Gravitation

JEE Main level, single correct, medium

24 of 223

Question 24 of 223

The ratio of the weights of a body on Earth’s surface to that on the surface of a planet is 9:49:4 The mass of the planet is 19th\dfrac{1}{9}th of that of the Earth. If RR is the radius of the Earth, what is the radius of the planet? (Take the planets to have the same mass density)

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Answer: (B) R2\dfrac{R}{2}

Weight =m(GMR2)=m\left(\dfrac{GM}{{R}^{2}}\right)
∴94=GMeRe2×Rp2GMp∴\dfrac{9}{4}=\dfrac{G{M}_{e}}{{R}_{e}^{2}}\times \dfrac{{R}_{p}^{2}}{G{M}_{p}}
Given that, Mp=Me9{M}_{p}=\dfrac{{M}_{e}}{9}
∴94=Me(Me9)×Rp2Re2∴\dfrac{9}{4}=\dfrac{{M}_{e}}{\left(\dfrac{{M}_{e}}{9}\right)}\times \dfrac{{R}_{p}^{2}}{{R}_{e}^{2}}
Rp2=Re2×14{R}_{p}^{2}={R}_{e}^{2}\times \dfrac{1}{4}
Rp=Re2{R}_{p}=\dfrac{{R}_{e}}{2}
As, Re=R{R}_{e}=R
⇒Rp=R2\Rightarrow {R}_{p}=\dfrac{R}{2}

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