Getting back to the old things #4

Consider a hypothetical planet which is a very long solid cylinder of radius \(R\) and has uniform density \(\rho\). There is no atmosphere above the surface of the planet.

Now, somewhere from its curved surface, a small object is projected radially outward such that it reaches up to a maximum distance of \(3R\) from the axis.

If the speed of projection can be represented as \(a R\sqrt{\pi \rho G \ln b}\), where \(a\) and \(b\) are positive integers and \(G\) is the universal gravitational constant, then find \(a+b\).

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