Question

The distance of two planets from the sun are $ {10}^{13}$ m and $ {10}^{12}$ m respectively. The ratio of time periods of the planets is

• A. $\sqrt{10}$
• B. $ 10 \sqrt{10}$
• C. 10
• D. $ \frac{1}{\sqrt{10}}$

Answer 1

The Correct Option is B

The problem requires us to find the ratio of the time periods of two planets orbiting the Sun, given their distances from the Sun. We will use Kepler's Third Law of Planetary Motion to solve this question.

Kepler's Third Law states:

{T^2 \propto R^3}

where T is the time period of the planet, and R is the average distance from the Sun.

For two planets with distances R_1 and R_2 from the Sun and time periods T_1 and T_2, the law can be expressed as:

\left(\frac{T_1}{T_2}\right)^2 = \left(\frac{R_1}{R_2}\right)^3

We are given:

• R_1 = 10^{13} \, \text{m}
• R_2 = 10^{12} \, \text{m}

Plug these values into the equation:

\left(\frac{T_1}{T_2}\right)^2 = \left(\frac{10^{13}}{10^{12}}\right)^3

This simplifies to:

\left(\frac{T_1}{T_2}\right)^2 = (10)^3

Calculate further:

\left(\frac{T_1}{T_2}\right)^2 = 1000

Taking the square root on both sides gives:

\frac{T_1}{T_2} = \sqrt{1000}

Thus:

\frac{T_1}{T_2} = 10 \sqrt{10}

Therefore, the correct answer is the ratio of the time periods, 10 \sqrt{10}.