Question

Three point masses each of mass in are placed at the vertices of an equilateral triangle of side a. The gravitational potential due to these masses at the centroid of the triangle is

Answer 1

Answer:

The gravitational potential due to the masses at the centroid of the triangle is

-3√3(Gm/a)

Explanation:

Given data:

Three point masses each of mass m are placed at the vertices of an equilateral triangle of side a.

To find:

The gravitational potential due to these masses at the centroid of the triangle =?

Solution:

Let us consider a triangle ABC which is an equilateral triangle.

The length of each side is given as a. Let us consider O as the centroid of the triangle.

We know that the distance of centroid from each vertex of the triangle is same.

This is given by

OA= a/√3

OB = a/√3

OC = a/√3

r= OA = OB = OC = a/√3

The gravitational potential on point O due to mass at point A is given by

V = -Gm/r

Thus

VA = -Gm (√3/a)

Similarly, gravitational potential on point O due to mass at B

VB = -Gm (√3/a)

Similarly, gravitational potential on point O due to mass at C

VC = -Gm (√3/a)

This potential due to all the three masses is given by

V = 3(-Gm (√3/a))

V = -3√3(Gm/a)

Hence,

The gravitational potential due to the masses at the centroid of the triangle is

-3√3(Gm/a)