Animated Solution for Physics - Gravitation: Two bodies of masses $m$ and $4m$ are placed at a distance $r$. The gravitational potential at a point on the line joining them, where the gravitational field is zero, is

Finding the Neutral Point

Imagine two masses, $m$ and $4m$, separated by a distance $r$. We are tasked with finding a specific point $P$ on the line joining them where the net gravitational field is exactly zero. This point is often called the neutral point.

Let's assume this point $P$ is located at a distance $x$ from the smaller mass $m$. Consequently, its distance from the larger mass $4m$ will be $r-x$.

For the net gravitational field to be zero at point $P$, the gravitational pull from mass $m$ must perfectly cancel out the pull from mass $4m$. Since these two forces act in opposite directions along the line joining the masses, their magnitudes must be equal.

We can set up the equation by equating the magnitudes of the gravitational fields:

Now, let's solve for $x$. We can start by canceling the common terms, $G$ and $m$, from both sides of the equation:

Taking the square root of both sides simplifies things beautifully. Since $P$ lies between the masses, we only consider the positive root:

Cross-multiplying gives us a simple linear equation:

So, the neutral point is located at a distance of $\frac{r}{3}$ from mass $m$, and $\frac{2r}{3}$ from mass $4m$.

Now that we have the exact location of point $P$, we need to find the total gravitational potential there. Remember, unlike the gravitational field, gravitational potential is a scalar quantity. This means we don't need to worry about directions; we just add the potentials due to each mass algebraically.

The total potential $V_P$ at point $P$ is the sum of the potential due to $m$ and the potential due to $4m$:

Let's substitute our value of $x=\frac{r}{3}$ into the potential equation:

Simplifying the fractions, we get:

Adding these two terms together yields our final answer:

This is the total gravitational potential at the point where the gravitational field is zero. It's a beautiful demonstration of how vector fields and scalar potentials interact in a simple two-body system!