Universal Law Of Gravitation

The mutual gravitational attraction obeys $F=G\frac{m_1{m}_2}{r^2}$, so the force depends on the product of the two masses divided by the square of their separation. When each mass is doubled, the numerator $m_1{m}_2$ increases by a factor of $2\times 2=4$. Simultaneously doubling the separation multiplies $r^2$ by a factor of $2^2=4$ in the denominator. The new force is therefore $F^{\prime }=G\frac{(2m_1)(2m_2)}{(2r{)}^2}=G\frac{4m_1{m}_2}{4r^2}=G\frac{m_1{m}_2}{r^2}=F$, leaving the force unchanged. The option $2F$ is wrong because it accounts only for the mass increase and ignores the distance change. The option $F/2$ is wrong as it mismatches the powers of the two effects. The option $4F$ is wrong because it doubles the mass effect while ignoring the denominator entirely. This conceptual cancellation is a classic JEE proportionality test. A sanity check confirms the equal fourfold changes in numerator and denominator exactly offset.