Gravitation

JEE Main level, single correct, medium

57 of 223

Question 57 of 223

The radii of two planets AA and BB are RR and 4R4R and their densities are ρ\rho and ρ3\dfrac{\rho }{3} respectively. The ratio of acceleration due to gravity at their surfaces (gA:gB)\left({g}_{A}:{g}_{B}\right) will be

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Answer: (D) 3:43:4

Acceleration due to gravity is written as g=GMr2=G×43×π×r3×ρr2=(4πG3)ρrg=\dfrac{GM}{{r}^{2}} =\dfrac{G\times \dfrac{4}{3}\times \pi \times {r}^{3}\times \rho }{{r}^{2}} =\left(\dfrac{4\pi G}{3}\right)\rho r
Given:
rA=RrB=4RρA=ρρB=ρ3{r}_{A}=R {r}_{B}=4R {\rho }_{A}=\rho {\rho }_{B}=\dfrac{\rho }{3}
Hence, the ratio is
gAgB=R×ρ4R×ρ3=34\dfrac{{g}_{A}}{{g}_{B}}=\dfrac{R\times \rho }{4R\times \dfrac{\rho }{3}}=\dfrac{3}{4}.

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