Question

The distances of Neptune and Saturn from the Sun are respectively ${10}^{13}$ and ${10}^{12}$ meters and their periodic times are respectively $T_n$ and $T_S$. If their orbits are assumed to be circular, the value of $\frac{T_n}{T_S}$ is :

• A. $100$
• B. $10\sqrt{10}$
• C. $\frac{1}{10\sqrt{10}}$
• D. $10$

Answer 1

Answer: (B)

$10\sqrt{10}$

Time period $T^2=\frac{4{\pi }^2}{GM}{R}^3$
i.e $T^2\propto R^3$
$\Rightarrow {\left(\frac{T_n}{T_S}\right)}^2={\left(\frac{R_n}{R_S}\right)}^3$
Given: $\frac{R_n}{R_S}=\frac{{10}^{13}}{{10}^{12}}=10$
${\left(\frac{T_n}{T_S}\right)}^2={10}^3$
$\Rightarrow \frac{T_n}{T_S}={10}^{3/2}=10\sqrt{10}$