A particle of mass M moves with constant speed along a circular path of radius r under the action of a force F. Its speed is

Options

A. `sqrt((Fr)/(m))`
B. `sqrt(F/r)`
C. `sqrt(Fmr)`
D. `sqrt(F/mr)`

Explanation

To solve the problem, we need to find the speed of a particle of mass \( M \) moving with constant speed along a circular path of radius \( r \) under the action of a force \( F \).

### Step-by-Step Solution:

1. **Understand the Forces Acting on the Particle**:
- The particle is moving in a circular path, which means it is experiencing centripetal force directed towards the center of the circle.
- The net force acting on the particle is the centripetal force, which is provided by the force \( F \).

2. **Centripetal Force Formula**:
- The formula for centripetal force \( F_c \) required to keep a mass \( M \) moving in a circle of radius \( r \) at speed \( v \) is given by:
\[ F_c = \frac{Mv^2}{r} \]

3. **Equating the Forces**:
- Since the particle is moving with constant speed under the action of force \( F \), we can equate the force \( F \) to the centripetal force:
\[ F = \frac{Mv^2}{r} \]

4. **Rearranging the Equation**:
- To find the speed \( v \), we rearrange the equation:
\[ v^2 = \frac{Fr}{M} \]

5. **Taking the Square Root**:
- Taking the square root of both sides gives us the speed \( v \):
\[ v = \sqrt{\frac{Fr}{M}} \]

### Final Answer:
The speed \( v \) of the particle is:
\[ v = \sqrt{\frac{Fr}{M}} \]