Two planets have the same surface gravity, but planet B has twice the radius of planet A. If planet A has mass m, what is the mass of planet B?
A) m/4
B) $\mathrm { m } / \sqrt { 2 }$
C) 4m
D) m
E) $m \sqrt { 2 }$

Question

Two planets have the same surface gravity, but planet B has twice the radius of planet A. If planet A has mass m, what is the mass of planet B? 
A)  m/4 
B) $\mathrm { m } / \sqrt { 2 }$
C)  4m 
D)  m 
E) $m \sqrt { 2 }$

Quizplus · Accepted Answer

The surface gravity $g$ is given by $g = \frac{Gm}{r^2}$, where $G$ is the gravitational constant, $m$ is the mass of the planet, and $r$ is the radius of the planet. If planet B has twice the radius of planet A, its radius is $2r$. For the surface gravity to be the same on both planets, the mass of planet B must be such that the increase in radius (which would decrease the gravity) is offset by an increase in mass. Setting the gravities equal gives $\frac{Gm}{r^2} = \frac{Gm_B}{(2r)^2}$, solving for $m_B$ gives $m_B = 4m$, indicating that planet B must have four times the mass of planet A to have the same surface gravity.

Two Planets Have the Same Surface Gravity, but Planet B m/2\mathrm { m } / \sqrt { 2 }m/2​

Two Planets Have the Same Surface Gravity, but Planet B $\mathrm { m } / \sqrt { 2 }$