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Gravitational Force

Two planets o...

Question

Two planets of same density and with radii $R_{1}$ and $R_{2}$ such that $R_{1} = 2 R_{2}$ have gravitation fields on the surface $g_{1}$ and $g_{2}$ and escape velocities $v_{1}$ and $v_{2}$ respectively. Then find the value of $\frac{v_{1}}{v_{2}}$ .

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Solution

$g = \frac{G M}{R^{2}} = \frac{G (\frac{4}{3} π R^{3} ρ)}{R^{2}}$

$∴ g \propto R$
$∴ \frac{g_{1}}{g_{2}} = \frac{R_{1}}{R_{2}} = 2$
$v = \sqrt{2 g R} \propto \sqrt{R (R)}$ or $V \propto R$

$\frac{v_{1}}{v_{2}} = \frac{R_{1}}{R_{2}} = 2$ .

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