Question

Three mass points each of mass $m$ are placed at the vertices of an equilateral triangle of side $1$ . What is the gravitational potential at the centroid of the triangle.

Answer 1

Let the distance of the centroid from each vertex is $r$ and it can be written as,

$r=\frac{l}{\sqrt{3}}$

The gravitational potential at centroid due to mass $m$ at A is given as,

$V_A=\frac{-Gm}{r}$

$=\frac{-\sqrt{3}Gm}{l}$

The gravitational potentials at B and C are given as,

$V_B=\frac{-\sqrt{3}Gm}{l}$

$V_C=\frac{-\sqrt{3}Gm}{l}$

The total potential due to all the three masses is given as,

$V=V_A+V_B+V_C$

$V=\frac{-\sqrt{3}Gm}{l}-\frac{\sqrt{3}Gm}{l}-\frac{\sqrt{3}Gm}{l}$

$V=\frac{-3\sqrt{3}Gm}{l}$

Thus, the gravitational potential at the centroid of the triangle is $\frac{-3\sqrt{3}Gm}{l}$.