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 $R_2=\left(R_1\right){\left(\frac{T_2}{T_1}\right)}^{3/2}$
$=\left(R_1\right){\left(\frac{16}{2}\right)}^{3/2}$
$=4R_1=4R\left(given{R}_1=R\right)\dots \left(i\right)$
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 $v_{o2}=\frac{1}{2}{v}_{o1}=\frac{1}{2}{v}_o$