Question

A planet has the same average density as Earth but only $1/8$ the mass of Earth. If $g_p$ and $g_e$ are the surface gravities on the planet and Earth, then $\frac{g_p}{g_e} =$ ............ (Specify your answer up to one digit after decimal.)
 

Hint:

When density is constant, mass & radius scale as $M \propto R^3$.

Answer 1

Correct Answer: 0.4

Step 1: Relate mass and radius using density.
Same density implies $\frac{M_p}{R_p^3} = \frac{M_e}{R_e^3}$. Given $M_p = \frac{1}{8} M_e$, we get $R_p^3 = \frac{1}{8} R_e^3$, so $R_p = \frac{1}{2} R_e$.

Step 2: Use gravitational formula.
$g = \frac{GM}{R^2}$. Compute ratio:
$\frac{g_p}{g_e} = \frac{\frac{GM_p}{R_p^2}}{\frac{GM_e}{R_e^2}} = \frac{M_p}{M_e}\cdot\frac{R_e^2}{R_p^2}$.

Step 3: Substitute values.
$\frac{g_p}{g_e} = \frac{1}{8} \cdot \frac{R_e^2}{(R_e/2)^2} = \frac{1}{8}\cdot 4 = \frac{1}{2} = 0.5$.