Please help me!

Question

If you doubled the mass and then tripled the radius of a planet, by what factor would the acceleration due to gravity at its surface change?

Answer 1

The formula for gravitational acceleration

g = M*G/r²

where M is the mass of the planet, r is the radius of the planet, and G is the universal gravitational constant (G = 6.673×10^-11 N.m²/Kg², but you'll see that we won't need that value to answer the question).

so if we double the mass and triple the radius, our equation turns into

g_new = (2M)*G/(3r)² = (2M)*G/(9r²)

note, all we did was replace the M by 2*M and the r by 3*r (which, was squared in the parenthesis). If we want to see the factor by which the acceleration due to gravity changes, we can simply divide

g_new/g = 2MG/(9r²) ÷ MG/r² = 2MG/(9r²) × r²/(MG) = 2MGr² / (9r² MG) = 2/9 (G, M, and r² all cancel out)

The new planet has an acceleration due to gravity that's 2/9 the strength of gravity of the original planet. If the planet was earth, our acceleration would change from 9.81 m/s² —> 2.18 m/s²