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A stick of length $L$ and mass $M$ lies on a frictionless horizontal surface on which it is free to move in any ways. A ball of mass $m$ moving with speed $v$ collides elastically with the stick as shown in the figure. If after the collision the ball comes to rest, then what should be the mass of the ball ?
$m=2M$
$m=M$
$m=M/2$
$m=M/4$
Solution
Applying conservation of angular momentum
$\frac{\mathrm{mvL}}{2}=\frac{\mathrm{ML}^{2}}{12} \omega$
$\omega=\frac{6 \mathrm{mv}}{\mathrm{ML}}$ $…(1)$
conservation of linear momentum
$\mathrm{mv}=\mathrm{Mv}_{2}$
$\mathrm{v}_{2}=\frac{\mathrm{mv}}{\mathrm{M}}$ $…(2)$
Applying conservation of kinetic energy (elastic collision)
$\frac{1}{2} \mathrm{mv}^{2}=\frac{1}{2} \mathrm{Mv}_{2}^{2}+\frac{1}{2} \mathrm{I} \omega^{2}$ $…(3)$
solving eq. $(1),(2) \&(3)$
$\mathrm{m}=\frac{\mathrm{M}}{4}$
