Question

The distance between the Sun and Earth is \( R \). The duration of a year if the distance between the Sun and Earth becomes \( 3R \) will be:

• A. \( \sqrt{3} \) years
• B. \( 3 \) years
• C. \( 9 \) years
• D. \( 3\sqrt{3} \) years

Hint:

Kepler’s third law states: \[ T^2 \propto R^3 \] which helps determine orbital periods when the radius changes.

Answer 1

The Correct Option is D

Step 1: Kepler's third law
\[ T^2 \propto R^3 \] Step 2: Calculate the new time period
\[ \left( \frac{T_2}{T_1} \right)^2 = \left( \frac{R_2}{R_1} \right)^3 \] Substitute values: \[ T_2 = \left( \frac{3R}{R} \right)^{3/2} \times 1 \] \[ = 3\sqrt{3} \text{ years} \] The new time period is \( 3\sqrt{3} \) years.