Question Regarding the conservation of momentum in an inelastic collision of two rods I am tasked with solving this question but am facing some intuition difficulty.
consider this system:

The empty circle signifies a nail that is stuck in the wall.
I am unsure if there is conversion of angular momentum around the nail- On the one hand, it seems its force is parallel to the force that will cause rotational motion after the collision of the bodies.
On the other hand, If we consider the CoM, which would be located between the nail and the bottom rod, the force which is acting on it seems to create torque, seeing as r has a component which is perpendicular to the force.
Am I missing something? Where should my intuition come from?
(side note: I believe that the angular momentum around the CoM is NOT conserved but am not quite sure so any input on that would be welcome)
 A: In an isolated system, angular momentum is always conserved.
The nail provides a connection between your system of two rods and the outside world.  So your system of two rods is not isolated.  You can't guarantee conservation of energy (you said already that the collision is not elastic).  The nail will exert an external force on the system of two rods, so you should not expect conservation of linear momentum.  And you have already intuited that the nail will exert a torque on the system, so you can't rely on conservation of angular momentum, either.
However, the total angular momentum in a system depends on your choice of origin.  (An exercise that you should do if you haven't: prove to yourself that an isolated object moving in a straight line at constant speed has constant angular momentum, by choosing some centers of rotation both on and off of its line of motion.) The same holds for an external torque: the magnitude of the torque $\vec \tau = \vec r \times \vec F$ depends on the displacement $\vec r$ between the point of application of the external force $\vec F$ and your arbitrary choice of a center of rotation.
If you can find a point where the torque $\vec\tau$ from the nail is guaranteed to be zero, the angular momentum in that frame will be the same before and after the collision. You have correctly identified this privileged center of rotation as the nail, which sets $\vec r = 0$.
