A particle executes circular motion under a central attractive force inversely proportional to distance R. The speed of the particle is

Options

A. Dependent on R
B. Dependent on `R^(2)`
C. Independent of R
D. Dependent on `1//R`

Explanation

To solve the problem regarding the speed of a particle executing circular motion under a central attractive force that is inversely proportional to the distance \( R \), we can follow these steps:

### Step 1: Understanding the Forces
In circular motion, a particle experiences a centripetal force that keeps it moving in a circular path. This centripetal force is provided by the central attractive force acting on the particle.

### Step 2: Expressing the Centripetal Force
The centripetal force \( F_c \) required for circular motion is given by the formula:
\[ F_c = \frac{mv^2}{r} \]
where \( m \) is the mass of the particle, \( v \) is the speed of the particle, and \( r \) is the radius of the circular path.

### Step 3: Relating the Attractive Force to Distance
According to the problem, the central attractive force \( F \) is inversely proportional to the distance \( R \):
\[ F = \frac{k}{R} \]
where \( k \) is a constant of proportionality.

### Step 4: Setting the Forces Equal
Since the central attractive force provides the necessary centripetal force, we can set them equal:
\[ \frac{mv^2}{r} = \frac{k}{R} \]

### Step 5: Solving for Speed
Rearranging the equation to solve for \( v \):
\[ mv^2 = \frac{kr}{R} \]
\[ v^2 = \frac{kr}{mR} \]
\[ v = \sqrt{\frac{k}{m} \cdot \frac{r}{R}} \]

### Step 6: Analyzing the Dependency on \( R \)
From the expression derived, we can see that \( v \) depends on \( R \) through the term \( \frac{r}{R} \). However, if we consider that the speed \( v \) is constant for a given circular path, it implies that the speed does not change with varying \( R \) if \( k \) and \( m \) are constants.

### Conclusion
Thus, the speed of the particle is independent of the distance \( R \) in terms of its constant value during circular motion. Therefore, the correct answer is that the speed of the particle is independent of \( R \).