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I was taught the elastic potential energy was stored in the spring but then one practice problem on my textbook suggests the potential energy is in the object that is compressing the spring:

A $42 \,\text{kg}$ teenager balances briefly on a pogo stick, causing the spring in the stick to compress downward by $0.18 \,\text{m}$. Determine the elastic potential energy of the teenager.

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I looked it up and it seems the potential energy is of the system? I don't really understand. So does this mean both the spring and the object has this same elastic potential energy? Or just the object?

Qmechanic
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Lemonnny
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3 Answers3

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I don't think physics really cares about what it means for energy to be "stored in" something. What we call forces can do work over paths, and in certain cases we can associate this work with a function that is path-independent.

The other current answer talks about gravitational potential energy being a property of a single mass, and yet if we had a frame where the mass was at rest and the Earth moved towards it, would we instead assign the energy to the Earth? Usually we talk about potential energy being a part of the system; something that emerges from conservative forces that, by Newton's third law, must be interactions. But even then, that's just a way to talk about forces doing work.

In the case of the spring, the spring force does work, and it's a force that involves an interaction between whatever the two ends of the spring is attached to (or you can include the spring itself if it has mass). Can you say the energy is "stored in the spring"? Sure. Can you say the energy is in the system involving what the ends of the spring is connected to? Sure. That's why we have more objective formalism: forces do work.

BioPhysicist
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I think this is semantics, and it depends what you consider your "system" to be. If you abstract and ideal spring, then you system has a position dependent potential:

$$ U(x) = \frac 1 2 kx^2 $$

and you ideal mass $m$ then has potential energy based on it's position.

You may be tempted to say the energy is in the bonds of the atoms in the ideal spring, but there is a problem with that: ideal springs don't have atoms...they don't have mass, and they don't have losses.

IRL ofc, the spring has atoms and energy stored in the bonds, but now your system has gone from 1 DoF to $1+N_A$ DoF, and your spring has fluctuations and dissipation, which is too much, so we use ideal springs for basic problems.

JEB
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Firstly, let us define the term elastic potential energy and briefly talk about the fundamentals behind why it arises.

Elastic potential energy is the energy stored in a stretched or compressed object (this stretching/compression has been carried out by an external force applied to that object), such as a spring. Fundamentally, it relates to the chemical bonds within that material that holds all of its atoms/molecules together.

Suppose we have a vertical spring of force/spring constant $k \,\,Nm^{-1}$ with its original length. A mass (an external object) is suspended from the bottom of the spring,causing an external force to stretch this spring downwards, leading to it extending by $x$ metres. As you already know, the elastic potential energy is equal to $\frac{1}{2}kx^2$ Joules.

But is this elastic potential energy stored in the object - the mass stretching the spring - or the spring itself? Well, it is clear that the mass has lost gravitational potential energy (GPE) due to its height being $x$ metres lower than its initial position when it was first suspended on the spring - if the mass has a mass of $m$, the value of this "lost" GPE is $mgx$.

Now, let us assume that there are no energy losses in this system and the only conversion is between GPE and elastic potential energy. The "lost" GPE of the mass has been converted to the elastic potential energy of the spring - energy is conserved and physics works! Hence, the elastic potential energy (at least as far as I know) is definitely stored in the spring that stretches, and is provided by the conversion from the GPE of the mass to the elastic potential energy of the spring.

In terms of the question in your textbook, I think it should say the elastic potential energy of the spring in the pogo stick, not the teenager - if you do any further investigation into this, do correct me if I'm wrong. But anyways, since the teenager loses $42g\cdot 0.18=7.56g$ Joules of GPE, the elastic potential energy stored in the spring must also be $7.56g$ Joules, which (if we assume $g=9.8\,\,ms^{-2}$) is equal to $74.088$ Joules - ideally, this should be written to 2 significant figures (the same number of s.f. as $0.18$m), hence the answer would be $74$ Joules.

I hope that this answer has been helpful.

Physoverlord
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