A satellite is launched into a circular orbit of radius 'R' around earth while a second satellite is launched into an orbit or radius 1.02 R. The percentage difference in the time periods of the two satellites is

Options

A. 0.7
B. `1.0`
C. 1.5
D. `3.0`

Explanation

To find the percentage difference in the time periods of two satellites in different circular orbits around Earth, we can follow these steps:

### Step 1: Understand the relationship between time period and radius
According to Kepler's third law, the square of the time period (T) of a satellite is directly proportional to the cube of the radius (R) of its orbit. Mathematically, this can be expressed as:
\[ T^2 \propto R^3 \]
This implies:
\[ T \propto R^{3/2} \]

### Step 2: Express the time period for both satellites
Let the time period of the first satellite (radius R) be \( T_1 \) and the time period of the second satellite (radius 1.02R) be \( T_2 \).
Using the proportionality, we can write:
\[ T_1 = k \cdot R^{3/2} \]
\[ T_2 = k \cdot (1.02R)^{3/2} \]
where \( k \) is a constant.

### Step 3: Calculate \( T_2 \)
Now, we can calculate \( T_2 \):
\[ T_2 = k \cdot (1.02^{3/2}) \cdot R^{3/2} \]

### Step 4: Find the ratio of the time periods
To find the percentage difference, we first need to find the ratio of \( T_2 \) to \( T_1 \):
\[ \frac{T_2}{T_1} = \frac{k \cdot (1.02^{3/2}) \cdot R^{3/2}}{k \cdot R^{3/2}} = 1.02^{3/2} \]

### Step 5: Calculate \( 1.02^{3/2} \)
Now, we can calculate \( 1.02^{3/2} \):
\[ 1.02^{3/2} = (1.02^{1.5}) \approx 1.03 \] (using a calculator or approximation)

### Step 6: Find the percentage difference in time periods
The percentage difference in time periods can be calculated as:
\[ \text{Percentage Difference} = \left( \frac{T_2 - T_1}{T_1} \right) \times 100\% \]
Substituting \( T_2 = 1.03 T_1 \):
\[ \text{Percentage Difference} = \left( \frac{1.03 T_1 - T_1}{T_1} \right) \times 100\% = (0.03) \times 100\% = 3\% \]

### Final Answer
The percentage difference in the time periods of the two satellites is **3%**.

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