Q.If the mass and radius of the Earth are both doubled, then the acceleration due to gravity $g'$

Every object dropped near the Earth's surface -- no matter how heavy or light -- speeds up downward at exactly the same rate. This universal acceleration, $g$, follows directly from combining Newton's law of gravitation with his second law.

Derivation. The gravitational force on a mass $m$ at Earth's surface is $F=GM_em/R_e^2$. Setting this equal to $ma$ from Newton's second law and cancelling the test mass $m$ from both sides gives

The mass $m$ cancels out completely -- so the acceleration any object experiences depends only on the mass and radius of the planet doing the pulling, never on the falling object's own mass. This matches Galileo's purely experimental finding, made 400 years before Newton's theory existed to explain it, that all objects fall at the same rate (ignoring air resistance). …