Question

Derive the expression for centripetal force.

Answer 1

Answer:

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 = k ma vb rc…..... (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

[M1L1T-2] = Ma [LT-1]bLc

=> [M1L1T-2] = MaLbLcT-b

=> [M1L1T-2] = MaLb+c T-b

on comparing both sides we get

a = 1

b+c = 1

-b = -2 => b = 2

Then c = -1

Onputting these values in (i),

we get

F = k m1 v2 r-1

Or

F=kmv2r

This is the required relation for centripetal force.