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If G is the u...

Question

If G is the universal gravitational constant and $ρ$ is the uniform density of a spherical planet. Then shortest possible period of rotation around a planet can be

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Solution

The correct option is D

The fastest possible rate of rotation around a planet is that for which the gravitational force on material at the equator just barely provides the centripetal force needed for the rotation.Let M be the mass of the planet R its radius and m the mass of a particle on its surface.Then

$M R ω_{m a x}^{2} = \frac{G M m}{R^{2}}$

or $ω_{m a x} = \sqrt{\frac{G M}{R^{3}}} = \sqrt{\frac{G (\frac{4}{3} π R^{3} ρ)}{R^{3}}}$

= $2 \sqrt{\frac{π G ρ}{3}} = \frac{2 π}{T_{m i n}}$

$∴$ $T_{m i n} = \sqrt{\frac{3 π}{G ρ}}$

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If G is the universal gravitational constant and ρ is the uniform density of a spherical planet. Then shortest possible period of rotation around a planet can be

If G is the universal gravitational constant and $ρ$ is the uniform density of a spherical planet. Then shortest possible period of rotation around a planet can be