Question

Two planets revolve around the sun in elliptical orbits with their major axes in the ratio 2 : 3. Their periods of revolution around the sun are in the ratio

• A. 2 : 3
• B. 3 : 2
• C. \(2\sqrt{3} : 3\sqrt{2}\)
• D. \(2\sqrt{2} : 3\sqrt{3}\)
• E. 4 : 9

Hint:

Remember that \(x^{3/2}\) is just \(x\sqrt{x}\). So \((2/3)^{3/2} = (2/3)\sqrt{2/3}\). Multiplying inside and outside gives \(2\sqrt{2} / 3\sqrt{3}\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Kepler's Third Law of Planetary Motion relates the orbital period of a planet to the length of the semi-major axis of its orbit.

Step 2: Key Formula or Approach:
Kepler's Third Law: \(T^2 \propto a^3\), where \(T\) is the orbital period and \(a\) is the semi-major axis (or major axis).

Step 3: Detailed Explanation:

Step 1: Setup the ratio from Kepler's Law.
\[ \left(\frac{T_1}{T_2}\right)^2 = \left(\frac{a_1}{a_2}\right)^3 \]

Step 2: Substitute the given ratio of major axes \(\frac{a_1}{a_2} = \frac{2}{3}\).
\[ \left(\frac{T_1}{T_2}\right)^2 = \left(\frac{2}{3}\right)^3 = \frac{8}{27} \]

Step 3: Take the square root of both sides.
\[ \frac{T_1}{T_2} = \sqrt{\frac{8}{27}} = \frac{\sqrt{8}}{\sqrt{27}} \]
\[ \frac{T_1}{T_2} = \frac{2\sqrt{2}}{3\sqrt{3}} \]

Step 4: Final Answer:
The periods of revolution are in the ratio \(2\sqrt{2} : 3\sqrt{3}\).