To solve this problem, we need to find the gravitational potential at the point where the gravitational field is zero between two masses $m$ and $4m$ placed a distance $r$ apart. Let's break it down step by step.

Step 1: Locate the Point Where Gravitational Field is Zero

Let the two masses be placed along the x-axis. Place mass $m$ at $x=0$ and mass $4m$ at $x=r$. Let the point where gravitational field is zero be at a distance $x$ from mass $m$. At this point, the gravitational fields due to both masses must be equal in magnitude and opposite in direction.

Gravitational field due to $m$ is $\frac{Gm}{x^2}$ (towards $m$).

Gravitational field due to $4m$ is $\frac{G\left(4m\right)}{{(r-x)}^2}$ (towards $4m$).

Set them equal for zero net field:

$\frac{Gm}{x^2}=\frac{G\left(4m\right)}{{(r-x)}^2}$

Cancel $G$ and $m$ (assuming $m\ne 0$):

$\frac{1}{x^2}=\frac{4}{{(r-x)}^2}$

Take square root of both sides:

$\frac{1}{x}=\frac{2}{r-x}$

Solve for $x$:

$r-x=2x$

$r=3x$

$x=\frac{r}{3}$

So, the point is at $x=\frac{r}{3}$ from mass $m$ and $r-x=\frac{2r}{3}$ from mass $4m$.

Step 2: Calculate Gravitational Potential at This Point

Gravitational potential is a scalar quantity. The total potential at the point is the sum of potentials due to each mass.

Potential due to a point mass $M$ at a distance $d$ is $-\frac{GM}{d}$.

Potential due to mass $m$ at distance $\frac{r}{3}$: $V_1=-\frac{Gm}{r/3}=-\frac{3Gm}{r}$

Potential due to mass $4m$ at distance $\frac{2r}{3}$: $V_2=-\frac{G\left(4m\right)}{2r/3}=-\frac{4Gm}{2r/3}=-\frac{4Gm\times 3}{2r}=-\frac{12Gm}{2r}=-\frac{6Gm}{r}$

Total potential $V=V_1+V_2=-\frac{3Gm}{r}+(-\frac{6Gm}{r})=-\frac{9Gm}{r}$

Final Answer

The gravitational potential at the point where the gravitational field is zero is $-\frac{9Gm}{r}$.

Related Topics and Formulae

Gravitational Field

The gravitational field due to a point mass $M$ at a distance $r$ is given by $\overrightarrow{E}=-\frac{GM}{r^2}\hat{r}$, where $\hat{r}$ is the unit vector pointing from the mass to the point. It is a vector quantity.

Gravitational Potential

The gravitational potential due to a point mass $M$ at a distance $r$ is $V=-\frac{GM}{r}$. It is a scalar quantity and represents the work done per unit mass to bring a test mass from infinity to that point.

Superposition Principle

Both gravitational field and potential follow the superposition principle. For multiple masses, the net field is the vector sum of fields due to each mass, and the net potential is the scalar sum of potentials due to each mass.