A fixed U-shaped smooth wire has a semi-circular bending between A and B as shown in Fig. A bead of mass m moving with uniform speed v through the wire enters the semicircular bend at A and leaves at B. The average force exerted by the bead on the part AB of the wire is

Choosing the positive x—y axis as shown in the Fig., the momentum of the bead at A is p→i=+mv→. The momentum of the bead at B is p→f=−mv→. Therefore, the magnitude of the change in momentum between A and B is Δp→=p→f−p→i=−2mv→. The time taken by the bead to reach from A to B is Δt=π.d/2v=πd2v. Therefore, the average force exerted by the bead on the wire is Fav=ΔpΔp=2mv/πd2v=4mv2πd