Correct formula for height of a satellite from earths surface is _______.

\(\Rightarrow \frac{\mathrm{GM}}{(\mathrm{R}+\mathrm{h})}=\mathrm{v}^2 \ldots(1)\)

\(\Rightarrow \mathrm{v}=(\mathrm{R}+\mathrm{h}) \omega\)

\(\Rightarrow \mathrm{v}=(\mathrm{R}+\mathrm{h}) \frac{2 \pi}{\mathrm{T}} .\)

\(\Rightarrow \frac{\mathrm{GM}}{\mathrm{R}^2}=\mathrm{g}\)

\(\Rightarrow \mathrm{GM}=\mathrm{gR}^2 .\)

Put value from \((2) \& (3) in eq. (1)\)

\(\Rightarrow \frac{\mathrm{gR}^2}{(\mathrm{R}+\mathrm{h})}=(\mathrm{R}+\mathrm{h})^2\left(\frac{2 \pi}{\mathrm{T}}\right)^2\)

\(\Rightarrow \frac{\mathrm{T}^2 \mathrm{R}^2 \mathrm{~g}}{(2 \pi)^2}=(\mathrm{R}+\mathrm{h})^3\)

\(\Rightarrow\left[\frac{\mathrm{T}^2 \mathrm{R}^2 \mathrm{~g}}{(2 \pi)^2}\right]^{1 / 3}-\mathrm{R}=\mathrm{h}\)