Question

The force F acting on a body moving in a circular path depends on mass of the body(m),velocity(v) and radius (r) of the circular path.
Obtain the expression for the force by
dimensional analysis method. (k=1)​

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 m^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

[M¹L¹T‐²] = M^a [LT‐¹]^b L^c

=> [M¹L¹T‐²] = M^a L^b L^c T^-b

=> [M¹L¹T-²] = M^a L^b+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 v¹ r‐¹

Or

F=k(mv²)/r

This is the required relation for centripetal force.