A mass `M` is broken into two parts of masses `m_(1)` and `m_(2)`. How are `m_(1)` and `m_(2)` related so that force of gravitational attraction between the two parts is maximum?

Answer

To solve the problem of how the masses \( m_1 \) and \( m_2 \) are related so that the gravitational force between them is maximized, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Masses**:
Let the total mass \( M \) be divided into two parts, \( m_1 \) and \( m_2 \). Therefore, we have:
\[ m_1 + m_2 = M \]

2. **Express One Mass in Terms of the Other**:
We can express \( m_2 \) in terms of \( m_1 \):
\[ m_2 = M - m_1 \]

3. **Write the Gravitational Force Formula**:
The gravitational force \( F \) between the two masses is given by Newton's law of gravitation:
\[ F = \frac{G m_1 m_2}{r^2} \]
Substituting \( m_2 \) from step 2, we get:
\[ F = \frac{G m_1 (M - m_1)}{r^2} \]

4. **Differentiate the Force with Respect to \( m_1 \)**:
To find the maximum gravitational force, we need to take the derivative of \( F \) with respect to \( m_1 \) and set it to zero:
\[ \frac{dF}{dm_1} = \frac{G}{r^2} \left( M - 2m_1 \right) = 0 \]

5. **Solve for \( m_1 \)**:
Setting the derivative equal to zero gives:
\[ M - 2m_1 = 0 \implies 2m_1 = M \implies m_1 = \frac{M}{2} \]

6. **Find \( m_2 \)**:
Using the relation \( m_2 = M - m_1 \):
\[ m_2 = M - \frac{M}{2} = \frac{M}{2} \]

7. **Conclusion**:
Thus, we find that:
\[ m_1 = m_2 = \frac{M}{2} \]
Therefore, the two masses are equal when the gravitational force between them is maximized.

### Final Relation:
The relation between the two masses \( m_1 \) and \( m_2 \) for maximum gravitational attraction is:
\[ m_1 = m_2 \]