Invariance of elastic potential energy

[e2m2a]

At non-relativistic speeds is the elastic potential energy of a compressed spring frame-invariant? That is, would all reference frames agree on how much elastic potential energy is stored in the spring?

 

[dextercioby]

At non-relativistic speeds is the elastic potential energy of a compressed spring frame-invariant? That is, would all reference frames agree on how much elastic potential energy is stored in the spring?

Rather than asking this question and waiting for an answer, can you state your opinion first? And what mathematical formulas would you use to support it?

 

[e2m2a]

It is my opinion that at non-relativistic speeds, all reference frames would agree on the elastic potential energy of the spring. I assert this because the elastic potential energy is given by: 1/2 k (delta l) squared, where k is the spring constant and delta I is the change in length of the spring. Since k is a constant, all reference frames will agree on this value, and all reference frames would agree on delta l or the change in the length of the spring. But am I missing something?

 

[dextercioby]

No, you are not missing anything. Since time is absolute, the length between two points is an invariant when hypothetically measured, thus this purely potential energy k delta l^2 should be invariant under Galilean relativity.

 

[pervect]

It is my opinion that at non-relativistic speeds, all reference frames would agree on the elastic potential energy of the spring. I assert this because the elastic potential energy is given by: 1/2 k (delta l) squared, where k is the spring constant and delta I is the change in length of the spring. Since k is a constant, all reference frames will agree on this value, and all reference frames would agree on delta l or the change in the length of the spring. But am I missing something?

I'd agree with this, though as it stands it's a statement about non-relativistic physics in the relativity forum. Are you interested in a discussion of the relativistic case at all? My thoughts on this would be that in the relativistic case, we represent the compressed spring by its stress energy tensor $T^{ab}$. We generally don't try to separate out "potential energy" from "kinetic energy", in the relativistic case we concentrate on how $T^{ab}$ transforms via the tensor transformation rules.

There is a lot more that could be said and maybe should be said, but it's not clear that you're interested in the relativistic case - or what your background knowledge of tensors might be. I will say that I don't think there is a good way to treat the issue of a relativistic compressed spring without tensors (in particular the stress-energy tensor).

[add]I'll add that I believe Rindler has a treatment of the relativistically moving compressed spring (I think the text uses a rod, which is makes no difference to the analysis) which uses the stress-energy tensor in "Essential Relativity".

 

[e2m2a]

Thank you for your responses. Maybe I should have posted this in the general physics forum because I am not interested in the relativistic case.

 

[Nugatory]

Note: this thread has been moved to General Physics.