Introduction:
Dimensional analysis is a technique used to obtain the relationships between physical quantities by examining their dimensions. In this case, we will use dimensional analysis to derive the relation between the centripetal force, mass, velocity and radius of a body moving in a circular path.

Centripetal force:
The centripetal force is the force that acts on an object moving in a circular path, towards the center of the circle. The magnitude of this force is given by the formula:

Fc = (mv^2)/r

Where Fc is the centripetal force, m is the mass of the object, v is the velocity of the object and r is the radius of the circular path.

Dimensions:
The dimensions of the variables in the formula for centripetal force are as follows:

Fc - [M L T^-2]
m - [M]
v - [L T^-1]
r - [L]

Dimensional analysis:
To derive the relation between the variables, we need to equate the dimensions on both sides of the formula. This gives us:

[M L T^-2] = [M] x [L T^-1]^2 x [L^-1]

Simplifying the dimensions on the right-hand side, we get:

[M L T^-2] = [M] x [L^2 T^-2] x [L^-1]

Cancelling out the common L and T terms on both sides, we get:

[M] = [M]

This shows that the dimensionally correct relation between the centripetal force, mass, velocity and radius is:

Fc = k x (m x v^2)/r

Where k is a dimensionless constant.

Conclusion:
Using dimensional analysis, we have derived the relation between the centripetal force, mass, velocity and radius of a body moving in a circular path. This relation is useful in understanding the behavior of objects in circular motion and can be applied in various fields such as physics, engineering and astronomy.