Identifying the right system in an energy problem

Question

What’s the best way to intuitively identify the right “system” for a problem involving energies?

For instance, let’s say you have a block that compresses a spring and it goes up a ramp with friction. If you wanted to analyze energies here, what would be the best reference system to do that with - block-spring, spring-earth, block-earth, spring-block etc?

Additionally, what does it broadly mean for the “Earth” to be involved in a system - I’ve seen this a lot in potential energy problems but I’ve never really understood what it means.

It depends on what you need to calculate. In your example you did not indicate what is supposed to be determined. Knowing that, you would define your system in such a way as to simplify the problem. — Bob D, Jun 16 '21 at 19:36
This may help - Basic energy question. In short, you can choose a frame of reference. — mmesser314, Jun 16 '21 at 20:21
Also this may help - Potential energy of a spring vs potential energy of the spring-mass system — mmesser314, Jun 16 '21 at 20:23
@BobD If I wanted to calculate the mechanical energy lost to friction, what system should I define? — Sam Lewis, Jun 17 '21 at 05:35

score 2 · Answer 1 · answered Jun 16 '21 at 20:42

You've picked a terrible example by including friction. A proper treatment involving conservation of energy requires keeping track of Avogadro's number ($N_A$) of atomic coordinates that are heated by friction via some combination of $N_A!$ possible little tiny elastic collisions.

Nevertheless, since you only have one degree-of-freedom (the spring displacement), that is pretty much your "system". The potential energy is then a function of that:

$$ U=U_k+U_g = \frac 1 2 kx^2 +mgx\sin{\theta} $$

where $\theta$ is the incline of the ramp. The reason you don't consider the ramp as part of your system is that it is fixed. It doesn't move. It doesn't exchange energy, it is just a constraint of motion: everything happens on the ramp.

It does heat up from friction, but that heat energy is leaving the system (your system isn't closed), but it's never coming back. To capture that, you just need to include energy loss as a function of movement:

$$\frac{dE}{dx} =\mu mg \cos{\theta}$$

which is a property of motion in your system, that there is a physical ramp does not matter.

Last but not least is $g$. It tells you the Earth is there, with gravity. If the Earth is part of you system, then you need to include its motion in response to the moving mass. To do this, you factor out the center of mass motion (it's solved problem), and work in the stationary center-of-mass frame.

The dynamic coordinate then becomes the separation of the block and the Earth, using the reduced mass, defined by:

$$\mu =\frac{1}{\frac 1 m+ \frac 1 {M_E}}\approx m(1 - \frac m{M_E}) $$

which is a 1 part in $10^{24}$ correction.

That is pointless, so in most cases for physics problems, having the Earth present means the space of the system is restrict to $z>0$, and there is a position dependent energy (per unit mass) given by:

$$ \frac {U(x, y, z)} m = gz $$

Identifying the right system in an energy problem

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