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Question

The Jupiter's period of revolution around the Sun is $$12$$ times that of the Earth. Find the ratio gravitational force exerted on Earth to that on Jupiter


A
27.47
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B
127.47
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C
16.2
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D
116.2
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Solution

The correct option is A $$27.47$$
$$ T_{earth} = 2\pi \left(\dfrac{R_e^3 }{GM_S}\right)^{0.5} $$
$$ T_{jupiter} = 2\pi \left(\dfrac{R_j^3 }{GM_S}\right)^{0.5} $$
Given ratio, $$ \dfrac{T_j}{T_e} = 12 $$
$$ \Rightarrow 2\pi \left(\dfrac{R_j^3 }{GM_S}\right)^{0.5} \times \left(\dfrac{GM_S}{R_e^3 }\right)^{0.5} \times \dfrac{1}{2\pi}=12 $$
So, $$\dfrac{R_j}{R_e} = 12^{0.67} $$

$$ Gravitational\ force\ F \propto \dfrac{1}{R} $$
$$ \Rightarrow \dfrac{F_e}{F_j} = \left(\dfrac{R_j}{R_e}\right)^2 = 12^{\dfrac{4}{3}} = 27.47 $$

Physics

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