Two small and heavy spheres, each of mass M, are placed a distance r apart on a horizontal surface. The graviational potential at the mid-point on the line joining the centre of the spheres is :-

Options

A. Zero
B. `- (GM)/r`
C. `- (2 GM)/r`
D. `- (4GM)/r`

Answer

To find the gravitational potential at the midpoint between two spheres of mass \( M \) placed a distance \( r \) apart, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Position of the Midpoint**:
- The two spheres are placed at a distance \( r \) apart. The midpoint between them is at a distance of \( \frac{r}{2} \) from each sphere.

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

3. **Calculate the Gravitational Potential from Each Sphere**:
- For Sphere 1 (located at one end), the distance to the midpoint is \( \frac{r}{2} \):
\[ V_1 = -\frac{G M}{\frac{r}{2}} = -\frac{2 G M}{r} \]
- For Sphere 2 (located at the other end), the distance to the midpoint is also \( \frac{r}{2} \):
\[ V_2 = -\frac{G M}{\frac{r}{2}} = -\frac{2 G M}{r} \]

4. **Calculate the Total Gravitational Potential at the Midpoint**:
- The total gravitational potential \( V_{\text{total}} \) at the midpoint is the sum of the potentials due to both spheres:
\[ V_{\text{total}} = V_1 + V_2 = -\frac{2 G M}{r} + -\frac{2 G M}{r} = -\frac{4 G M}{r} \]

5. **Final Result**:
- Therefore, the gravitational potential at the midpoint between the two spheres is:
\[ V = -\frac{4 G M}{r} \]

### Conclusion:
The gravitational potential at the midpoint on the line joining the centers of the two spheres is \( -\frac{4 G M}{r} \).

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