I saw the following problem from the USAPhO:
A uniform pool ball of radius $r$ begins at rest on a pool table. The ball is given a horizontal impulse $J$ of fixed magnitude at a distance $\beta r$ above its center, where $−1 \leq \beta \leq 1$. The coefficient of kinetic friction between the ball and the pool table is $\mu$. Find an expression for the final speed of the ball as a function of $J$, $m$, and $\beta$.
Two solutions are presented. In the first solution, we are told to "consider an axis perpendicular to the initial impulse and coplanar with the table." Then we apply conservation of angular momentum momentum with the initial state immediately after the impulse and the final state after the ball has achieved rolling without slipping (and hence has constant velocity)
From my understanding, at any given moment, the axis always passes through the point of contact between the ball and the ground. This is the part the confuses me. Since the axis is not fixed, why is angular momentum $\ell$ conserved between the initial and final states? I noticed that $\ell$ at some time when the ball is still rolling without slipping is not the same as the initial $\ell$. (If the ball has angular velocity $\omega$ at some time, then $\ell = I\omega$, which varies as $\omega$ varies.) I believe this is related to the fact that during the rolling without slipping phase, we are not in an inertial reference frame.
(In particular, the problem is is USAPhO 2008 Quarterfinal, problem 2(a). You can find the problem and its solution here: http://www.aapt.org/physicsteam/2012/exams.cfm)