A particle moves in a circular orbit under the action of a central attractive force inversely proportional to the distance `r`. The speed of the particle is

Options

A. propotional to `r^(2)`
B. independent of `r`
C. propotional to `r`
D. propotional to `1//r`

Explanation

To solve the problem, we need to analyze the motion of a particle moving in a circular orbit under the influence of a central attractive force that is inversely proportional to the distance \( r \).

### Step-by-Step Solution:

1. **Understanding the Central Force**:
The central attractive force \( F \) acting on the particle is given by:
\[ F = \frac{k}{r} \]
where \( k \) is a constant and \( r \) is the distance from the center of the circular orbit.

2. **Centripetal Force Requirement**:
For a particle moving in a circular path of radius \( r \) with speed \( v \), the required centripetal force \( F_c \) is given by:
\[ F_c = \frac{mv^2}{r} \]
where \( m \) is the mass of the particle.

3. **Setting the Forces Equal**:
Since the central force provides the necessary centripetal force, we can set these two forces equal:
\[ \frac{mv^2}{r} = \frac{k}{r} \]

4. **Simplifying the Equation**:
We can multiply both sides by \( r \) (assuming \( r \neq 0 \)):
\[ mv^2 = k \]

5. **Solving for Speed \( v \)**:
Rearranging the equation gives us:
\[ v^2 = \frac{k}{m} \]
Taking the square root of both sides, we find:
\[ v = \sqrt{\frac{k}{m}} \]

6. **Conclusion**:
The speed \( v \) of the particle is independent of the radius \( r \) and depends only on the constant \( k \) and the mass \( m \) of the particle.

### Final Answer:
The speed of the particle is given by:
\[ v = \sqrt{\frac{k}{m}} \]