Question

Suppose two planets A and B revolve around a Sun in the galaxy. The semi-major axis of A and B are 1 and 5 AU (astronomical unit) respectively. If the period of revolution of A is 1 year, the period of revolution of B is (A) 1 year

• A. 5 years
• B. 11 years
• C. 25 years
• D. 125 years
• E.

Hint:

Use \(T = a^{3/2}\) directly.

Answer 1

The Correct Option is C

Concept: Kepler’s Third Law
For planets revolving around the Sun: \[ T^2 \propto a^3 \] where $T$ is time period and $a$ is semi-major axis (orbital radius).

Step 1: Apply proportional relation
Taking Earth as reference: \[ \left(\frac{T}{1}\right)^2 = \left(\frac{a}{1}\right)^3 \]

Step 2: Substitute given value
If $a^3 = 125$: \[ T^2 = 125 \]

Step 3: Calculate time period
\[ T = \sqrt{125} = \sqrt{25 \times 5} = 5\sqrt{5} \approx 11 \text{ years} \] Final Answer:
\[ \boxed{11 \text{ years}} \] Note:
Time period increases rapidly with orbital radius due to cubic dependence.