The density of the core of a planet is $p_{1}$ and that of the outer part(shell) is $p_{2}$ . The radii of the core and that of the planet are R and 2R respectively. The acceleration due to gravity at the surface of the planet is same as at a depth R. Find the ratio of $\frac{ρ_{1}}{ρ_{2}}$

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1 : 3

2 : 5

5 : 3

7 : 3

answer is D.

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Detailed Solution

Let $m_{1}$ be the mass of the core and $m_{2}$ be the mass of outer shell $g_{1} = g_{2}$

$\Rightarrow \frac{{Gm}_{1}}{R^{2}} = \frac{G (m_{2} + m_{1})}{4 R^{2}}$

$\Rightarrow 4 m_{1} = m_{1} + m_{2} \Rightarrow 3 m_{1} = m_{2}$

$\Rightarrow 3 [\frac{4}{3} π R^{3} ρ_{1}] = \frac{4}{3} π [8 R^{3} - R^{3}] ρ_{2}$

$\Rightarrow 3 ρ_{1} = 7 ρ_{2} \Rightarrow ρ_{1} : ρ_{2} = 7 : 3$

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