The radii of two planets are respectively R1 and R2 and their densities are respectively -1 and -2- The ratio of the accelerations due to gravity at their surfaces is

The correct option is D g_1:g_2=R_1ρ_1:R_2ρ_2 g_1g_2=GM_1R^2_1GM_2R^2_2 =Gρ _14/3π R_1 ^3R^2_1Gρ _24/3π R_2 ^3R^2_2 =R_1ρ _1R_2ρ _2 ∴ g ...

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Standard X

Physics

Acceleration Due to Gravity

The radii of ...

Question

The radii of two planets are respectively $R_{1}$ and $R_{2}$ and their densities are respectively $ρ_{1}$ and $ρ_{2}$ . The ratio of the accelerations due to gravity at their surfaces is

g_{1} : g_{2} = \frac{ρ_{1}}{R_{1}^{2}} : \frac{ρ_{2}}{R_{2}^{2}}

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g_{1} : g_{2} = R_{1} R_{2} : ρ_{1} ρ_{2}

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g_{1} : g_{2} = R_{1} ρ_{2} : R_{2} ρ_{1}

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g_{1} : g_{2} = R_{1} ρ_{1} : R_{2} ρ_{2}

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Solution

The correct option is D $g_{1} : g_{2} = R_{1} ρ_{1} : R_{2} ρ_{2}$
$\frac{g_{1}}{g_{2}} = \frac{\frac{G M_{1}}{R_{1}^{2}}}{\frac{G M_{2}}{R_{2}^{2}}}$

$= \frac{\frac{G ρ_{1} \frac{4}{3} π R_{1}^{3}}{R_{1}^{2}}}{\frac{G ρ_{2} \frac{4}{3} π R_{2}^{3}}{R_{2}^{2}}}$

$= \frac{R_{1} ρ_{1}}{R_{2} ρ_{2}}$

$∴ g_{1} : g_{2} = R_{1} ρ_{1} : R_{2} ρ_{2}$

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