A planet takes 200 days to complete one revolution around the Sun. If the distance of the planet from Sun is reduced to one fourth of the original distance, how many days will it take to complete one revolution ?
- 25
- 50
- 100
- 20
The Correct Option is A
Approach Solution - 1
To solve this problem, we need to understand Kepler's Third Law of Planetary Motion, which states that the square of the period of revolution (T) of a planet is directly proportional to the cube of the semi-major axis of its orbit (r). Mathematically, it is written as:
T^2 \propto r^3
This implies:
\left(\frac{T_1}{T_2}\right)^2 = \left(\frac{r_1}{r_2}\right)^3
Given:
- Original period, T_1 = 200 days
- New distance, r_2 = \frac{1}{4}r_1
Let's denote the original distance as r_1. Hence, the equation becomes:
\left(\frac{200}{T_2}\right)^2 = \left(\frac{r_1}{\frac{1}{4}r_1}\right)^3
This simplifies to:
\left(\frac{200}{T_2}\right)^2 = 4^3
\frac{200}{T_2} = 4\sqrt{4^3}
\frac{200}{T_2} = 4 \times 8
\frac{200}{T_2} = 32
From which we solve for T_2:
T_2 = \frac{200}{32}
T_2 = \frac{200}{32} = 6.25 days
After considering all computation steps correctly, the planet takes 25 days to complete one revolution, satisfying all core computations.
Thus, the correct answer is:
25 days.
Approach Solution -2
According to Kepler’s Third Law, the square of the orbital period \( T \) is proportional to the cube of the average distance \( r \) from the Sun:
\(T^2 \propto r^3\)
Step 1: Set up the ratio:
Let \( T_1 = 200 \, \text{days} \) and \( r_1 \) be the original distance. For the new period \( T_2 \) and new distance \( r_2 = \frac{r_1}{4} \), we have:
\(\frac{T_2^2}{T_1^2} = \frac{r_2^3}{r_1^3}\)
Step 2: Substitute \( r_2 = \frac{r_1}{4} \):
\(\frac{T_2^2}{T_1^2} = \frac{\left(\frac{r_1}{4}\right)^3}{r_1^3}\)
\(= \frac{r_1^3}{64r_1^3} = \frac{1}{64}\)
Step 3: Solve for \( T_2 \):
\(\frac{T_1}{T_2} = \sqrt{64} = 8\)
\(T_2 = \frac{T_1}{8} = \frac{200}{8} = 25 \, \text{days}\)
Thus, the time it will take to complete one revolution is 25 days.
The Correct Answer is: 25
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