A particle of mass 'm' is moving on a circular path of radius 'r' with uniform speed 'v'. Rate of change of linear momentum is

Options

A. A. Proportional to radius 'r'
B. B. Proportional to `v^2`
C. C. Zero
D. D. Independent of speed

Answer

To find the rate of change of linear momentum for a particle of mass 'm' moving in a circular path of radius 'r' with uniform speed 'v', we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Linear Momentum**:
The linear momentum (P) of an object is given by the product of its mass (m) and its velocity (v):
\[ P = m \cdot v \]

2. **Identifying the Nature of Motion**:
Since the particle is moving in a circular path with uniform speed, even though the speed is constant, the direction of the velocity is continuously changing. This means there is an acceleration acting on the particle, known as centripetal acceleration.

3. **Centripetal Force and Acceleration**:
The centripetal acceleration (a) for an object moving in a circle of radius 'r' at speed 'v' is given by:
\[ a = \frac{v^2}{r} \]
The force required to maintain this circular motion (centripetal force, F) is:
\[ F = m \cdot a = m \cdot \frac{v^2}{r} \]

4. **Rate of Change of Linear Momentum**:
The rate of change of linear momentum (which is also the net force acting on the object) can be expressed as:
\[ \frac{dP}{dt} = F \]
Substituting the expression for centripetal force, we have:
\[ \frac{dP}{dt} = m \cdot \frac{v^2}{r} \]

5. **Final Expression**:
Therefore, the rate of change of linear momentum is:
\[ \frac{dP}{dt} = \frac{mv^2}{r} \]

### Conclusion:
The rate of change of linear momentum for a particle of mass 'm' moving in a circular path of radius 'r' with uniform speed 'v' is given by:
\[ \frac{dP}{dt} = \frac{mv^2}{r} \]