Mass `M` is split into two parts `m` and `(M-m)`, which are then separated by a certain distance. What is the ratio of `(m//M)` which maximises the gravitational force between the parts ?

Options

A. `1:4`
B. `1:2`
C. `4:1`
D. `2:1`

Answer

To find the ratio of \( \frac{m}{M} \) that maximizes the gravitational force between the two parts of mass \( m \) and \( (M - m) \), we can follow these steps:

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

### Step 2: Simplify the expression
We can express the force as:
\[ F = \frac{G}{r^2} \cdot (mM - m^2) \]
Let's denote \( k = \frac{G}{r^2} \) (a constant with respect to \( m \)):
\[ F = k(mM - m^2) \]

### Step 3: Differentiate the force with respect to \( m \)
To find the maximum force, we need to differentiate \( F \) with respect to \( m \) and set the derivative equal to zero:
\[ \frac{dF}{dm} = k(M - 2m) \]
Setting the derivative to zero for maximization:
\[ M - 2m = 0 \]

### Step 4: Solve for \( m \)
From the equation \( M - 2m = 0 \), we can solve for \( m \):
\[ 2m = M \quad \Rightarrow \quad m = \frac{M}{2} \]

### Step 5: Find 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
Thus, the ratio of \( \frac{m}{M} \) that maximizes the gravitational force between the two parts is:
\[ \frac{m}{M} = \frac{1}{2} \]