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

The correct option is B its radius of orbit is 4R and orbital velocity is $\frac{v_o}{2}$.

According to the Kepler's third law $T^2\propto R^3$

$\therefore \frac{T_1}{T_2}=\left({\frac{R_1}{R_2}}^{3/2}\right)$

or $R_2=\left(R_1\right){\left(\frac{T_2}{T_1}\right)}^{3/2}$

$=\left(R_1\right){\left(\frac{16}{2}\right)}^{2/3}=4R_1=4R$ (given $R_1=R$) ....(i)

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 (i))

or $v_{o2}=\frac{1}{2}{v}_{o1}=\frac{1}{2}{v}_o$