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The density of core of a planet is $ρ_{1}$ and that of outer shell is $ρ_{2}$ . The radius of core is R and that of planet is 2R. Gravitational field at outer surface of planet is same as at the surface of core. What is the ratio $\frac{ρ_{1}}{ρ_{2}}$ .

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$\frac{3}{5}$

$\frac{5}{3}$

$\frac{7}{3}$

$\frac{3}{4}$

answer is C.

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Detailed Solution

Let $m_{1}$ is mass of core and $m_{2}$ is of outer portion

$m_{1} = \frac{4}{3} {πR}^{3} ρ_{1}, m_{2} = \frac{4}{3} π [{(2 R)}^{3} - R^{3}] ρ_{2}$

Given that: $\frac{{Gm}_{1}}{R^{2}} = \frac{G (m_{1} + m_{2})}{{(2 R)}^{2}} \Rightarrow \frac{ρ_{1}}{ρ_{2}} = \frac{7}{3}$

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