If a ball collides with and sticks to an unhinged uniform rod on a smooth table (the path of the ball does not necessarily pass through the center of the uniform rod), is angular momentum conserved about the center of mass of the rod and ball or about the center of the rod? Why? Is angular momentum about both conserved, and, if so, how can we calculate this?
Angular momentum should be conserved in any inertial frame of reference - if you move with the center of mass, the motion you see will be rotation about the center of mass; if you move with a different frame of reference, you will see rotation about a different axis.
So the short answer is "it doesn't matter".
Whether they "give the same answer" depends on whether you do the calculations properly...
Angular momentum depends on the axis about which you calculate it. That means you'll get a different number depending on the axis, but no matter the choice, it's always conserved (assuming no external forces act on the system).
For example, imagine a ball of mass $m$ and velocity $v$ hits and sticks to the rod a distance $d$ from its center:
- If you choose the rotation axis to be the center of the stationary rod, you would calculate the angular momentum before the collision to be $mvd$. After the collision the system ang. mom. is still $mvd$ about the fixed point where the rod was initially.
- Now say you choose an axis at the point where the ball hits the rod. The initial ang. mom. is zero, so the final ang. mom. is also zero around that point. After the collision, the system is rotating, but its linear momentum is offset from the chosen axis, so there are two contributions to ang. mom. that exactly cancel.