centroids and weighted position vectorsCentroidsAndWeightedPositionVectors2026-08-21 00:46:402026-08-21 00:46:40Definitioncenter of massgrouping theorem% this is the default PhysicsLibrary preamble. as your knowledge
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% define commands here\section*{Centroids and Weighted Position Vectors}
Let points $P_i$ have scalar weights $m_i$, with
\[
M=\sum_i m_i\ne0.
\]
Their weighted centroid $P^*$ is defined by
\[
\sum_i m_i\overrightarrow{P^*P_i}=\mathbf0.
\tag{1}
\]
In position-vector form,
\[
\boxed{\mathbf r_{P^*}=\frac{\sum_i m_i\mathbf r_i}{\sum_i m_i}.}
\tag{2}
\]
This is exactly the modern center of mass \PMLinkescape{formula} when the $m_i$ are masses.
\begin{center}
\includegraphics[width=.68\textwidth]{brand_fig_12.png}
\textit{Figure 12, modernized: a weighted centroid with one negative weight.}
\end{center}
For equal weights,
\[
\mathbf r_G=\frac1n\sum_i\mathbf r_i,
\]
which is the arithmetic mean of the position vectors.
For two weighted points $A,B$ with weights $a,b$,
\[
\mathbf r_{P^*}=\frac{a\mathbf{r}_A+b\mathbf{r}_B}{a+b},
\]
so the point divides $AB$ inversely to the weights.
\subsection*{Grouping theorem}
If a collection of weighted points is split into two \PMLinkescape{groups} with centroids
$P',P''$ and total weights $M',M''$, then the centroid of the entire set is
the centroid of $P',P''$ weighted by $M',M''$:
\[
\mathbf r_{P^*}=\frac{M'\mathbf r_{P'}+M''\mathbf r_{P''}}{M'+M''}.
\]
This property is useful for composite bodies and hierarchical center-of-mass
calculations.
\section*{Source problems}
\begin{enumerate}
\item Find the centroid of $(0,1,3)$, $(-3,0,4)$, $(3,-2,0)$ with weights
$2,3,1$.
\item In the tetrahedral midpoint construction of Figure 7f, prove the two
midpoint-joining segments meet at the centroid of $A,B,C,D$.
\item If $G$ is the centroid of $A,B,C$, prove that the centroid of
$A,B,C,D$ divides $DG$ in the ratio $3:1$.
\item If $P^*,Q^*$ are the centroids of two sets of $n$ points
$P_1,\ldots,P_n$ and $Q_1,\ldots,Q_n$, prove
\[
\sum_{i=1}^n\overrightarrow{P_iQ_i}=n\overrightarrow{P^*Q^*}.
\]
\item Give a construction for the centroid of five equally weighted points
using the grouping theorem.
\end{enumerate}
\section*{Modern notation references}
The notation and terminology in this modernized article follow standard
present-day mechanics and vector-analysis usage, particularly:
\begin{enumerate}
\item J. R. Taylor, \emph{Classical Mechanics}, University Science Books, 2005.
\item D. Kleppner and R. Kolenkow, \emph{An Introduction to Mechanics}, 2nd ed., Cambridge University Press, 2014.
\item H. Goldstein, C. Poole, and J. Safko, \emph{Classical Mechanics}, 3rd ed., Addison--Wesley, 2002.
\end{enumerate}
\section*{Source}
This article is a modernized restatement of the corresponding Public \PMlinkescape{Domain} article in
Louis Brand, \emph{Vectorial Mechanics}, John Wiley \& Sons, New York, 1930,
Chapter I, ``Vector Algebra.'' The original 1930 edition is the source basis.