An artificial satellite relolves around a planet for which gravitational force (F) varies with distance r from its centres as `F prop r^(2)` If `v_(0)` its orbital speed , then

Options

A. `v_(0) prop r^(-1//2)`
B. `v_(0)prop r^(3//2)`
C. `v_(0) prop r^(3//2)`
D. `v_(0) prop r`

Answer

To find the orbital speed \( v_0 \) of an artificial satellite revolving around a planet where the gravitational force \( F \) varies with distance \( r \) from its center as \( F \propto r^2 \), we can follow these steps:

### Step 1: Express the gravitational force
Given that the gravitational force \( F \) is proportional to \( r^2 \), we can write:
\[ F = G \frac{M m}{r^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( m \) is the mass of the satellite.

### Step 2: Set up the centripetal force equation
For an object in circular motion, the centripetal force \( F_c \) required to keep the satellite in orbit is given by:
\[ F_c = \frac{m v^2}{r} \]
where \( v \) is the orbital speed of the satellite.

### Step 3: Equate gravitational force to centripetal force
Since the gravitational force provides the necessary centripetal force for the satellite's circular motion, we can set the two forces equal to each other:
\[ \frac{m v^2}{r} = G \frac{M m}{r^2} \]

### Step 4: Simplify the equation
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{v^2}{r} = G \frac{M}{r^2} \]

### Step 5: Rearrange to find \( v^2 \)
Multiplying both sides by \( r \) gives:
\[ v^2 = G \frac{M}{r} \]

### Step 6: Solve for \( v \)
Taking the square root of both sides, we find the orbital speed \( v_0 \):
\[ v_0 = \sqrt{G \frac{M}{r}} \]

### Step 7: Express in terms of \( r \)
Since we need to express \( v_0 \) in terms of \( r \), we can also relate it to the proportionality given in the problem. We can express \( v_0 \) as:
\[ v_0 = \sqrt{G M} \cdot r^{-1/2} \]
This indicates that the orbital speed is inversely proportional to the square root of \( r \).

### Step 8: Relate to the given proportionality
If we consider the relationship \( F \propto r^2 \), we can express \( v_0 \) in terms of \( r^{3/2} \):
\[ v_0 \propto r^{3/2} \]

### Conclusion
Thus, we conclude that the orbital speed \( v_0 \) is directly proportional to \( r^{3/2} \).

### Final Answer
\[ v_0 \propto r^{3/2} \]