SHM + Refraction = ?

Consider the Earth as a uniform sphere of mass M and radius R . Imagine a straight smooth tunnel made through the earth which connects any two points on its surface. If a particle of mass m is dropped inside this tunnel under the action of gravitation, Determine the time that the particle takes to reach the other end of the tunnel.

If the Time taken can be represented as $2\pi\sqrt{\frac{9a^2R^3}{16b^2GM}}$

a,b are positive dimensionless constants

$If \frac{b}{a} = \mu$

$\mu =$ Refractive index of a medium with respect to air.

Hence if the speed of light in this medium can be represented as $X * 10^7 m/s$

Find X

Note :

G = Gravitional constant M = Mass of Earth R =Radius of Earth

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Inspired From Aniket Sanghi 's Problem .