Question

Suppose the gravitational potential due to a small system is k/r² at a distance r from it. What will be the gravitational field? Can you think of any such system? What happens if there were negative masses?

Answer 1

The gravitational potential due to the system is given as \[V = \frac{k}{r^2}\] 

Gravitational field due to the system :
\[E = - \frac{dV}{dr}\]
\[ \Rightarrow E = - \frac{d}{dr}\left( \frac{k}{r^2} \right) = - \left( - \frac{2k}{r^3} \right)\]
\[ \Rightarrow E = \frac{2k}{r^3}\] 
We can see that for this system , \[E \propto \frac{1}{r^3}\]
This type of system is not possible because \[F_g\] is always proportional to inverse of square of distance(experimental fact). 
If there were negative masses, then this type of system is possible.
This system is a dipole of two masses, i.e., two masses, one positive and the other negative, separated by a small distance.
In this case, the gradational field due to the dipole is proportional to \[\frac{1}{r^3}\]