Prove that f=mv2/r

Let us consider a particle travel with constant speed from A to B in a circular path of radius r in a short time Δt

making an angular displacement Δθ as shown in figure.

distance S travelled is the arc length AB and it is given by,  S = v×Δt ..................(1)

Arc distance S also written geometrically as,   S = r×Δθ  ...........................(2)

from (1) and (2), we write ,  Δθ/Δt  = v/r  ...............(3)

Eqn.(3) is rate of angular displacement or angular velocity.

For small angular displacement made in short time Δt, we can write = dθ/dt = v/r  ......................(4)

In right side figure, velocity vectors at A and B are drawn by merging their initial points.

The angle between these vectors also Δθ.

from the right side figure, we write,  sin (Δθ/2) = (Δv/2) / v    or  Δv/2 = v sin(Δθ/2)  .................(5)

For small angle Δθ,  eqn.(5) is written as, Δv/2 = v Δθ/2   or Δv/Δθ = v   or     dv/dθ = v  .....................(6)

using eqn.(4) and eqn(6),    acceleration directed towards centre, dv/dt  = ( dv/dθ ) (dθ/dt) = v×(v/r) = ( v² / r )

If m is mass, then   force F = mass×acceleration = m×(v² / r )