Question

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

• A. \(\sqrt3\) years
• B. 3 years
• C. 9 years
• D. 3\(\sqrt3\) years

Answer 1

The Correct Option is D

To determine the new duration of a year when the distance between the Sun and Earth increases to \(3R\), we need to apply Kepler's Third Law of Planetary Motion. This law states:

T^2 \propto R^3

Where \( T \) is the orbital period (duration of a year) and \( R \) is the average distance from the Sun.

Given: \( R_{\text{initial}} = R \) and \( T_{\text{initial}} = 1 \text{ year} \). At this distance, the Earth's orbital period is 1 year.

If the new distance \( R_{\text{new}} = 3R \), we need to find \( T_{\text{new}} \).

According to Kepler's Third Law:

\frac{T_{\text{new}}^2}{T_{\text{initial}}^2} = \frac{R_{\text{new}}^3}{R_{\text{initial}}^3}

\frac{T_{\text{new}}^2}{1^2} = \frac{(3R)^3}{(R)^3}

Simplifying the right side, we get:

\frac{T_{\text{new}}^2}{1} = \frac{27R^3}{R^3} = 27

Thus, T_{\text{new}}^2 = 27

Taking the square root on both sides:

T_{\text{new}} = \sqrt{27} = 3\sqrt{3} \text{ years}

Therefore, the duration of the year when the distance between the Sun and Earth becomes \(3R\) is 3\(\sqrt{3}\) years.

This confirms that the correct answer is: 3 \(\sqrt{3}\) years.