If a planet rotates too fast, rocks from its surface will start flying off its surface. If density of a homogeneous planet is $ρ$ and material is not flying off its surface then its time period of rotation must be greater then

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$\sqrt{\frac{2 π}{ρ G}}$

$\sqrt{\frac{π}{2 ρ G}}$

$\sqrt{\frac{3 π}{ρ G}}$

$\sqrt{\frac{3 π}{2 ρ G}}$

answer is C.

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Detailed Solution

The rocks start flying away from the equator of the planet if
$\begin{array}{l} ω^{2} R \geq g \Rightarrow ω \geq \sqrt{\frac{g}{R}} \\ \Rightarrow \frac{2 π}{T} \geq \sqrt{\frac{g}{R}} \Rightarrow \frac{T}{2 π} \leq \sqrt{\frac{R}{g}} \\ T \leq 2 π \sqrt{\frac{R}{g}} \end{array}$
For rocks to not fly away
$\begin{array}{l} T \geq 2 π \sqrt{\frac{R}{g}} = 2 π \sqrt{\frac{R^{3}}{GM}} \\ = 2 π \sqrt{\frac{R^{2}}{G \cdot \frac{4}{3} {πR}^{3} \cdot ρ}} = \sqrt{\frac{3 π}{Gρ}} \end{array}$