What is orbital velocity ? Obtain an expression for it.

Definition: The minimum velocity required for an object to revolve around a planet in a circular orbit is called orbital velocity Consider an object of mass “m” revolves with velocity V0 around a planet of mass “M” of radius “R” at certain height “h” from surface of planet.
Centripetal force on object $=\frac{{\mathrm{mv}}_0^2}{\mathrm{R}+\mathrm{h}}$...(1)

Gravitational force on object $=\frac{\mathrm{GMm}}{(\mathrm{R}+\mathrm{h}{)}^2}$....(2)

(1) & (2)$\because \frac{{\mathrm{mv}}_0^2}{\mathrm{R}+\mathrm{h}}=\frac{\mathrm{GMm}}{(\mathrm{R}+\mathrm{h}{)}^2}\Rightarrow {\mathrm{V}}_0^2=\frac{\mathrm{GM}}{\mathrm{R}+\mathrm{h}}$

${\mathrm{V}}_0=\sqrt{\frac{\mathrm{GM}}{\mathrm{R}+\mathrm{h}}}$

When h is very small, the hight ‘h’ is neglected

${\mathrm{V}}_0=\sqrt{\frac{\mathrm{GM}}{\mathrm{R}}}$

We know$\mathrm{g}=\frac{\mathrm{GM}}{{\mathrm{R}}^2}\Rightarrow \mathrm{GM}={\mathrm{gR}}^2$

${\mathrm{V}}_0=\sqrt{\frac{{\mathrm{gR}}^2}{\mathrm{R}}}\Rightarrow {\mathrm{V}}_0=\sqrt{\mathrm{gR}}$