Ranking the surface gravity of three planets

Problem

Three hypothetical planets: A (mass $M$, radius $R$, reference); B (mass $2M$, radius $2R$); C (mass $M$, radius $R/2$). Rank the surface gravitational acceleration from smallest to largest.

Solution

Set-up. Surface gravity is $g = GM/R^2$. Taking planet A as the unit, we compute the factor $M/R^2$ for each: $\begin{aligned} \text{A}\ (M,\,R):\quad & \frac{M}{R^2} = 1 \\ \text{B}\ (2M,\,2R):\quad & \frac{2M}{(2R)^2} = \frac{2M}{4R^2} = \frac{1}{2} = 0{,}5 \\ \text{C}\ (M,\,R/2):\quad & \frac{M}{(R/2)^2} = \frac{M}{R^2/4} = 4 \end{aligned}$

Ranking. Comparing $0{,}5 < 1 < 4$: $\ev{g_C > g_A > g_B}$

Radius matters more than mass: since it appears squared in the denominator, halving it (planet C) quadruples $g$, whereas doubling both mass and radius together (planet B) makes $g$ decrease.

Links

Topics: Gravitation Concepts: Weight force · Law of universal gravitation Skills: Ranking reasoning · Symbolic set-up