Gravitational force for a negative mass Consider a hypothetical ball of mass -1kg .It is released from air .now will the ball go up or down towards earth?
my approach: since $\vec{F}=-\frac{GMm}{r^2}\hat{r}$ hence getting an extra minus sign  should make the force repulsive, however it is said that gravitational force is always attractive in nature. Although I have heard it only in context of positive mass , so I am not very sure. Kindly predict what will happen in  the given situation.
 A: A counter argument - an object with negative inertial mass will accelerate in the opposite direction to an applied force, since $\vec F = m \vec a$. So if the force of gravity on the ball is repulsive then the force on the ball is upwards but the ball will fall towards the earth because it accelerates in the opposite direction to the applied force.
This assumes, of course, that a negative gravitational mass implies a negative inertial mass, and vice versa.
The earth, having a normal positive mass, will be repulsed too (assuming Newton's Third Law still holds) and will accelerate away from the ball. Because of its large mass the earth will only move a small distance before the ball catches up with it.
However, if we had two masses of $1$ kg and $-1$ kg in empty space, we would seem to have created a situation in which the negative mass continually accelerates towards the positive mass, and the positive mass continually accelerates away from the negative mass. This looks very much like a source of free energy, so - oh dear - we have just broken the Second Law of Thermodynamics.
The simple fact of the matter is that we do not know how an object with negative mass would behave because we have never observed such an object. In the absence of experimental data, we can only guess.
A: In precision short-range gravitation experiments, the test masses are very uniform metal discs with precisely-drilled holes in them.  The discs are suspended above and below each other, close and parallel, and then turned; the non-uniform mass distribution causes a twisting force which is measured with a torsion fiber.
In the analysis of these experiments, the discs with holes are treated mathematically as uniform discs superimposed with negative masses to make the holes.  The mathematics is exactly like in electromagnetism, where you are free to superimpose the fields produced by different charges.  Positive-mass regions are attracted to positive-mass regions, and negative-mass regions are attracted to negative-mass regions: gravity will make two discs with the same hole patterns want to twist so that the holes line up with the holes.  That can also be interpreted as negative and positive masses interacting repulsively: the holes in one disc want to avoid the solid regions on the other disc.
A: Newton's law of universal gravitation was formulated for "positive" masses. You have to think about the fact that the concept of "negative mass" can be considered in analogy as a negative charge, and then use an extension of Newton's law for negative masses similar to the Coulomb's law. But "negative mass" remains a theoric concept that you have to specify better, because, right know it's more a problem of mental experiment than a real one.
