Prove that when a body is sub divided into two equal masses then the gravitational force between the two is maximum.

Question

Gravitational force question

 

Answer 1

Let k be a proper fraction, and let the original Mass be divided into two parts: kM & (1-k)M.

Then using Newton's formula for the force of gravitational attraction between two bodies, we have

F = G(kM)(1-k)M/d²  

If we desire to maximize this force between the two parts, we treat GM²/d² as constant,

and attempt to maximize f(k) = k(1-k) i.e. maximize f(k) = k - k².

 

This being a quadratic in k, the maximum occurs at the vertex since its graph is a downwardly concave parabola.

We use the formula for the vertex's "x" coordinate, x_vertex = -b/(2a)

which gives us k = -1/(2(-1)) which is k = ½.

 

Q.E.D.