A mass M is split into two parts m and M - m. Which are then separated by a certain distance. The ratio m/M which maximizes the gravitational force between the parts is,
(A) 1:4
(B) 1:3
(C) 1:2
(D) 1:1

We are given a mass M, which is split into two parts: m and M - m. These two parts are then separated by a certain distance, and we are asked to find the ratio of m/M that maximizes the gravitational force between the two parts.

Gravitational Force
The gravitational force between two masses m and M - m, separated by a distance r, is given by Newton's law of gravitation:

F = G * (m * (M - m)) / r²

Where:

G is the gravitational constant
m is one part of the mass
(M - m) is the other part of the mass
r is the distance between the two parts
We are tasked with finding the value of m/M that maximizes the force F.

Step 1: Express the force in terms of m/M
Let x = m/M. Then, m = xM and M - m = M(1 - x).

Substitute these expressions into the gravitational force equation:

F(x) = G * (xM * M(1 - x)) / r² = G * M² * x(1 - x) / r²

Step 2: Maximize the force
To find the value of x that maximizes F(x), we take the derivative of F(x) with respect to x and set it equal to 0.

dF(x)/dx = G * M² * (1 - 2x) / r²

Set the derivative equal to 0:

1 - 2x = 0

Solve for x:

x = 1/2

Step 3: Find the ratio
The value of x = 1/2 means that m = M/2 and M - m = M/2.

Thus, the ratio of m/M that maximizes the gravitational force is 1:1.

Final Answer:
The correct answer is (D) 1:1.