Two point masses m are kept r distance apart. What will be the potential at 0.25r distance from any one mass towards the other mass ?

Explanation

To find the gravitational potential at a point located 0.25r from one mass towards another mass, we can follow these steps:

### Step 1: Understand the Setup
We have two point masses, both of mass \( m \), separated by a distance \( r \). We need to find the gravitational potential at a point \( P \) which is located at a distance of \( 0.25r \) from one of the masses (let's call it mass 1) and \( 0.75r \) from the other mass (mass 2).

### Step 2: Write the Formula for Gravitational Potential
The gravitational potential \( V \) due to a point mass \( m \) at a distance \( d \) is given by the formula:
\[ V = -\frac{Gm}{d} \]
where \( G \) is the gravitational constant.

### Step 3: Calculate the Potential Due to Mass 1
The distance from mass 1 to point \( P \) is \( 0.25r \). Therefore, the potential \( V_1 \) at point \( P \) due to mass 1 is:
\[ V_1 = -\frac{Gm}{0.25r} = -\frac{Gm}{\frac{1}{4}r} = -\frac{4Gm}{r} \]

### Step 4: Calculate the Potential Due to Mass 2
The distance from mass 2 to point \( P \) is \( 0.75r \). Therefore, the potential \( V_2 \) at point \( P \) due to mass 2 is:
\[ V_2 = -\frac{Gm}{0.75r} = -\frac{Gm}{\frac{3}{4}r} = -\frac{4Gm}{3r} \]

### Step 5: Combine the Potentials
The total gravitational potential \( V \) at point \( P \) is the sum of the potentials due to both masses:
\[ V = V_1 + V_2 = -\frac{4Gm}{r} - \frac{4Gm}{3r} \]

### Step 6: Simplify the Expression
To combine these fractions, we need a common denominator. The common denominator for \( r \) and \( 3r \) is \( 3r \):
\[ V = -\frac{12Gm}{3r} - \frac{4Gm}{3r} = -\frac{12Gm + 4Gm}{3r} = -\frac{16Gm}{3r} \]

### Final Answer
Thus, the gravitational potential at point \( P \) is:
\[ V = -\frac{16Gm}{3r} \]