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

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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.

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