Consider a planet with radius \(r_{0}\) located somewhere far away in the universe. Interestingly, it is a perfect sphere of uniform density. There is an infinitely tall cylinder on the planet. It is filled with a mixture of hydrogen and helium such that the molar mass of the mixture is \(M\). The pressure , acceleration due to gravity and the temperature at the bottom of the cylinder, i.e., at the surface of the planet are \(P_{0}\) , \(g_{0}\) and \(T_{0}\) respectively.

Also the temperature of the mixture in the cylinder varies according the observed formula given by,

\[T = \frac{T_{0}}{(1 + \frac{h}{r_{0}})}\]

where h is the height from the surface of the planet i.e., the base of the cylinder.

Find the value of \(\frac{dg}{dT}\) at the center of mass of the cylinder.

**Details and Assumptions:**

The cylinder is only affected by the planet's gravitational field and not by field of any other celestial body.

The area of cross section of the cylinder is too small to be affected by any gravitational aspect.

\(r_{0} = \frac{4}{3} \times 10^3 km\)

\(M = 2.5 g/mol\)

\(T_{0} = 1000 K\)

\(g_{0} = 10 m/s^2\)

\(R = \frac{25}{3} J/K-mol \) , where \(R\) is the universal gas constant.

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