The bodies of mass m and 4m are placed at a distance of 6m apart. P is the point on the line joining two bodies where gravitational field is zero. The gravitational potential at this point is

Options

A. `-(3)/(2) Gm`
B. `-(5)/(2) Gm`
C. `-(9)/(2) Gm`
D. `-(7)/(2) Gm`

Explanation

To solve the problem, we need to find the gravitational potential at point P where the gravitational field is zero between two masses, m and 4m, separated by a distance of 6m.

### Step-by-Step Solution:

1. **Identify the Positions of the Masses:**
Let the mass m be at point A and the mass 4m be at point B. The distance between A and B is 6m.

2. **Define the Distance to Point P:**
Let point P be at a distance x from mass m (at point A). Therefore, the distance from mass 4m (at point B) to point P will be (6 - x).

3. **Set Up the Gravitational Field Equation:**
The gravitational field intensity (E) at point P due to mass m is given by:
\[ E_m = -\frac{Gm}{x^2} \]
The gravitational field intensity at point P due to mass 4m is:
\[ E_{4m} = -\frac{G(4m)}{(6 - x)^2} \]
At point P, the net gravitational field is zero, so we set the magnitudes equal:
\[ \frac{G(4m)}{(6 - x)^2} = \frac{Gm}{x^2} \]

4. **Simplify the Equation:**
Cancel out G and m from both sides:
\[ \frac{4}{(6 - x)^2} = \frac{1}{x^2} \]

5. **Cross Multiply:**
\[ 4x^2 = (6 - x)^2 \]

6. **Expand the Right Side:**
\[ 4x^2 = 36 - 12x + x^2 \]

7. **Rearrange the Equation:**
\[ 4x^2 - x^2 + 12x - 36 = 0 \]
This simplifies to:
\[ 3x^2 + 12x - 36 = 0 \]

8. **Divide by 3:**
\[ x^2 + 4x - 12 = 0 \]

9. **Use the Quadratic Formula:**
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 4, c = -12 \):
\[ x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (-12)}}{2 \cdot 1} \]
\[ x = \frac{-4 \pm \sqrt{16 + 48}}{2} \]
\[ x = \frac{-4 \pm \sqrt{64}}{2} \]
\[ x = \frac{-4 \pm 8}{2} \]
This gives two solutions:
\[ x = 2 \quad \text{(valid, since distance cannot be negative)} \]
\[ x = -6 \quad \text{(not valid)} \]

10. **Calculate the Distance from 4m:**
The distance from mass 4m to point P is:
\[ 6 - x = 6 - 2 = 4 \]

11. **Calculate the Gravitational Potential at Point P:**
The gravitational potential (V) at point P due to mass m is:
\[ V_m = -\frac{Gm}{x} = -\frac{Gm}{2} \]
The gravitational potential at point P due to mass 4m is:
\[ V_{4m} = -\frac{G(4m)}{(6 - x)} = -\frac{G(4m)}{4} = -Gm \]
Therefore, the total gravitational potential at point P is:
\[ V_P = V_m + V_{4m} = -\frac{Gm}{2} - Gm = -\frac{Gm}{2} - \frac{2Gm}{2} = -\frac{3Gm}{2} \]

### Final Answer:
The gravitational potential at point P is:
\[ V_P = -\frac{3Gm}{2} \]