Question

Fig. shows a bead of mass $m$ moving with uniform speed $v$ through a $U-$shaped smooth wire the wire has a semicircular bending between $A$ and $B$. Calculate the average force exerted by the bead on the part $AB$ of the wire.

Answer 1

Change in linear momentum of the bead $\Delta P=mv-(-mv)=2mv$

Time taken $\Delta t=\frac{\pi d/2}{v}=\frac{\pi d}{2v}$

Average force $F=\frac{\Delta P}{\Delta t}=\frac{2mv}{\pi d/2v}=\frac{4m{v}^2}{\pi d}$