# What would you expect in a graph showing the stiffness of a spring over the force exerted onto it?

Asked by Canonball (17) April 26th, 2010

Trying to work out the k (stiffness) of a spring (F=xk). F: force x: extension.

Attaching weights onto a spring and measuring its extension would you expect in a graph showing F in the x direction and k in the y direction, for the result to be proportional for there to be a horizontal line?

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My thoughts, and I’m just assuming, are that the graph would have two axes. One (probably the x-axis) would be the force exerted on the spring, measured in some unit. The other (probably the y-axis) would be the stiffness of the spring, measured in some unit. As the force exerted on the spring increases, I think that the stiffness of the spring would change accordingly.

jfos (7370)

I would expect you to do your own homework.

MrItty (17381)

Uhh, k is a constant descriptor of the spring’s stiffness. k is not a function of F. It will not change with F. It will change when you change the spring. x the deflection of the spring is a function of F (this should be obvious). As you apply more force to the spring, the spring will deflect x amount depending on the force applied and the stiffness of the spring as depicted in Hooke’s law. You should really read more about that.

lilikoi (10084)

I don’t really understand your question, you seem to be a bit confused about how this equation works. Either that, or you need to work on wording your questions more clearly. Assuming it’s the first one, here’s a brief explanation of what I think you’re talking about, but I’m going to try not to do it all for you, because this does sound like a homework question.

In the equation F=kx, k is the “spring constant”, so it doesn’t change no matter what, as long as you’re only talking about one spring. Therefore, the graph you describe, with k on one of the axes, would need to be talking about multiple springs. This is a weird graph, a graph with F and x on your axes would be much more common.

So, we’re making a graph that relates the spring constant k to the distance that a spring gets stretched. What’s missing from this graph? Force! Therefore, you must be talking about a graph of constant force. Your question now is: “if I exert the same force on a bunch of different springs with different stiffness, how far does each spring stretch?” Or, “if a spring is more stiff, does it stretch more or less than a less stiff spring with the same force applied to it?”

I hope this helps.

if it doesn’t help, it’s probably because you really want to be doing a force vs distance graph, not a spring constant vs distance graph

hannahsugs (3238)

@MrItty
Um sorry I was just wondering, not homework.

Canonball (17)

@hannahsugs No, you’re never going to plot k vs x or k vs F. That makes no sense. k is independent of x and F, as you said, not a function of either.

lilikoi (10084)

I’m wary of answering the whole question because I’m not sure if this is homework or just a curiosity. But either way, it pains me too much to ignore something math related. Here is a link to a mathematical definition of proportional. Hopefully you can finish up from there.

I thought Hook’s law states that K is constant so I expect the result to be a horizontal line

Canonball (17)

@Canonball uh-huh. Please explain to me the real-life situation that would make any non-HighSchool physics student curious about the answer to this question as phrased. I’d really love to hear it.

MrItty (17381)

@Canonball Hooke’s law does state k is constant. You expect the results of what to be similar to what???

lilikoi (10084)

[pure nonsense—I apologize]

So what’s the problem? The force should increase with the stiffness of the spring, right?

Heh. Stiffness of the spring. That’s what she said?

Haleth (19538)

@Haleth No, the deflection increases with increased force for a constant stiffness. The force required to achieve uniform deflection across springs of different stiffnesses increases with stiffness of the spring.

lilikoi (10084)

@lilikoi: just because k is constant for ONE spring, doesn’t mean that you couldn’t make a graph of how k relates to x for a given force. For example, it wouldn’t be imposible to create a “spring” (some sort of mechanical system that obey’s Hooke’s law) that has variable k. You could have an experimental question like: “i want my spring to stretch 10 cm when I apply a force of 3 N. What spring constant must I use?” A k vs x graph would solve this problem.

I agree, it’s not what you would usually choose to graph for hooke’s law, but it’s not “impossible” just because k is constant for a given spring.

hannahsugs (3238)

@hannahsugs There is NO relationship between k and x. k is NOT a variable. You would not gain anything from trying to plot k vs x or F vs x to determine the stiffness you want based on a known F and x. It would be a simple algebraic equation that you could solve for in 5 seconds. When you plot “a” vs “b”, you are explicitly stating by definition of the plot that a is a function of b. k is not a function of x or F therefore it would be illogical and meaningless to try to plot it as a function of x or F.

lilikoi (10084)

So is K constant or not?

Canonball (17)

K is constant.

lilikoi (10084)

@lilikoi: What do you mean, there is NO relationship? You seem to be assuming that you only have ONE spring. That is NOT what I am saying. I am saying, if you pick a fixed F, there definitely IS a relationship between k and x, you just have to allow for MANY different springs. How would the following question be answerable if there was NO relationship?

No matter what, I am going to apply a force of 5 newtons on whatever spring I choose. I want the spring to stretch 1.5 meters. I have a bunch of different springs that I could use. What spring constant k should i pick to satisfy the parameters of my problem?

hannahsugs (3238)

@hannahsugs Uh, in your scenario, the value for k would always be the same single number….

lilikoi (10084)

@Canonball: k is constant for a given spring. If you have multiple springs of different stiffnesses, k can vary among the springs. For EACH spring, k is constant.

hannahsugs (3238)

In this case just assume I have one.

Canonball (17)

Yes that’s true. But for a known F and x (as was the case in your scenario), k will always be the same number.

lilikoi (10084)

@lilikoi: yes i know. As a matter of fact, it would be about ~3.3 newton/meters. My point is that you could pick ANY distance x (ie, now i want it to stretch 3 meters, not 1.5), and get a different k. You could make a graph of k vs x by picking an infinite number of distances x that you “want” and graphing the resulting k for all possible values of x.

hannahsugs (3238)

@lilikoi : yes, we seem to be seeing eye to eye now. My point was, in the original question posed by @Canonball, x wasn’t “known”!

hannahsugs (3238)

@Canonball: if you only have one spring, your graph makes no sense. k would be constant.

hannahsugs (3238)

@hannahsugs
So you would expect the line too….? Be proportional or a horizontal line?

Canonball (17)

@MrItty
I like learning in my own time.

Canonball (17)

@Canonball: Just in case this is a homework question, i don’t feel comfortable answering that. If k is constant, and you have a basic understanding of graphs, you should be able to figure it out from there.

also, there’s no such thing as a “proportional” line. Lines are curved, straight, horizontal, vertical, but not “proportional”. They can demonstrate a proportional relationship, but a line isn’t “proportional” itself

hannahsugs (3238)

@hannahsugs
Um ok well this isn’t for any type of homework I just want to know but ok.

Canonball (17)

@Canonball: if k is on the y-axis, and it is CONSTANT, is the value of y on your graph ever going to change? How would you draw a graph with a constant y-value?

hannahsugs (3238)

Ok I’ll be honest I have a graph of this right here and it’s not what I expected, I was expecting a straight horizontal line and I didn’t get that, it increased with more weight.

Canonball (17)

@Canonball: where does weight come into this graph? I thought we covered this already. Please tell us what the axes of your graph are.

hannahsugs (3238)

@hannahsugs Yes I think we are.

@Canonball OK. You have one spring. By making a graph what are you trying to show?

lilikoi (10084)

http://tinypic.com/r/ej8nf4/5

Canonball (17)

@Canonball Did you not see the force variable in Hooke’s law? Again, what are you trying to show?

lilikoi (10084)

@Canonball: First of all, the x-axis shows weight, not distance (weight is related to force, F). However, this graph still doesn’t make sense to me. Where did it come from? who made it? Maybe this one will make more sense to you.

i have to go now, i’ll check back on this thread later. good luck!

hannahsugs (3238)

@hannahsugs
Why would I make that graph to work out K? Would it be better to work out K making that graph and finding the gradient?

Canonball (17)

Just made another graph using the result and it’s perfect.

Canonball (17)

@Canonball

I asked you why you are making the graph. What are you trying to show??? If you could tell me that, I could tell you what you need to do to accomplish it. Are you trying to solve for K? In that case, you don’t need a graph. Apply a force to the spring, measure the deflection, and then use Hooke’s Law to calculate K.

In the link she provided, the graph was not made to “work out K”. It was made to depict the relationship between force and mass of a particular spring.

So again, what are you trying to show?

lilikoi (10084)

@lilikoi
I want to work out the constant stiffness of the spring.

Canonball (17)

@Canonball

Okay. The problem I see with your graph is that for a single particular spring, the stiffness does not change with weight. Therefore, something is wrong with the graph you uploaded. The stiffness should be a horizontal line across all the weights.

That aside, you have a plot of k vs x. This implies quite strongly that both k and x are known. If you are trying to find the stiffness, why is it shown in the graph, i.e. how do you already know it?

lilikoi (10084)

.

lilikoi (10084)

I just learned (or perhaps relearned) that individual springs can have variable stiffness. I’m thinking that is beyond the scope of the original question, but thought I should point out that what I said before was wrong!

lilikoi (10084)

@lilikoi: Yes! this is the sort of complication i was trying to allow for earlier, but I didn’t want to confuse the issue any more than necessary. Also, Hooke’s law can be used to discuss systems that aren’t “springs” like metal coils, such as vibrations between the atoms of a molecule or compression in a stiff rod. When it comes to basic physics, it’s dangerous to talk in absolutes, there’s almost always a complication or an exception that you might not know about!

hannahsugs (3238)

@hannahsugs Yes, so true. I’ve used Hooke for other things, too.

lilikoi (10084)