A mass M splits into two parts m and (M - m) , which are then separated by a certain distance. Wha ratio (m/M) maximize the gravitational force between the path ?

Options

A. `(2)/(3)`
B. `(3)/(4)`
C. `(1)/(2)`
D. `(1)/(3)`

Explanation

To solve the problem of maximizing the gravitational force between two parts of a mass \( M \) that splits into two parts \( m \) and \( (M - m) \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Gravitational Force**:
The gravitational force \( F \) between two masses \( m \) and \( (M - m) \) separated by a distance \( r \) is given by Newton's law of gravitation:
\[ F = \frac{G \cdot m \cdot (M - m)}{r^2} \]
where \( G \) is the gravitational constant.

2. **Rearranging the Force Equation**:
We can express the force in a more convenient form:
\[ F = \frac{G}{r^2} \cdot m \cdot (M - m) \]

3. **Differentiating the Force with Respect to \( m \)**:
To find the value of \( m \) that maximizes the force, we differentiate \( F \) with respect to \( m \):
\[ \frac{dF}{dm} = \frac{G}{r^2} \cdot \left( M - 2m \right) \]

4. **Setting the Derivative to Zero**:
To find the maximum, we set the derivative equal to zero:
\[ M - 2m = 0 \]
Solving for \( m \):
\[ 2m = M \implies m = \frac{M}{2} \]

5. **Finding the Ratio \( \frac{m}{M} \)**:
Now, we can find the ratio of \( m \) to \( M \):
\[ \frac{m}{M} = \frac{\frac{M}{2}}{M} = \frac{1}{2} \]

### Conclusion:
The ratio \( \frac{m}{M} \) that maximizes the gravitational force between the two parts is \( \frac{1}{2} \).