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A U-shaped smooth wire has a semi-circul...

A 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 semiculcular bent at A and leaves at B. Find the average force exerted by the bead on the part AB of the wire.

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Choosing the position X-Y axis as shown in the figure, the momentum of the bead at A is `vec(P)_(i)= +mvec(v)`. The momentum of the bead at B is `vec(P)_(f)=-mvec(v)`.
Therefore, the magnitude of the change in momentum between A and B is
`Delta vec(p) = vec(p)_(f) - vec(p)_(i) = -2m vec(v)`
i.e. `Delta p=2mv` along positive X-axis.
The time interval taken by the bead to reach from A to B is
`Delta t=(pi.d//2)/(v) =(pi d)/(2 v)`
Therefore, the average force exerted by the bead on the wire is
`F_(av)=(Delta P)/(Delta t)=(2mv)/((pid)/(2v))=(4mv^(2))/(pi d)`.
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