# Invariance of elastic potential energy

• e2m2a
In summary, the elastic potential energy of a compressed spring is frame-invariant at non-relativistic speeds, meaning that all reference frames would agree on the amount of energy stored in the spring. This is supported by the formula 1/2 k (delta l)^2, where k is the spring constant and delta l is the change in length of the spring, which is a constant for all reference frames. While the discussion also touched on the relativistic case, it was determined that tensors are necessary to accurately analyze the behavior of a relativistically moving compressed spring. However, as the conversation was not focused on the relativistic case, the thread was moved to the general physics forum.
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?

e2m2a said:
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?

Last edited:
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?

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.

e2m2a said:
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".

Greg Bernhardt and dextercioby
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.

Note: this thread has been moved to General Physics.

## 1. What is the concept of invariance of elastic potential energy?

The invariance of elastic potential energy refers to the principle that the potential energy stored in an elastic material remains constant regardless of the change in its shape or orientation. This means that the elastic potential energy is independent of any external factors, such as the size, shape, or position of the material.

## 2. How does invariance of elastic potential energy apply to real-life situations?

The concept of invariance of elastic potential energy is applicable in various real-life situations, such as in the design and functioning of springs, rubber bands, and other elastic materials. It is also important in understanding the behavior of structures under stress, such as bridges and buildings, and in the development of new materials for different applications.

## 3. What is the relationship between Hooke's Law and the invariance of elastic potential energy?

Hooke's Law states that the force applied to an elastic material is directly proportional to the amount of deformation it undergoes. This is also related to the concept of invariance of elastic potential energy, as it shows that the potential energy stored in an elastic material is directly related to the applied force and the amount of deformation.

## 4. How does invariance of elastic potential energy impact the energy conservation principle?

The invariance of elastic potential energy is a manifestation of the principle of energy conservation, which states that energy cannot be created or destroyed, only transferred or converted from one form to another. In the case of elastic materials, the potential energy is converted into kinetic energy when the material returns to its original shape after being deformed, thus demonstrating the conservation of energy.

## 5. Are there any limitations to the concept of invariance of elastic potential energy?

While the invariance of elastic potential energy is a useful concept in understanding the behavior of elastic materials, it does have its limitations. It assumes that the material is perfectly elastic and does not account for factors such as friction, temperature, and the material's internal structure. In real-world situations, these factors can affect the behavior of elastic materials and may cause deviations from the principle of invariance.

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