homework and exercises - Is momentum conserved in the collision of a ball with a hanging rod?

Suppose we have a situation like

A ball of some mass $m$ with some velocity collides with rod hinged at point $A$. Is momentum conserved in this situation? I know that hinge will give impulsive force on rod but that is internal force when we take ball and rod together as a system so that shouldn't stop us using conservation of momentum equation

Answer

Generally speaking linear momentum cannot be conserved in this case as hinge will exert a (external) force on rod at all but one position.

Note:Angular momentum is definitely conserved about hinged point since the force applied by hinge produces zero torque here.

We may want to find that one position for which linear momentum conservation holds.We proceed as follows:

$mv_od=mv_{final}d+(M\dfrac{l^2}{3})\omega$ -----(1)

$mu=mv_{final}+M(\dfrac{l}{2}\omega)$-------------(2) [holds since force by hinge is zero for this position $d$]

Solving we get $d=\dfrac{2l}{3}$