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# The mean radius of earth is R, its angular velocity on its own axis is 'w' and acceleration due to gravity is g. What will be the radius of orbit of a geostationary satellite ?

$(a)\;\bigg(\frac{R^2}{w^2}\bigg)^{1/3} \quad (b)\;\bigg(\frac{Rg}{w^2}\bigg)^{1/3} \quad (c)\;\bigg(\frac{R^2w^2}{g}\bigg)^{1/3} \quad (d)\;\bigg(\frac{R^2g}{w}\bigg)^{1/3}$

Can you answer this question?

Satellite orbitary with a orbital velocity $v_0$, radius of orbit r
$v_0=\sqrt {\large\frac{GM}{r}}$
$\qquad= \sqrt {\large\frac{gR^2}{r}}$
Time period $=\large\frac{2 \pi r}{v_0}$
$\qquad= \large\frac{2 \pi r}{\bigg(\Large\frac{gR^2}{r}\bigg)^{1/2}}$
$\qquad= \large\frac{2 \pi r^{3/2}}{\sqrt {gR^2}}$
$T=\large\frac{2 \pi}{w}$
$\therefore \large\frac{2 \pi}{w}=\frac{2 \pi r^{3/2}}{\sqrt {gR^2}}$
$r^{3/2} =\large\frac{\sqrt {gR^2}}{w}$
$r^3= \large\frac{gR^2}{w^2}$
$r= \bigg( \large\frac{gR^2}{w}\bigg)^{1/3}$
Hence d is the correct answer.
answered Aug 24, 2013 by
edited Jun 30, 2014