Question

The distance of planet Jupiter from the Sun is 5.2 times that of Earth. Find the period of revolution of Jupiter around the Sun.

Answer 1

Hint: To solve this question, we have to use Kepler’s third law of planetary motion, the square of time period of revolution of a planet is directly proportional to the cube of its distance from the sun.
Mathematically, the above statement can be written as:
${T^2}\alpha {R^3}$
So,
$\dfrac{{T_{Jupiter}^2}}{{T_{Earth}^2}} = \dfrac{{R_{Jupiter}^3}}{{R_{Earth}^3}}$

Complete step by step solution:
The German Astronomer Johannes Kepler gave Kepler’s three laws of Planetary motion, which as the name suggests gives the laws governing the revolution of planets around the Sun.
Now, his third law of planetary motion states that
“The square of time period of revolution of a planet is directly proportional to the cube of its distance from the sun”.
Mathematically, the above statement can be written as:
${T^2}\alpha {R^3}$
Assuming the Time period of revolution of Earth to be T_EARTH and its distance from the Sun to be R_EARTH.
Similarly, the Time period of revolution of Jupiter will be T_JUPITER and its distance from the Sun will be R_JUPITER.

Now, since ${T^2}\alpha {R^3}$,
So,
$\dfrac{{T_{Jupiter}^2}}{{T_{Earth}^2}} = \dfrac{{R_{Jupiter}^3}}{{R_{Earth}^3}}$
But it is given in the question that,
${R_{Jupiter}} = 5.2{R_{Earth}}$
Inserting the value of RJupiter in the above equation,
We get,
\[ \Rightarrow \dfrac{{T_{Jupiter}^2}}{{T_{Earth}^2}} = \dfrac{{{{\left( {5.2{R_{Earth}}} \right)}^3}}}{{R_{Earth}^3}}\]
\[ \Rightarrow \dfrac{{T_{Jupiter}^2}}{{T_{Earth}^2}} = \dfrac{{140.6R_{Earth}^3}}{{R_{Earth}^3}}\]
\[ \Rightarrow \dfrac{{T_{Jupiter}^2}}{{T_{Earth}^2}} = 140.6\]
\[ \Rightarrow T_{Jupiter}^2 = 140.6T_{Earth}^2\]
\[ \Rightarrow {T_{Jupiter}} = \sqrt {140.6T_{Earth}^2} \]
\[ \Rightarrow {T_{Jupiter}} = 11.86{T_{Earth}}\]

Note: This is a tricky question often asked in competitive examinations. Solving such questions requires in-depth knowledge of the Gravitation. Chapter and Kepler’s Laws of Planetary Motion Also, this is a calculation intensive problem. Silly mistakes must be avoided at all costs.