Introduction:
Dimensional analysis is a useful tool in physics to derive relationships between different physical quantities. In this case, we are trying to find the expression for the force acting on a particle moving in a circle of radius r with velocity v, given the mass of the particle is m.

The Centripetal Force:
The force that keeps a particle moving in a circle is called the centripetal force. It acts towards the center of the circle and is given by the equation:

F = m * a_c

where F is the centripetal force, m is the mass of the particle, and a_c is the centripetal acceleration.

The Centripetal Acceleration:
The centripetal acceleration is the acceleration of an object moving in a circle and is given by the equation:

a_c = v^2 / r

where a_c is the centripetal acceleration, v is the velocity of the particle, and r is the radius of the circle.

Expressing the Force using Dimensional Analysis:
To express the force in terms of mass, velocity, and radius, we can substitute the expression for the centripetal acceleration into the equation for the centripetal force:

F = m * (v^2 / r)

Final Expression:
The final expression for the force acting on a particle of mass m, moving with velocity v, in a circle of radius r can be written as:

F = m * v^2 / r

This equation shows that the force is directly proportional to the mass of the particle and the square of its velocity, while inversely proportional to the radius of the circle.

Summary:
Using dimensional analysis, we derived the expression for the force acting on a particle moving in a circle of radius r with velocity v. The force is given by F = m * v^2 / r, where F is the centripetal force, m is the mass of the particle, v is the velocity, and r is the radius of the circle. This equation shows the relationship between these quantities and highlights the factors that affect the magnitude of the centripetal force.