The ratio of distance of two satellites from the centre of earth is `1:4`. The ratio of their time periods of rotation will be

Options

A. `1:4`
B. `4:1`
C. `1:8`
D. `8:1`

Explanation

To find the ratio of the time periods of two satellites based on their distances from the center of the Earth, we can follow these steps:

### Step 1: Understand the relationship between time period and distance
The time period \( T \) of a satellite in orbit is related to its distance \( r \) from the center of the Earth by the formula:
\[ T \propto r^{3/2} \]
This means that the square of the time period is proportional to the cube of the distance from the center of the Earth.

### Step 2: Set up the ratio of distances
Let the distances of the two satellites from the center of the Earth be \( r_1 \) and \( r_2 \). According to the problem, the ratio of their distances is given as:
\[ \frac{r_1}{r_2} = \frac{1}{4} \]
This implies \( r_1 = r \) and \( r_2 = 4r \) for some distance \( r \).

### Step 3: Apply the time period formula
Using the relationship \( T \propto r^{3/2} \), we can express the time periods \( T_1 \) and \( T_2 \) for the two satellites:
\[ T_1 \propto r_1^{3/2} \]
\[ T_2 \propto r_2^{3/2} \]

### Step 4: Calculate the ratio of time periods
Now, we can write the ratio of the time periods:
\[ \frac{T_1}{T_2} = \frac{r_1^{3/2}}{r_2^{3/2}} = \left(\frac{r_1}{r_2}\right)^{3/2} \]
Substituting the ratio of distances:
\[ \frac{T_1}{T_2} = \left(\frac{1}{4}\right)^{3/2} \]

### Step 5: Simplify the expression
Calculating \( \left(\frac{1}{4}\right)^{3/2} \):
\[ \left(\frac{1}{4}\right)^{3/2} = \frac{1^{3/2}}{4^{3/2}} = \frac{1}{(2^2)^{3/2}} = \frac{1}{2^{3}} = \frac{1}{8} \]

### Conclusion
Thus, the ratio of the time periods of the two satellites is:
\[ \frac{T_1}{T_2} = \frac{1}{8} \]
This means the time period of the first satellite is to the time period of the second satellite as 1 is to 8.

### Final Answer
The ratio of their time periods of rotation is \( 1:8 \).

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