Two satellites are at distances of 2R and 4R respectively from the surface of earth. The ratio of their time periods will be :

Options

A. `sqrt((125)/(27))`
B. `sqrt((27)/(125))`
C. `sqrt((54)/(125))`
D. `sqrt((8)/(125))`

Explanation

To find the ratio of the time periods of two satellites located at distances of 2R and 4R from the surface of the Earth, we can follow these steps:

### Step 1: Identify the distances from the center of the Earth
The distance of the first satellite (let's call it Satellite 1) from the center of the Earth is:
\[ r_1 = R + 2R = 3R \]
The distance of the second satellite (Satellite 2) from the center of the Earth is:
\[ r_2 = R + 4R = 5R \]

### Step 2: Use Kepler's Third Law
According to Kepler's Third Law of planetary motion, the square of the time period (T) of a satellite is directly proportional to the cube of the semi-major axis (r) of its orbit:
\[ T^2 \propto r^3 \]
This can be expressed as:
\[ \frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3} \]

### Step 3: Substitute the distances into the equation
Now, substituting the values of \( r_1 \) and \( r_2 \):
\[ \frac{T_1^2}{T_2^2} = \frac{(3R)^3}{(5R)^3} \]

### Step 4: Simplify the equation
Calculating the cubes:
\[ (3R)^3 = 27R^3 \]
\[ (5R)^3 = 125R^3 \]
Thus, we have:
\[ \frac{T_1^2}{T_2^2} = \frac{27R^3}{125R^3} = \frac{27}{125} \]

### Step 5: Take the square root to find the ratio of time periods
Taking the square root of both sides gives:
\[ \frac{T_1}{T_2} = \sqrt{\frac{27}{125}} = \frac{\sqrt{27}}{\sqrt{125}} = \frac{3\sqrt{3}}{5\sqrt{5}} \]

### Final Step: Express the ratio
Thus, the ratio of the time periods of the two satellites is:
\[ T_1 : T_2 = \sqrt{27} : \sqrt{125} = 3\sqrt{3} : 5\sqrt{5} \]