# The density of case of a planet is $s_1$ and that of the outer shell is $s_2$.The radius of case and that of the planet are R and 2R respectively. Gravitational acceleration at the surface of the planet is same as at a depth R. Find $\large\frac{s_1}{s_2}$

$\begin{array}{1 1} \large\frac{1}{3} \\\large\frac{7}{3} \\ \large\frac{4}{3} \\ \large\frac{8}{3} \end{array}$

Let $m$ be the mass of core and $m_2$ mass of outer shell
$g_A=g_B$ (given)
$\large\frac{Gm_1}{R^2}= \large\frac{G(m_1+m_2)}{(2R)^2}$
=> $4m_1=(m_1 +m_2)$
$4 \bigg[ \large\frac{4}{3} $$\pi R^3 s_1\bigg]=\large\frac{4}{3}$$\pi R^3 s_1+\large\frac{4}{3}$$[(2R)^3-R^3]s_2]$
$4s_1=s_1 +7s_2$
$\therefore \large\frac{s_1}{s_2}=\frac{7}{3}$
Hence b is the correct answer
edited Jul 3, 2014