Question

The time period of an earth satellite in a circular orbit of radius $R$ is $2$ days and its orbital velocity is $v_o$. If time period of another satellite in a circular orbit is $16$ days then

• A. its radius of orbit is $4R$ and orbital velocity is $v_o$
• B. its radius of orbit is $4R$ and orbital velocity is $\frac{v_o}{2}$
• C. its radius of orbit is $2R$ and orbital velocity is $v_o$
• D. its radius of orbit is $2R$ and orbital velocity is $\frac{v_o}{2}$

Answer 1

Answer: (B)

According to Keplers third law
$T^{2} \propto R^{3}$
$\therefore \frac{T_{1}}{T_{2}} = \left(\frac{R_{1}}{R_{2}}\right)^{3/2}$ or $R_{2} = \left(R_{1}\right) \left(\frac{T_{2}}{T_{1}}\right)^{3/2}$
$= \left(R_{1}\right)\left(\frac{16}{2}\right)^{3/2}$
$ = 4R_{1} = 4R \left(given R_{1} = R\right)\quad\ldots\left(i\right)$
Orbital velocity, $v_{o} = \sqrt{\frac{GM}{R}}$
$\therefore \frac{v_{o2}}{v_{o1}} = \sqrt{\frac{R_{1}}{R_{2}}}$
$= \sqrt{\frac{R_{1}}{4R_{1}}} = \frac{1}{2}$ (using $\left(i\right)$)
or $v_{o2} = \frac{1}{2}v_{o1} = \frac{1}{2}v_{o}$