Two satellite of mass `m` and `9 m` are orbiting a planet in orbits of radius `R`. Their periods of revolution will be in the ratio of

Options

A. `1:9`
B. `1:3`
C. `1:1`
D. `3:1`

Answer

To solve the problem of finding the ratio of the periods of revolution of two satellites of masses `m` and `9m` orbiting a planet at the same radius `R`, we can follow these steps:

### Step 1: Understand the relationship between period, radius, and mass
The period of revolution \( T \) of a satellite in orbit is given by the formula:
\[ T = 2\pi \sqrt{\frac{R^3}{GM}} \]
where:
- \( T \) is the period of revolution,
- \( R \) is the radius of the orbit,
- \( G \) is the gravitational constant,
- \( M \) is the mass of the planet.

### Step 2: Analyze the effect of mass of the satellite
From the formula, we can see that the period \( T \) depends on the radius \( R \) and the mass of the planet \( M \), but it does not depend on the mass of the satellite. Therefore, whether the satellite has mass \( m \) or \( 9m \), the period remains the same.

### Step 3: Calculate the periods for both satellites
Let’s denote the period of the first satellite (mass \( m \)) as \( T_1 \) and the period of the second satellite (mass \( 9m \)) as \( T_2 \). According to the formula:
\[ T_1 = 2\pi \sqrt{\frac{R^3}{GM}} \]
\[ T_2 = 2\pi \sqrt{\frac{R^3}{GM}} \]

### Step 4: Find the ratio of the periods
Now, we can find the ratio of the periods:
\[ \frac{T_1}{T_2} = \frac{2\pi \sqrt{\frac{R^3}{GM}}}{2\pi \sqrt{\frac{R^3}{GM}}} = 1 \]

### Conclusion
The periods of revolution of the two satellites are the same, hence the ratio of their periods is:
\[ \frac{T_1}{T_2} = 1:1 \]

### Final Answer
The ratio of the periods of revolution of the two satellites is \( 1:1 \).

---