Question

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

Answer 1

Let F = K m^a v^b r^c

Where K is the dimensionless constant of proportionality and a, b, c are the powers of m, v and r respectively to represent F.

Writing the dimensions of various quantities in equation (ii), we get

[M¹L¹T^-2] = [M]^a [LT^-1]^b [L]^c

= M^a L^b T^-b L^c = M^a L^b + T^-b

Applying the principle of homogeneity of dimensions we get

a = 1, b +c = 1

-b = -2 or, b = 2

From (ii) c = 1 -b = 1 -2 = -1

Putting the values of a,b and c, in (i) we get

F = K m¹ v² r^-1

or, F =K mv²/r