Question

Masses of two bodies are made four times each. How should the distance between them be changed, so that the gravitational force between them is doubled?

Answer 1

Initially, let the masses of the bodies be $m_1$ and $m_2$ and the separation between them be $r$.
The gravitational force acting between them is given by
$F=G\frac{m_1{m}_2}{r^2}$
Now the mass of both the bodies is made 4 times. Let the new separation be $r^{\prime }$.
As per the question, $F^{\prime }=2F$
$\Rightarrow G\frac{4m_{1}4m_2}{r^{\prime 2}}=2G\frac{m_1{m}_2}{r^2}$
$\Rightarrow \frac{8}{r^{\prime 2}}=\frac{1}{r^2}$
$\Rightarrow r^{\prime 2}=8r^2$
$\Rightarrow r^{\prime }=2\sqrt{2}r$