The mass of a planet that has a moon whose ti | vedclass

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Question

(A) For a moon orbiting a planet in a circular orbit,the gravitational force provides the necessary centripetal force.Let $M$ be the mass of the plane

Vedclass · Accepted Answer

$4{\pi ^2}{R^3}{G^{ - 1}}{T^{ - 2}}$

Vedclass · Answer

(A) For a moon orbiting a planet in a circular orbit,the gravitational force provides the necessary centripetal force.Let $M$ be the mass of the planet and $m$ be the mass of the moon.The gravitational force is $F_g = \frac{GMm}{R^2}$.The centripetal force is $F_c = m\omega^2 R$,where $\omega = \frac{2\pi}{T}$.Equating the two forces: $m\omega^2 R = \frac{GMm}{R^2}$.Substituting $\omega = \frac{2\pi}{T}$: $m \left(\frac{2\pi}{T}\right)^2 R = \frac{GMm}{R^2}$.Simplifying the equation: $\frac{4\pi^2}{T^2} R = \frac{GM}{R^2}$.Solving for $M$: $M = \frac{4\pi^2 R^3}{GT^2} = 4\pi^2 R^3 G^{-1} T^{-2}$.

The mass of a planet that has a moon whose time period and orbital radius are $T$ and $R$ respectively can be written as

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