Two planets have radii R1 and R2 and densities rho1 and rho2 respectively. What is the ratio of the acceleration due to gravity on their surfaces?

A. R_1 _1R_2 _2 — Since g = GM/R^2 and M = density * volume = rho * (4/3)pi*R^3, we find g is proportional to rho * R. Thus, the ratio is (R1*rho1)/(R2*rho2).

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$Two planets have radii R_{1} and R_{2} and densities ρ_{1} and ρ_{2} respectively. What is the ratio of the acceleration due to gravity g_{1} / g_{2} on their surfaces?$

A. $\frac{R _{1} ρ _{1}}{R _{2} ρ _{2}}$
B. $\frac{R _{1} ρ _{2}}{R _{2} ρ _{1}}$
C. $\frac{R _{1}^{2} ρ _{1}}{R _{2}^{2} ρ _{2}}$
D. $\frac{ρ _{1}}{ρ _{2}}$

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Correct answer: A — $\frac{R _{1} ρ _{1}}{R _{2} ρ _{2}}$

Solution

Since g = \frac{GM}{R ^{2}} and M = ρ \cdot \frac{4}{3} π R^{3}, we get g = \frac{Gρ \frac{4}{3} π R ^{3}}{R ^{2}} = \frac{4}{3} G π ρR . Thus, g \propto ρR .

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Two planets have radii R1​ and R2​ and densities ρ1​ and ρ2​ respectively. What is the ratio of the acceleration due to gravity g1​/g2​ on their surfaces?

Solution

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$Two planets have radii R_{1} and R_{2} and densities ρ_{1} and ρ_{2} respectively. What is the ratio of the acceleration due to gravity g_{1} / g_{2} on their surfaces?$