Two spheres each of mass `M` and radius `R` are separated by a distance of `r`. The gravitational potential at the midpoint of the line joining the centres of the spheres is

Explanation

To find the gravitational potential at the midpoint of the line joining the centers of two spheres, we can follow these steps:

### Step 1: Understand the Setup
We have two spheres, each with mass \( M \) and radius \( R \), separated by a distance \( r \). We need to find the gravitational potential at the midpoint of the line joining the centers of these spheres.

### Step 2: Identify the Midpoint
The midpoint between the two spheres is located at a distance of \( \frac{r}{2} \) from each sphere's center.

### Step 3: Gravitational Potential Formula
The gravitational potential \( V \) due to a mass \( M \) at a distance \( d \) is given by the formula:
\[ V = -\frac{GM}{d} \]
where \( G \) is the gravitational constant.

### Step 4: Calculate the Potential from Each Sphere
At the midpoint (let's call it point A), the distance from each sphere to point A is \( \frac{r}{2} \).

- The potential at point A due to the first sphere (let's call it \( V_1 \)):
\[ V_1 = -\frac{GM}{\frac{r}{2}} = -\frac{2GM}{r} \]

- The potential at point A due to the second sphere (let's call it \( V_2 \)):
\[ V_2 = -\frac{GM}{\frac{r}{2}} = -\frac{2GM}{r} \]

### Step 5: Total Gravitational Potential at Point A
Since gravitational potential is a scalar quantity, we can add the potentials from both spheres:
\[ V_{total} = V_1 + V_2 = -\frac{2GM}{r} + -\frac{2GM}{r} = -\frac{4GM}{r} \]

### Final Answer
Thus, the gravitational potential at the midpoint of the line joining the centers of the two spheres is:
\[ V = -\frac{4GM}{r} \]

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