Question

A satellite is launched in the equatorial plane in such a way that it can transmit signals upto ${60}^{\circ }$ latitude on the earth. Then the angular velocity of the satellite is :

• A. $\sqrt{\frac{GM}{8R^3}}$
• B. $\sqrt{\frac{GM}{2R^3}}$
• C. $\sqrt{\frac{GM}{4R^3}}$
• D. $\sqrt{\frac{3\sqrt{3}GM}{8R^3}}$

Answer 1

The correct option is A $\sqrt{\frac{GM}{8R^3}}$

We have,

Centripetal force $=$ Gravitational force

$\therefore mr{\omega }^2=\frac{GMm}{r^2}$

$\therefore \omega =\sqrt{\frac{GM}{r^3}}$

where, $r=R+h$

Also,

Range of projection of satellite is perpendicular to radius of Earth

$\therefore$ from figure

$sin{30}^o=\frac{R}{r}$

$\therefore r=2R$

$\therefore w=\sqrt{\frac{GM}{8R^3}}$