Question

A body is orbiting around the earth at mean radius $9$ times as great as the orbit of a geostationary satellite. In how many days it will complete one revolution around the earth and what is its angular velocity?

Answer 1

$\left(T_1^2/T_2^2\right)=\left(R_1^3/R_2^3\right)$
$\Rightarrow T_2^2=\left(\frac{R_2}{R_1}\right)T_1^2\Rightarrow T_2={\left(\frac{R_2}{R_1}\right)}^{3/2}{T}_1$
For the geostationary satellite, $T_1=1day(24hrs.)$
$\therefore T_2={\left(\frac{9R}{R}\right)}^{\frac{3}{2}}=27days$
So it will complete one revolution in $27$ days
$\omega =2\pi /T_2=\frac{2\pi }{27\times 24\times 3600}rad/s$
$2.693\times {10}^{-6}rad/sec$.