find the formula for centripetal force using dimensions if it depends on mass "m",velocity "v" and radius "r" of circular path

The centripetal force, F acting on a particle moving uniformly in a circle  depend upon the mass (m), velocity (v) and radius (r) of the circle. The formula for F using the method of dimensions.

Let F = km ^a v ^b r ^c … (i)

Where, k is the dimensionless constant of proportionality, and a, b, c are the powers of m, v, r respectively.

On writing the dimensions of various quantities in (i), we get

[M ¹ L ¹ T ⁻² ] = M ^a [LT ⁻¹ ] ^b L ^c

= M ^a L ^b T ^−b L ^c

M ¹ L ¹ T ⁻² = M ^a L ^b + c T ^−b

On applying the principle of homogeneity of dimensions, we get

a = 1,

b= 2,

b + c = 1 …(ii)

From (ii), c = 1 − b = 1 − 2 = −1

On putting these values in (i), we get

F = km ¹ v ² r ⁻¹

OR

This is the required relation for centripetal force.