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

Options

A. proportional to ` r^(2)`
B. Independent of `r`
C. proportional to r
D. proportional to 1/r

Explanation

To solve the problem, we need to analyze the motion of a particle 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. **Identify the Forces Acting on the Particle:**
The particle is moving in a circular orbit due to a central attractive force. This force can be expressed as:
\[ F = \frac{k}{r} \]
where \( k \) is a constant.

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

3. **Set the Attractive Force Equal to the Centripetal Force:**
Since the attractive force provides the necessary centripetal force, we can set these two forces equal to each other:
\[ \frac{mv^2}{r} = \frac{k}{r} \]

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

5. **Solve 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 of the particle \( v \) is independent of the radius \( r \) and depends only on the constant \( k \) and the mass \( m \).

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