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As the title indicates, I was wondering if we can use different coordinate systems for different objects. Specifically in the classic inclined pulley problem:enter image description here

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?

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  • $\begingroup$ It's preferable to use two coordinate systems, and then use some equation of constraint to link the two. There are other ways to solve a simple problem like this one such as taking the two masses and the rope as one object, but more complicated ones may not yield to such simplification. Having a local coordinate system for each object is a more systematic approach that can be applied to a much larger class of problems. $\endgroup$ – garyp Oct 31 '17 at 17:29
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Yes, you are free to choose different coordinate systems.

You just must tie them together correctly (the rope tension pulling along the $+x$-axis in $m_1$ equals the rope tension pulling upwards along the $-y$-axis in $m_2$).

Tips for choosing coordinate systems? My tip is to choose it so it minimizes the number of acceleration components. That often simplifies the work a lot (you could have chosen an x-axis tilted 45 degrees from the incline - but that would have caused a parallel and perpendicular acceleration component to be present. So don't).

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  • $\begingroup$ "My tip is to choose it so there are no acceleration components." Isn't it safer to state one tries to minimise the number of acceleration components? If a system is going to move then no choice of coordinate system is going to make that 'disappear'. But careful choice can 'confine' acceleration to (e.g.) $a_x$ or $a_y$ or $a_z$. $\endgroup$ – Gert Oct 31 '17 at 17:17
  • $\begingroup$ Perhaps you mean one acceleration component for each object? $\endgroup$ – garyp Oct 31 '17 at 17:30
  • $\begingroup$ @Gert. Garyp. You are both very right. I have updated that nerving detail. $\endgroup$ – Steeven Nov 1 '17 at 8:31
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I was wondering if we can use different coordinate systems for different objects.

Yes. The choice of coordinate system is always arbitrary, as no absolute frame of reference exists (see Galilean invariance for further reading).

So it's perfectly possible to derive the equation of motion (EoM) for $m_1$ in one coordinate system and for $m_2$ in another one.

Generally one chooses the coordinate system for one's own convenience. It would be mathematically more difficult to derive the EoMs with an $x$-axis at say $63$ degrees to the gradient and $y$-axis at $-52$ degrees to the gradient, so we tend to avoid that kind of awkward choices.

Also, do you have any tips on choosing a coordinate system?

As said, mathematical convenience should guide you. For radially symmetrical problems (involving rotation e.g.) we may choose a convenient polar coordinate system rather than a Cartesian one.

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