When the distance between two objects is reduced to half, the gravitational force between them increases by a factor of four. This is because the gravitational force follows the inverse-square law, meaning it is inversely proportional to the square of the distance between the objects. Mathematically, if the original distance is rrr, and the force is FFF, then reducing the distance to r2\frac{r}{2}2r results in a new force F′F'F′ given by:
F′=Gm1m2(r/2)2=Gm1m2r2/4=4×Gm1m2r2=4FF'=G\frac{m_1m_2}{(r/2)^2}=G\frac{m_1m_2}{r^2/4}=4\times G\frac{m_1m_2}{r^2}=4FF′=G(r/2)2m1m2=Gr2/4m1m2=4×Gr2m1m2=4F
Thus, the gravitational force becomes four times greater when the distance is halved
. This relationship is a direct consequence of the inverse-square law, which applies to gravity and other forces spreading out uniformly in three- dimensional space
