Two satellites `A` and `B` of the same mass are revolving around the earth in the concentric circular orbits such that the distance of satellite `B` from the centre of the earth is thrice as compared to the distance of the satellite `A` from the centre of the earth. The ratio of the centripetal force acting on `B` as compared to that on `A` is

Options

A. `1/3`
B. `3`
C. `1/9`
D. `1/(sqrt(3))`

Explanation

To find the ratio of the centripetal force acting on satellite B compared to that on satellite A, we can follow these steps:

### Step 1: Define the variables
Let:
- \( m \) = mass of satellite A = mass of satellite B (since they are the same)
- \( R_A \) = distance of satellite A from the center of the Earth
- \( R_B \) = distance of satellite B from the center of the Earth

According to the problem, \( R_B = 3 R_A \).

### Step 2: Write the expression for gravitational force
The gravitational force \( F \) acting on a satellite in orbit is given by the formula:
\[ F = \frac{G \cdot M \cdot m}{R^2} \]
where:
- \( G \) = universal gravitational constant
- \( M \) = mass of the Earth
- \( m \) = mass of the satellite
- \( R \) = distance from the center of the Earth to the satellite

### Step 3: Calculate the gravitational force for both satellites
For satellite A:
\[ F_A = \frac{G \cdot M \cdot m}{R_A^2} \]

For satellite B:
\[ F_B = \frac{G \cdot M \cdot m}{R_B^2} \]

### Step 4: Substitute \( R_B \) in terms of \( R_A \)
Since \( R_B = 3 R_A \), we can substitute this into the equation for \( F_B \):
\[ F_B = \frac{G \cdot M \cdot m}{(3 R_A)^2} = \frac{G \cdot M \cdot m}{9 R_A^2} \]

### Step 5: Find the ratio of the forces
Now, we can find the ratio of the centripetal forces (which are equal to the gravitational forces) acting on satellites B and A:
\[ \frac{F_B}{F_A} = \frac{\frac{G \cdot M \cdot m}{9 R_A^2}}{\frac{G \cdot M \cdot m}{R_A^2}} = \frac{1}{9} \]

### Conclusion
Thus, the ratio of the centripetal force acting on satellite B compared to that on satellite A is:
\[ \frac{F_B}{F_A} = \frac{1}{9} \]

### Final Answer
The final answer is \( \frac{1}{9} \).

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