Gravitation takes a toll

A very large number of small particles forms a spherical cloud. Initially, they are at rest, having a uniform mass density per unit volume \(\rho_0\), and occupy a region of radius \(r_0\). The cloud collapses due to gravitation; the particles do not interact with each other in any other way.

How much time passes until the cloud collapses entirely?

\[ t = \sqrt{ \dfrac \pi{96 G \rho_0}} \; . \]

The correct ans is of the form \(kt\) where \(k\) is a rational number. Report your answer as the value of \(k\) upto 2 decimals.

Source: A test.

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