IF p is the magnitude of linear momentum of a particle executing a uniform circular motion, then the ratio of centripetal force acting on the particle to its linear momentum is given by

Options

A. `r/v`
B. `(v^2)/(mr)`
C. `v/r`
D. v.r

Answer

To solve the problem of finding the ratio of centripetal force acting on a particle executing uniform circular motion to its linear momentum, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Linear Momentum (P)**:
- The linear momentum \( P \) of a particle is defined as the product of its mass \( m \) and its linear velocity \( v \):
\[ P = m \cdot v \]

2. **Finding Linear Velocity in Circular Motion**:
- For a particle in uniform circular motion, the linear velocity \( v \) can be expressed in terms of the radius \( r \) and angular velocity \( \omega \):
\[ v = r \cdot \omega \]
- Substituting this into the equation for linear momentum gives:
\[ P = m \cdot (r \cdot \omega) = m \cdot r \cdot \omega \]

3. **Calculating Centripetal Force (Fc)**:
- The centripetal force \( F_c \) required to keep the particle moving in a circular path is given by:
\[ F_c = m \cdot a_c \]
- The centripetal acceleration \( a_c \) for uniform circular motion is given by:
\[ a_c = r \cdot \omega^2 \]
- Therefore, substituting for acceleration, we have:
\[ F_c = m \cdot (r \cdot \omega^2) \]

4. **Finding the Ratio of Centripetal Force to Linear Momentum**:
- Now, we can find the ratio of centripetal force to linear momentum:
\[ \frac{F_c}{P} = \frac{m \cdot (r \cdot \omega^2)}{m \cdot (r \cdot \omega)} \]
- The mass \( m \) and radius \( r \) cancel out:
\[ \frac{F_c}{P} = \frac{\omega^2}{\omega} = \omega \]

5. **Expressing in Terms of Linear Velocity**:
- Since we need the answer in terms of linear velocity \( v \), we can express \( \omega \) in terms of \( v \) and \( r \):
\[ \omega = \frac{v}{r} \]
- Thus, substituting this into our ratio gives:
\[ \frac{F_c}{P} = \frac{v}{r} \]

### Final Answer:
The ratio of the centripetal force acting on the particle to its linear momentum is:
\[ \frac{F_c}{P} = \frac{v}{r} \]