Angular momentum about different points

Question

From what I have gathered, angular momentum in a system can be calculated about any point. However, I am having trouble grasping this, and below I suggest an imaginary situation where I try to apply this.

Imagine a rod of mass %m% and length %L% that hangs from a hinge that allows it to rotate. At its lowest point, it is struck by a bullet of mass m as well that travels at a velocity v0 . I want to calculate the rod's resulting angular speed using the conservation of angular momentum about different points. I can easily calculate the conservation of angular momentum around the axis of rotation, but I am having trouble calculating it about other points.

Here is a conversation that features an answer very similar to the problem I am proposing, except that in the attached problem there is no hinge that restricts the movement of the rod. About what point is angular momentum conserved?

I would be very thankful if someone could provide an answer or correct me where I am wrong.

In the image below, the rod's resulting angular velocity is calculated using its angular momentum about its hinge (point A). If we were to calculate its angular velocity about its center of mass (point C), we should get the same result. Keep in mind that when calculating the angular momentum about the COM, we have to add the orbital component to the spin component.

If the rod were unhinged, as in the linked discussion, we could easily calculte the velocity of the COM using the conservation of angular momentum. But in this example this is no the case. Using the obtained angular velocity, we can calculate the linear speed of the COM, and plug it into the orbital component. When solving for the angular velocity, we should get the same result as before, but that is not what I am getting. (https://i.sstatic.net/v6vjh.jpg)

Answer 1

Yes, this website is not for Physics. You should post your questions on the Physics Stack Exchange. But I'll answer it anyway because I happen to know the answer.

The reason the axis matters for the moment of inertia is because spinning objects around different points takes differing amounts of torque. Without a constraint, the object will always spin about its center of mass (which is the definition of center of mass). With a constraint, well, it will spin differently.

However, while already rotating, the moment of inertia along which you measure the rotation is at the same point as the axis of rotation. You can't just choose a new one. That would give it a different theoretical angular momentum, because, you're assuming it's spinning in a way that it's not.

In this scenario, if the rod is rotating about the hinge, the angular momentum is based off the moment of inertia calculated at that hinge. If you calculated the moment of inertia with point C, a random spot on the rod, you would get the angular momentum it would theoretically have if it was rotating about that point.

It probably seems rather obvious in hindsight. Everyone has these confusions though, don't worry about it. Just put it on the right website next time :)