As the title indicates, I was wondering if we can use different coordinate systems for different objects. Specifically in the classic inclined pulley problem:
For $m_1$ I usually choose my $x$-axis to be running parallel to the inclined plane and my $y$-axis perpendicular to the plane. However, then I'm not sure what coordinate system to use for $m_2$. I have seen books where they choose their $x$-axis as the actual rope, and the $y$-axis running perpendicular to that rope.
I was wondering if I could set up, in this case, two different coordinate systems for $m_1$ and $m_2$. Here's what I mean by that: for $m_1$ the $x$-axis would be parallel to the inclined plane and the y-axis perpendicular to it; for $m_2$, the $y$-axis would be running straight down, parallel to the rope and the $x$-axis would be then running parallel the ground. Can this work?
Also, do you have any tips on choosing a coordinate system?