Question

Three equal masses $2$ $m$ each are placed at the vertices an equilateral triangle PQR
$\left(i\right)$ What is the force acting on a mass $m$ placed at the centroid $G$ of the triangle?
$\left(ii\right)$ What is the force on mass $m$ if the mass at the vertex $P$ is quadrupled?
Take $PG=QG=RG=1$ $m$

Answer 1

$\left(1\right).$ $F=F_{GP}+F_{GQ}+F_{GR}$

$\Rightarrow F=\frac{G\cdot 2m\cdot m}{PG}+\frac{G\cdot 2m\cdot m}{GQ}+\frac{G\cdot 2m\cdot m}{GR}$

$\Rightarrow F=\frac{2G{m}^2}{1}[1+1+1]$

$\Rightarrow F=6G{m}^2$

$\left(2\right).$ $F=\frac{G\cdot m\cdot 8m}{1}+\frac{G\cdot m\cdot 2m}{1}+\frac{G\cdot m\cdot 2m}{1}$

$\Rightarrow F=G{m}^{2}12$