# Potential energy of a spring vs potential energy of the spring-mass system

What is the difference between potential energy of a spring and potential energy of the spring-mass system?

The reason I am asking this question is, is elastic potential energy a property of the spring alone? Or is it a property of the mass attached to the spring as well?

My textbook says that it is incorrect to say "potential energy of a block raised $$h$$ meters above the Earth's surface". The book says "potential energy associated with the block", or "potential energy of the block-Earth system" is correct because gravitational potential energy is a property of both the block and the Earth. It makes total sense.

Likewise, is elastic potential energy a property of both the mass and the spring? If it is (and even if it's not), what is the difference between potential energy of a spring and potential energy of the spring-mass system?

Potential energy is configuration energy: energy by virtue of the relative positions of the various parts of the system. So, as you imply, your textbook is quite right to stress that the gravitational PE of a raised block really 'belongs' to the Earth-block system, not just the block.

But in the case of a stretched spring the energy is due to the (increased) separation of the parts of the spring relative to each other, so the elastic potential energy simply 'belongs' to the spring as a whole – because it contains its parts! It would be confusing to bring in the mass.

One can sensibly talk about "the potential energy of the mass-spring (and Earth)" system when considering a mass hanging from a spring. This is because both elastic PE (of the spring) and gravitational PE (of the mass-Earth system) are involved.

• in the case of a stretched spring the energy is due to the (increased) separation of the parts of the spring relative to each other, so the elastic potential energy simply 'belongs' to the spring as a whole – because it contains its parts! You mean, when a spring is stretched, its 'parts' (very small segments) get separated relative to each other, with net force acting on all the segments is zero except for the last segment, which has a net force equal in magnitude and opposite in direction to the force applied to stetch it, is that what you are referring to?
– 4d_
Commented May 21, 2020 at 17:52
• @ π times e Yes, though I didn't mention forces. How wires extend elastically is easier to understand in terms of atoms than how springs extend, so forgive me if I consider wires. When we stretch them we increase the separation between each atom and its neighbour along the wire. This requires us to do work against the mutual attractive forces of the atoms. We are therefore storing potential energy all along the wire, even though we apply forces only to its ends. Commented May 21, 2020 at 18:00
• I understand it now. Thanks. Elstic wires work pretty much the same way as elastic strings, don't they?
– 4d_
Commented May 21, 2020 at 18:11
• Yes indeed. Just replace "neighbouring atoms in the wire" by "neighbouring bits of spring". [The stretching of a helical spring is largely because of elastic bending of the wire, as one might bend a ruler, as well as so-called 'shearing', so the separation changes vary across the cross-section of the wire. It's quite hard to explain without diagrams, but I doubt if you wanted to go into this level of detail!] Commented May 21, 2020 at 18:19
• In the case of a stretched spring the energy is due to the separation of the parts of the spring relative to each other, so the elastic potential energy simply 'belongs' to the spring as a whole – because it contains its parts! It's true for an ideal massless spring, right? What if the spring has mass? Then work done on each of its tiny segments is +ve when the spring is stretched, while work done on the last tiny segment is still negative. Adding up all the +ve work done by the tiny segments on one another and the -ve work done on the last segment is still -ve? Spring force is -ve for sure
– 4d_
Commented May 22, 2020 at 4:58

What is the difference between potential energy of a spring and potential energy of the spring-mass system?

The latter is definitely the correct expression, the former is a rather lazy abbreviation.

We can understand this in quite an intuitive way with the following thought experiment.

Two fairly small, identical masses $$m$$ are connected by a spring. Now we separate the masses so that the spring is under tension.

This system, as a whole, has now gained potential energy. This is evidenced by the fact that if we release either mass it will start moving towards the other. Work is being done and potential energy is being converted.

Now we gradually increase the mass of one of the masses (say the left one), until it has reached mass comparable of that of the Earth. With each increase we repeat also the separation experiment.

It is self-evident that as the mass of the left mass increases and increases, it will be less and less inclined to move when released (because its inertia has increased so much). By the time it has grown to the mass of the Earth its movement will be imperceptible.

But the gradual increase of mass of the left mass doesn't mean it's not the whole system that has potential energy. It just looks that way for very large left masses because they don't move much when released.

The same reasoning can be applied for two masses and their mutual gravitational attraction.

• Wow, thank you so much. Earth is just another mass attached to one of the ends of a spring. It's perfectly clear to me now. I have just one confusion right now. All three answers helped me conceptually understand what I asked. Not sure which one to accept as 'the answer'.
– 4d_
Commented May 21, 2020 at 18:17
• You're welcome.
– Gert
Commented May 21, 2020 at 18:21

Whenever you have one force, you always have another equal and opposite. Some of these forces convert one kind of energy into potential energy. For example, gravity. Two forces push Earth and a block apart, against gravity. If you remove the force, gravity accelerates the Earth and block toward each other. Potential energy is converted to kinetic energy.

A counter example is friction. If a block slides on the Earth, they exert forces on each other. They slow down and come to a stop. (The earth moves so little that we usually ignore it). In this case, kinetic energy is converted to heat, not potential energy. You can't get the kinetic energy back like you could for gravity.

So your text book is right. You store energy in a system where there are two opposing forces. But ...

A spring is like gravity. Two forces on the ends of the spring can compress it, storing potential energy in the spring. If you remove the force, you get the energy back. You really mean the system consisting of the spring and the things pushing on the ends. But it is common to talk about the potential energy of the spring.

Likewise, it is common to talk about the energy of a raised block.

It is important to understand the point, because you need to know how potential energy works. But it is also important to understand when people are doing physics right, even though they are speaking carelessly.

It is a little like English teachers getting picky about minor grammatical points. Except that sometimes the picky details matter. This is one difference between mathematicians and physicists. Mathematicians tend to be picky and exact. Physicists tend to be looser.

For mathematicians, theorems come from proofs. Proofs have to be perfectly correct and airtight. If not, a false theorem can be proven, and this can be used to prove other false theorems. It can literally bring down the entire structure of mathematics.

For physicists, laws of physics are statements about the behavior of the universe. It is often complex and impossible to get the exact answer. Physicists make approximations all the time. Sometimes in calculations. Sometimes in the laws them selves.

• What a beautiful explanation.. thanks a lot! So elastic potential energy does belong to the spring and the things pushing or pulling on the ends but it is generally referred to as the potential energy of the spring. That's what confused me because although my textbook says that gravitational potential energy belongs to the block-mass system, it doesn't say the same about elastic potential energy. Just says, "potential energy of the spring"
– 4d_
Commented May 21, 2020 at 18:09
• @ mmesser314 "You really mean the system consisting of the spring and the things pushing on the ends." But the spring still stores the energy when you remove the things pushing on the ends. It doesn't, of course, store the energy for more than a split second (at least not in the form of elastic potential) when the things pushing on the ends have been removed, but that surely isn't the point. Commented May 22, 2020 at 10:10
• @PhilipWood - I have been rethinking. I really need to change this. The system I described stores energy with internal forces. If you pick just the spring as your system, the forces that compress it are now external, but potential energy is still stored. The main point is still valid. The forces that compress the spring are what converts some other kind of energy into potential energy. And it is OK to talk about storing potential energy in the spring with or without mentioning the things that do the compressing. Commented May 22, 2020 at 14:35
• @mmesser314 We now seem to be in complete agreement! Commented May 22, 2020 at 15:51