A simple doubt regarding spring potential energy

Question

A conceptual problem. Suppose we have a spring whose stiffness constant is $k$. The spring is attached to a wall and the other end is fastened to a block. If we pull the block by an external force without changing its kinetic energy and increase the length of spring by $x$, then which of the following statement should be correct:

$\to$ The potential energy of spring increases by $\frac{1}{2}kx^2$

$\to$ The potential energy of spring+wall+block system increases by $\frac{1}{2}kx^2$.

I think the latter statement is correct because if we leave either of block or wall or both free then they use this potential energy stored to convert it into kinetic energy. However, the first statement also doesn't seem wrong as we can say that spring applies force on the block when we leave it doing positive work on it and thus giving its stored potential energy to the block in form of kinetic energy

Are both correct?

What is the difference between the two or are they both equivalent?

Answer 1

I think both statements are correct.

In general, given some random physical system with no particularly special properties, we should say that the energy (including the potential energy) is a property of the system. The potential energy is generally defined to be the energy that would be needed to construct the system, starting from pieces that are non interacting. We aren't guaranteed to be able to unambiguously assign potential energy to any one part of the system.

Having said that, in this specific case we often say the spring contains the potential energy. There are a few (closely related) reasons why we do this.

1. Mathematically, the formula for the potential energy depends only on properties of the spring: the spring constant $k$ and the difference between the spring's actual length and its equilibrium length.
2. Physically, the origin of the potential energy at a microscopic level is electrostatic attraction wanting to pull together atoms in the spring back to the equilibrium length. The block and wall play no direct role in these forces.
3. Attaching energy to the spring makes it clear that we could transfer the spring (stretched to the same amount) to any other arrangement (e.g., we could attach it to two blocks instead of to a block and a wall), and we would have the same potential energy. This property, especially, is one of many reasons the spring is an excellent learning tool. Here, it provides good intuition for how energy is stored in more complicated situations, for example energy stored in chemical bonds in oil. Much like the spring, the potential energy stored in some volume of oil is a constant regardless of what exactly we do with the energy in the oil.

Answer 2

The wall cannot move (relative to the room or earth) and cannot do work or store energy. If the spring is fixed in the stretched position and the block removed, the spring still has the ability (if attached to a different block and released) to do the work expressed by the formula (1/2)k$x^2$. The energy goes into the spring.