Question

The time periods of revolution of Earth and Jupiter are in the ratio 1:12. Then the radii of Jupiter and Earth are in the ratio .

• A. $1:24\sqrt{3}$
• B. $\sqrt[3]{3144}$$:1$
• C. $24\sqrt{3}:1$
• D. $1:$$\sqrt[3]{3144}$

Answer 1

The correct option is B $\sqrt[3]{3144}$$:1$

Given:
The ratio of the time periods of revolution of the planets Earth and Jupiter, ${\text{T}}_{\text{E}}:{\text{T}}_{\text{J}}=1:12$
Let the ratio of the radii of Earth and Jupiter be ${\text{R}}_{\text{E}}:{\text{R}}_{\text{J}}$.
According to the Kepler's third law of planetary motion, the cube of the average distance from the Sun to the planet (radius of the planet's orbit) is directly proportional to the square of the time period of revolution of the planet.
$⟹{\text{R}}^3\propto {\text{T}}^2$
$⟹\frac{({\text{R}}_{\text{E}}{)}^3}{({\text{R}}_{\text{J}}{)}^3}=\frac{({\text{T}}_{\text{E}}{)}^2}{({\text{T}}_{\text{J}}{)}^2}$

$⟹\frac{({\text{R}}_{\text{E}}{)}^3}{({\text{R}}_{\text{J}}{)}^3}=\frac{(1{)}^2}{(12{)}^2}=\frac{1}{144}$

$\therefore \frac{{\text{R}}_{\text{E}}}{{\text{R}}_{\text{J}}}=(\frac{1}{144}{)}^{1/3}=\frac{1}{(144{)}^{1/3}}$
$⟹{\text{R}}_{\text{J}}:{\text{R}}_{\text{E}}={144}^{1/3}:1$
${\text{R}}_{\text{J}}:{\text{R}}_{\text{E}}=$$\sqrt[3]{3144}$$:1$