Two satellites are revolving around the earth in circular orbits of same radii. Mass of one satellite is `100` times that of the other. Then their periods of revolutions are in the ratio

Options

A. `1:1`
B. `10:1`
C. `100:1`
D. `1:100`

Answer

To solve the problem, we need to determine the ratio of the periods of revolution of two satellites revolving around the Earth in circular orbits of the same radius, given that the mass of one satellite is 100 times that of the other.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two satellites, let's call them Satellite A and Satellite B.
- The mass of Satellite A is 100 times the mass of Satellite B.
- Both satellites are in circular orbits around the Earth with the same radius \( r \).

2. **Using Kepler's Third Law**:
- According to Kepler's Third Law of planetary motion, the square of the period of revolution \( T \) of a satellite is directly proportional to the cube of the semi-major axis of its orbit (which is the radius \( r \) for circular orbits).
- Mathematically, this can be expressed as:
\[ T^2 \propto r^3 \]
- This means:
\[ \frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3} \]

3. **Setting Up the Ratios**:
- Since both satellites are in orbits of the same radius, we have:
\[ r_1 = r_2 = r \]
- Therefore, we can write:
\[ \frac{T_1^2}{T_2^2} = \frac{r^3}{r^3} = 1 \]

4. **Finding the Period Ratio**:
- From the above equation, we can conclude:
\[ T_1^2 = T_2^2 \]
- Taking the square root of both sides gives:
\[ T_1 = T_2 \]
- Thus, the ratio of the periods \( T_1 : T_2 \) is:
\[ T_1 : T_2 = 1 : 1 \]

5. **Final Answer**:
- The periods of revolution of the two satellites are in the ratio \( 1 : 1 \).