{n}\sum_{i=1}^n\overrightarrow{OA_i}.$ (1)
We can of course take especially  $O = A_1$,  and thus

$\displaystyle \overrightarrow{A_1M} = \frac{1}{n}\sum_{i=1}^n\overrightarrow{A_1A_i} = \frac{1}{n}\sum_{i=2}^n\overrightarrow{A_1A_i}.$

In the special case of the triangle $ABC$ we have

$\displaystyle \overrightarrow{AM} = \frac{1}{3}(\overrightarrow{AB}+\overrightarrow{AC}).$ (2)
The centre of mass of a triangle is the common point of its medians.

Remark. An analogical result with (2) concerns also the homogeneous tetrahedron $ABCD$,

$\displaystyle \overrightarrow{AM} = \frac{1}{4}(\overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD}),$
and any $n$-dimensional simplex (cf. the midpoint of line segment:  $\overrightarrow{AM} = \frac{1}{2}\overrightarrow{AB}$).



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See Also: center of mass, center of mass of half-disc


Cross-references: position vector, centre of mass

This is version 1 of centre of mass of polygon, born on 2009-04-18.
Object id is 663, canonical name is CentreOfMassOfPolygon.
Accessed 264 times total.

Classification:
Physics Classification: 02.40.Yy (Geometric mechanics )
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