# Relationship between force and potential energy

• I
I am aware that the negative derivative of potential energy is equal to force. Why is the max force found when the negative derivative of potential energy is equal to zero?

Chandra Prayaga
Could you give an example problem or situation when this is done? Where did you see this? It is not correct.

I was told by someone that when you set the derivative to zero, you get the max force.

anorlunda
Staff Emeritus
The answer to your question does not depend on any physics. It is a matter of simple math. Maxima and minima occur at the places where the slope of the curve is zero. Also, the sign of the slope flips as we pass a maximum or minimum. #### Attachments

• Asymptotic and sophiecentaur
So if we set the derivative to zero, we can calculate the minima and maxima of a potential energy graph, how does that help us find the max force?

sophiecentaur
Gold Member
2020 Award
So if we set the derivative to zero, we can calculate the minima and maxima of a potential energy graph, how does that help us find the max force?
It doesn't. If there is a turning value of Potential Energy with distance, the force is zero. As the hyperphysics link tells you, the highest force is where the Potential Energy is changing fastest with distance.
If you were told otherwise by "someone" then perhaps they are not a reliable source of info (or they misinterpreted the question that you asked them).

• I_laff
anorlunda
Staff Emeritus
So if we set the derivative to zero, we can calculate the minima and maxima of a potential energy graph, how does that help us find the max force?

No. You don't set the derivative to zero, you find the potential energy where the derivative is zero. That place is a ax or min.

Dale
Mentor
2020 Award
Why is the max force found when the negative derivative of potential energy is equal to zero?
This is incorrect. When the negative gradient of the PE is zero then the force is zero.

ZapperZ
Staff Emeritus

Where exactly in the hyperphysics link does it claim that ".... max force found when the negative derivative of potential energy is equal to zero ... "? I do not see it, and you're asking us to correct an non-existing error.

Force is the gradient of the potential energy. This means that the quicker the potential energy changes over distance, the higher the absolute value of the force. So it is not the absolute value of the potential energy, but rather the change in the potential energy that corresponds to the force.

Zz.

Where exactly in the hyperphysics link does it claim that ".... max force found when the negative derivative of potential energy is equal to zero ... "? I do not see it, and you're asking us to correct an non-existing error.

Force is the gradient of the potential energy. This means that the quicker the potential energy changes over distance, the higher the absolute value of the force. So it is not the absolute value of the potential energy, but rather the change in the potential energy that corresponds to the force.

Zz.
I didn't say that Hyperphysics stated that the max force is when the derivative was set to 0. I said that I was told that it was, which confused me since it didn't make sense. My only mistake was assuming that the person who told me that the max force could be calculated by setting the derivative to zero was correct.

• Dale
ZapperZ
Staff Emeritus
I didn't say that Hyperphysics stated that the max force is when the derivative was set to 0. I said that I was told that it was, which confused me since it didn't make sense. My only mistake was assuming that the person who told me that the max force could be calculated by setting the derivative to zero was correct.

Look at Posts 1, 2, and 3 and read them again in sequence to see why it appears that you are using the Hyperphysics link to justify what you were told.

Zz.

I didn't say that Hyperphysics stated that the max force is when the derivative was set to 0. I said that I was told that it was, which confused me since it didn't make sense. My only mistake was assuming that the person who told me that the max force could be calculated by setting the derivative to zero was correct.
So the maximum occurs where the slope of the potential energy curve is steepest. You can find those points by differentiating the potential energy twice with respect to position and setting that equal to 0. Only in those points can the force be maximal. (there could also be a minimum, or an inflection point)

ZapperZ
Staff Emeritus
So the maximum occurs where the slope of the potential energy curve is steepest. You can find those points by differentiating the potential energy twice with respect to position and setting that equal to 0. Only in those points can the force be maximal. (there could also be a minimum, or an inflection point)

This is not correct either. 2nd derivative tells you how rapidly the slope is changing, and thus, how rapidly the force changes. It doesn't tell you how large the force is.

You can have an almost vertical straight line on the V vs x graph (i.e. 2nd derivative is zero but not at a max or min or inflection point), and yet, this is where a force can be a maximum.

Zz.

Look at Posts 1, 2, and 3 and read them again in sequence to see why it appears that you are using the Hyperphysics link to justify what you were told.

Zz.
I would've thought the quoted post below would've clarified what information was and was not from the Hyperphysics website.

I was told by someone that when you set the derivative to zero, you get the max force.

ZapperZ
Staff Emeritus
I would've thought the quoted post below would've clarified what information was and was not from the Hyperphysics website.

I get that you were "told" by someone, but when asked for a source in Post 2, you cited Hyperphysics, as if you were using that page to justified what you were told by this "someone". That's why I asked where specifically on that Hyperphysics page matches what you were told by this "someone".

Many of us are familiar with Hyperphysics. I even used it as an additional educational source for my students. I don't ever recall them making this type of error, and that is why I was particular concerned in trying to find where exactly in there that matches the erroneous info that you were told. If nothing there matches what you were told, why did you cite it without understanding it?

Zz.

Apologises if I misled you, I will be more careful when citing sources next time so it is obvious what is and is not from the source. I cited hyperphysics because it was from there that I read about the relationship between force and potential. The first reply to my post simply said that what I had stated was 'not correct'. They didn't specify what part of what I said was incorrect, so in reply to this I just posted where I got all my information from. I think my understanding of what hyperphysics was on about was adequate enough for me to cite it. Once again, I never said that hyperphysics stated the incorrect information I was told. You insinuated that this is what I meant from my posts, but that is not the case.

sophiecentaur
Gold Member
2020 Award
I thought we had already decided that this "person" was not a reliable source of information but that Hyperphysics is pretty well trustworthy.

• I_laff
This is not correct either. 2nd derivative tells you how rapidly the slope is changing, and thus, how rapidly the force changes. It doesn't tell you how large the force is.

You can have an almost vertical straight line on the V vs x graph (i.e. 2nd derivative is zero but not at a max or min or inflection point), and yet, this is where a force can be a maximum.
I don't get this at all. You say you disagree with me, but then you produce a case where the 2nd derivative is 0 and the force is a maximum, but this is not a max or min or inflection point? This appears to be a contradiction.
The force is the first derivative of the potential energy. The only place where the force can be a maximum is where the derivative of the force, or the second derivative of the potential energy is 0.

sophiecentaur
Gold Member
2020 Award
I don't get this at all. You say you disagree with me, but then you produce a case where the 2nd derivative is 0 and the force is a maximum, but this is not a max or min or inflection point? This appears to be a contradiction.
The force is the first derivative of the potential energy. The only place where the force can be a maximum is where the derivative of the force, or the second derivative of the potential energy is 0.
I think your confusion is that you didn't just stop after the relationship was described properly with Maths. There is no point in 'talking around' something as straightforward as the way Hyperphysics states it. Whatever "somebody" said about it, it just adds confusion when that statement is included in the formal definition. If you understand the Maths then just use it and get familiar with the way it applies. The content of the first post is correct so why are we still discussing it, twenty posts later? Ignore your unhelpful, unofficial source.

jbriggs444
Homework Helper
I don't get this at all. You say you disagree with me, but then you produce a case where the 2nd derivative is 0 and the force is a maximum, but this is not a max or min or inflection point? This appears to be a contradiction.
The force is the first derivative of the potential energy. The only place where the force can be a maximum is where the derivative of the force, or the second derivative of the potential energy is 0.
You are correct, @ZapperZ was confused.

ZapperZ
Staff Emeritus
I don't get this at all. You say you disagree with me, but then you produce a case where the 2nd derivative is 0 and the force is a maximum, but this is not a max or min or inflection point? This appears to be a contradiction.
The force is the first derivative of the potential energy. The only place where the force can be a maximum is where the derivative of the force, or the second derivative of the potential energy is 0.

This is what you said:

willem2 said:
ou can find those points by differentiating the potential energy twice with respect to position and setting that equal to 0. Only in those points can the force be maximal. (there could also be a minimum, or an inflection point)

And this is what I argued against (bold added here):

ZapperZ said:
You can have an almost vertical straight line on the V vs x graph (i.e. 2nd derivative is zero but not at a max or min or inflection point), and yet, this is where a force can be a maximum.

In other words, there isn't any need for there to be a LOCAL MAXIMUM OR LOCAL MINIMUM.

I'll give you another example since there straight-line curve didn't sink in with you: U = ½ kx2

What is the 2nd derivative of that? Is it zero?

Yet, there IS a maximum force for this potential when we do a simple Hooke's law experiment.

Zz.

I'll give you another example since there straight-line curve didn't sink in with you: U = ½ kx2

What is the 2nd derivative of that? Is it zero?

Yet, there IS a maximum force for this potential when we do a simple Hooke's law experiment.

If the potential is reallly ½ kx2 for all x, there is no place where the second derivative is 0, but there is also no place where the force has a maximum. Of course real springs will break at some point, but U = ½ kx2 won't be valid anymore before this happens.

ZapperZ
Staff Emeritus
If the potential is reallly ½ kx2 for all x, there is no place where the second derivative is 0, but there is also no place where the force has a maximum. Of course real springs will break at some point, but U = ½ kx2 won't be valid anymore before this happens.

But if you do a Hooke's law experiment, that IS the potential! This is the common harmonic oscillator potential! We even used this in quantum mechanics!

It means that the force has NO LOCAL MINIMUM OR MAXIMUM. But if you stand in front of the class and claim that no where in the oscillation is there a maximum force, you will be making a false statement! There ARE two points where the force on the spring-mass system is maximum, and those are the two ends/turning points when the spring is extended the most! On the graph, these are the maximum limits of "x".

This was why I insisted that one looks at the actual force curve, rather than blindly applying mathematical rules to find such "maximum force"! One can find maximum force even if the system has no local max/min.... IF one pays attention to the physics!

Zz.