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Two planets are at distance R1 and R2 fr...

Two planets are at distance `R_1 and R_2` from the Sun. Their periods are `T_1 and T_2` then `(T_1/T_2)^2` is equal to

A

`R_1/R_2`

B

`(R_1/R_2)^2`

C

`(R_1/R_2)^3`

D

`(R_2/R_1)^3`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we will use Kepler's Third Law of planetary motion, which states that the square of the period of orbit of a planet is directly proportional to the cube of the semi-major axis of its orbit. ### Step-by-Step Solution: 1. **Understanding the Relationship**: According to Kepler's Third Law, we have the relationship: \[ T^2 \propto R^3 \] This means that for two planets orbiting the Sun at distances \( R_1 \) and \( R_2 \) with periods \( T_1 \) and \( T_2 \), we can write: \[ \frac{T_1^2}{R_1^3} = \frac{T_2^2}{R_2^3} \] 2. **Rearranging the Equation**: From the above relationship, we can rearrange to express the ratio of the squares of the periods: \[ \frac{T_1^2}{T_2^2} = \frac{R_1^3}{R_2^3} \] 3. **Taking the Square Root**: To find the ratio of the periods, we take the square root of both sides: \[ \left(\frac{T_1}{T_2}\right)^2 = \frac{R_1^3}{R_2^3} \] 4. **Final Expression**: Therefore, we can express the ratio of the squares of the periods in terms of the distances: \[ \left(\frac{T_1}{T_2}\right)^2 = \frac{R_1^3}{R_2^3} \] ### Conclusion: Thus, the final answer is: \[ \left(\frac{T_1}{T_2}\right)^2 = \frac{R_1^3}{R_2^3} \]
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