Question

The diameters of two planets are in the ratio 4:1 and their mean densities are in the ratio 1:2. The accelerations due to gravity on the planets are in the ratio?

Answer 1

By using:
g=GM/R²
so g₁=GM₁/R₁(gravity of planet 1)
g₂=GM₂/R₂(gravity of planet 2)
also M₁/M₂=V₁d₁/v₂d₂
AS v=4/3 π r³
M₁/M₂=V₁d₁/v₂d₂= R₁³ d₁/R₂³d₂--------equ(1)
Therefore,
g₁/g₂=R₁d₁/R₂d₂=(4/1)x(1/2) =2:1
∴The accelerations due to gravity on the planets are in the ratio of 2:1

Answer 2

Acceleration due to gravity is given by the equation g = GM/R2 (at surface)
so g1 = GM1/R12 and g2 = GM2/R22 (gravities at planets 1 and 2)
also M1/M2 = V1ρ1/ V2ρ2 = R13 ρ1/ R23 ρ2 (as V = 4/3 π r3)
Therefore, g1/g2 = R1 ρ1/ R2 ρ2 = (4/1)*(1/2) = 2:1 answer..