A mass M splits into two parts m and (M-m), which are separated by a cetain distance. The ratio `m"/"M` which maximizes the gravitational force between the parts is

Options

A. `1 : 4`
B. `1 : 3`
C. `1 : 2`
D. `1 : 1`

Answer

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

### Step 1: Write the expression for 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.

### Step 2: Rewrite the force in terms of \( m \)
The expression can be rewritten as:

\[ F = \frac{G \cdot (mM - m^2)}{r^2} \]

### Step 3: Differentiate the force with respect to \( m \)
To find the value of \( m \) that maximizes the force, we need to take the derivative of \( F \) with respect to \( m \) and set it to zero:

\[ \frac{dF}{dm} = \frac{G}{r^2} \cdot (M - 2m) \]

### Step 4: Set the derivative equal to zero
Setting the derivative equal to zero gives:

\[ M - 2m = 0 \]

### Step 5: Solve for \( m \)
From the equation \( M - 2m = 0 \), we can solve for \( m \):

\[ 2m = M \implies m = \frac{M}{2} \]

### Step 6: Find the ratio \( \frac{m}{M} \)
Now, we can find the ratio \( \frac{m}{M} \):

\[ \frac{m}{M} = \frac{\frac{M}{2}}{M} = \frac{1}{2} \]

### Conclusion
Thus, the ratio \( \frac{m}{M} \) that maximizes the gravitational force between the two parts is:

\[ \frac{m}{M} = \frac{1}{2} \]

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