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Question

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


A
g1:g2=ρ1R21:ρ2R22
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B
g1:g2=R1R2:ρ1ρ2
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C
g1:g2=R1ρ2:R2ρ1
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D
g1:g2=R1ρ1:R2ρ2
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Solution

The correct option is D $${g}_{1}:{g}_{2}={R}_{1}{\rho}_{1}:{R}_{2}{\rho}_{2}$$
$$\dfrac{g_1}{g_2}=\dfrac {\dfrac{GM_1}{R^2_1}}{\dfrac{GM_2}{R^2_2}}$$

$$=\dfrac {\dfrac{G\rho _1\frac{4}{3}\pi R_1 ^3}{R^2_1}}{\dfrac{G\rho _2\frac{4}{3}\pi R_2 ^3}{R^2_2}}$$

$$=\dfrac{R_1\rho _1}{R_2\rho _2}$$

$$\therefore g_1:g_2=R_1\rho _1:R_2\rho _2$$

Physics

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