A paticle of mass `m` is executing uniform circular motion on a path of radius `r`. If `p` is the magnitude of its linear momentum, then the radial force acting on the particle is

Options

A. pmr
B. `(rm)/(p)`
C. `(mp^(2))/(r)`
D. `(p^(2))/(rm)`

Explanation

To solve the problem, we need to determine the radial force acting on a particle of mass \( m \) that is executing uniform circular motion with a radius \( r \) and has a linear momentum \( p \).

### Step-by-Step Solution:

1. **Understand the Concept of Radial Force**:
The radial force (also known as centripetal force) acting on an object in uniform circular motion is given by the formula:
\[ F_r = \frac{mv^2}{r} \]
where \( m \) is the mass of the particle, \( v \) is the linear velocity, and \( r \) is the radius of the circular path.

2. **Relate Linear Momentum to Velocity**:
The linear momentum \( p \) of the particle is defined as:
\[ p = mv \]
From this, we can express the velocity \( v \) in terms of momentum:
\[ v = \frac{p}{m} \]

3. **Substitute Velocity into the Radial Force Formula**:
Now, we will substitute \( v \) in the radial force formula:
\[ F_r = \frac{m\left(\frac{p}{m}\right)^2}{r} \]
Simplifying this gives:
\[ F_r = \frac{m \cdot \frac{p^2}{m^2}}{r} = \frac{p^2}{mr} \]

4. **Final Expression for Radial Force**:
Thus, the radial force acting on the particle can be expressed as:
\[ F_r = \frac{p^2}{mr} \]

### Conclusion:
The radial force acting on the particle is \( \frac{p^2}{mr} \).