Question

Let planet A and planet B be two spherical planets such that the gravitational field at the surface of A is 8.0 times stronger than the gravitational field at the surface of B. Also, the radius of planet A is double the radius of planet B. Which planet has a higher density, and by what factor?

Answer 1

(1) Express fields of the planets in terms of their mass and radii.

(2) Express their masses in terms of densities and radius.

$E_A=8E_B,\\\space\\ (1)\space G\dfrac{M_A}{R_A^2}=8G\dfrac{M_B}{R_B^2},\\\space\\ G\dfrac{M_A}{[2R_B]^2}=8G\dfrac{M_B}{R_B^2},\\\space\\ G\dfrac{M_A}{4}=8GM_B,\\\space\\ (2)\space \dfrac{\rho_A\pi[2R_B]^3}{3}=8·\rho_B·\dfrac 43\pi R_B^3,\\\space\\ \dfrac{\rho_A}{3}=8·\rho_B·\dfrac 43 ,\\\space\\ \rho_A=32\rho_B.$