# Graph relating load force and spring extension in Hookes Law

I've come across a graph in my physics book, similar to this one.

But wouldn't the extension increase per load increase be reduced after the elastic limit.

The graph I have in mind is like this

Both graphs were found online, but which is correct, or are they referring to two different elastic objects, if so which?

• Hi Andrew, if the pictures are small, you are better off including them in the post, people don't always like clicking on links. Thanks – user108787 Sep 15 '16 at 21:45
• The upper graph is yielding (leading towards an elastic limit and destruction of the spring). The lower graph limiting where the coils have reached an elastic limit of their linear range. – docscience Sep 16 '16 at 0:37
• ... In one case the spring yields to the applied force by breaking or deforming, the other just limiting the force by its linear elastic range. Other design features of the spring, O.D. vs thickness, etc. could result in either outcome – docscience Sep 16 '16 at 0:40

The elastic limit means that if you extend the material beyond that point and then unload the material it will suffer a permanent deformation.
So you extend your material beyond the elastic limit and then remove the force.

Which graph gives a reasonable unloading curve such that at zero force there is a positive extension?

You have to be a little careful about the energy stored in the material.
You should use the area of the unshaded triangle rather than the area of the shaded triangle.
Now as long as the graph is a straight line through the origin it does not matter but as when this is not so it does.

The area of the unshaded triangle represents the work done by the load in extending the material by $50.0$ cm.
The assumption is then made that all of that work done by the load is stores as elastic potential energy in the material.

Now continue beyond the elastic limit "under" the graph which still represents the work done by the load in extending the material but this is no longer the elastic potential energy stored in the spring.
The elastic potential energy stored in the material is the area "under" the blue graph and the area between those two graphs represents the work done by the load in permanently extending the material - that is, permanently breaking bonds.

• But if the first graph is correct, doesn't it give the impression that loading additional weight past the elastic limit, will extend the spring much more than usual. Also does your graph continue upwards from the right, or do we stop the line there and assume that's the point where the spring has broken. – Andrew Brick Sep 16 '16 at 11:16
• Look at that graph carefully as you increase the load the extension increases faster. As to where the graph stops it all depends what the material is and what shape it is. If it was a spring you can imagine the coils getting uncoiled and the spring getting longer and longer. If the material was steel then it would require a much larger load for the spring to actually break. – Farcher Sep 16 '16 at 11:49

No, after the elastic limit, the extension should increase more per unit load. I think you can find this out in any standard physics textbook such as Halliday and Resnick (Fundamentals of Physics)

• yes, but I still come across several sources which say the correct graph looks like the second image, even this video – Andrew Brick Sep 15 '16 at 22:13
• This video gives a graph which seems incorrect at least to me. I cannot think of any physical reason about why the graph of an elastic object would look like this. – Kaitou1412 Sep 16 '16 at 8:22