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

Explanation

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

### Step 1: Understand the relationship between the time period and the radius of the orbit.
The time period \( T \) of a satellite in a circular orbit is given by Kepler's third law, which states that:
\[ T^2 \propto r^3 \]
This means that the square of the time period is directly proportional to the cube of the radius of the orbit.

### Step 2: Set up the relationship for both satellites.
Let the radius of the first satellite's orbit be \( R \) and its time period be \( T_1 \). For the second satellite, which is in an orbit of radius \( 1.02R \), let the time period be \( T_2 \).

According to Kepler's third law:
\[ T_1^2 \propto R^3 \]
\[ T_2^2 \propto (1.02R)^3 \]

### Step 3: Express the time periods in terms of the radius.
From the proportionality, we can write:
\[ T_1^2 = kR^3 \quad \text{(1)} \]
\[ T_2^2 = k(1.02R)^3 = k(1.02^3R^3) \quad \text{(2)} \]
where \( k \) is a constant.

### Step 4: Calculate \( T_2 \) in terms of \( T_1 \).
From equation (1) and (2):
\[ T_2^2 = 1.02^3 kR^3 \]
Taking the square root to find \( T_2 \):
\[ T_2 = \sqrt{1.02^3} \cdot \sqrt{kR^3} = \sqrt{1.02^3} \cdot T_1 \]

### Step 5: Calculate \( \sqrt{1.02^3} \).
Calculating \( 1.02^3 \):
\[ 1.02^3 = 1.061208 \]
Thus,
\[ T_2 \approx 1.0305 T_1 \]

### Step 6: Find the percentage difference in the time periods.
The percentage difference in time periods can be calculated as:
\[ \text{Percentage difference} = \frac{T_2 - T_1}{T_1} \times 100\% \]
Substituting \( T_2 \):
\[ \text{Percentage difference} = \frac{(1.0305 T_1 - T_1)}{T_1} \times 100\% \]
\[ = (1.0305 - 1) \times 100\% \]
\[ = 0.0305 \times 100\% \approx 3.05\% \]

### Final Answer:
The percentage difference in the time periods of the two satellites is approximately **3.05%**.

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