Relationship between force and potential energy

  • #26
sophiecentaur
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One can find maximum force even if the system has no local max/min....
The use of the term "maximum force" can mean one of two things. 1. The Force / Displacement Law has a local maximum or 2. You chose to limit the range of forces or displacements in your experiment.
Why is this verbal jiggery pokery being involved with a perfectly good mathematical description? If you want to be realllllly precise then include some more formal definitions and caveats such a that the function is continuous and differentiable within the limits being considered etc. etc. but would that really help?
Mods please help and put this thread out of its misery.
 
  • #27
ZapperZ
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The use of the term "maximum force" can mean one of two things. 1. The Force / Displacement Law has a local maximum or 2. You chose to limit the range of forces or displacements in your experiment.
Why is this verbal jiggery pokery being involved with a perfectly good mathematical description? If you want to be realllllly precise then include some more formal definitions and caveats such a that the function is continuous and differentiable within the limits being considered etc. etc. but would that really help?
Mods please help and put this thread out of its misery.
I don't understand your complain. This IS a physics question and not purely a mathematics question, is it not? After all, this was attached to the concept of "potential energy".

I gave a very simple example that did not conform to what was claimed, i.e. using the ONLY the 2nd derivative of the potential energy to find the "maximum force". I showed a specific example where using blind mathematics, you cannot get such a thing from something that is a COMMON example in intro physics. And yet, in the physical scenario, there ARE points where one does get "maximum forces".

Someone who reads Post #14, and then learn about harmonic oscillator, will get VERY confused.

Zz.
 
  • #28
sophiecentaur
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Someone who reads Post #14, and then learn about harmonic oscillator, will get VERY confused.
That is out of context but not wrong. The restoring force is proportional to the displacement away from the rest position. Any given oscillator falls into category 2 in my post. The maximum displacement is arbitrary and is chosen of the particular experiment and is nothing to do with the Force Law. Confusion could be caused but only because of lack of description. The PE variation with time is sinusoidal and the maximum of force corresponds to the maximum of PE. But this is just rehearsing what we know already and the sheer number of these posts is contributing more to confusion than any statement in any particular post.
As with many questions and solutions to many problems, the initial statement about the physical situation has to be clear and bombproof. Introducing SHM is a red herring because it introduces Time into the situation and the term "maximum" can either imply dy/dx = 0 or dy/dt = 0. There's your potential confusion.
 
  • #29
ZapperZ
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That is out of context but not wrong. The restoring force is proportional to the displacement away from the rest position. Any given oscillator falls into category 2 in my post. The maximum displacement is arbitrary and is chosen of the particular experiment and is nothing to do with the Force Law. Confusion could be caused but only because of lack of description. The PE variation with time is sinusoidal and the maximum of force corresponds to the maximum of PE. But this is just rehearsing what we know already and the sheer number of these posts is contributing more to confusion than any statement in any particular post.
As with many questions and solutions to many problems, the initial statement about the physical situation has to be clear and bombproof. Introducing SHM is a red herring because it introduces Time into the situation and the term "maximum" can either imply dy/dx = 0 or dy/dt = 0. There's your potential confusion.
I did not introduce time. You did. The PE variation is also in "x", the extension of the spring. This is the ONLY scenario that I have used, which is the only parameter that the OP brought up. This is not a question on the dynamics of the system. So if anyone is pushing this out beyond the original confines, it isn't me.

Post #14 implied that there is a general rule of determining maximum force, simply by applying the 2nd derivative of U. I disputed that by giving a very simple and common example and showed why that general rule does not work on something that every general physics student has seen. I do not understand why this is a problem!

Zz.
 
  • #30
sophiecentaur
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I did not introduce time. You did.
Iirc, you introduced the possible problems when students come across the Harmonic Oscillator. Is there an earlier mention of time? If you use the term "derivative" then the independent variable needs to be stated. Up till your post the dx was all we were concerned with and it would be best to avoid mention (implied or otherwise) of dynamics.
You were right to challenge the use of the second derivative as it doesn't help but the OP was almost driven by the thread into over-thinking the issue.
Haha - it didn't help that the original statement in the first post was wrong.
I will get on my soap box and shout the praises of Worked Examples from The Book. Doing a couple of those can usually put people right, far better than any amount of chat.
 
  • #31
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Iirc, you introduced the possible problems when students come across the Harmonic Oscillator. Is there an earlier mention of time? If you use the term "derivative" then the independent variable needs to be stated. Up till your post the dx was all we were concerned with and it would be best to avoid mention (implied or otherwise) of dynamics.
You were right to challenge the use of the second derivative as it doesn't help but the OP was almost driven by the thread into over-thinking the issue.
Haha - it didn't help that the original statement in the first post was wrong.
I will get on my soap box and shout the praises of Worked Examples from The Book. Doing a couple of those can usually put people right, far better than any amount of chat.
The ONLY usage of the word "derivative" was in relation to finding the gradient of "U". Until you brought it up, the derivative with respect to any other variables, such as time, was NEVER an issue. So if anyone here is confused, it might be you.

BTW, unless you have mixed things up, the OP and the person who posted Post #14 that I am disputing are two different people.

Zz.
 
  • #32
sophiecentaur
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Until you brought it up, the derivative with respect to any other variables, such as time, was NEVER an issue
If time is not an issue then how would a student be confused in the context of a harmonic oscillator? The maximum displacement in SHM is arbitrary and depends on where it's been let go.- That's my second category (2) of Maximum and it doesn't involve differentiation with respect to displacement. I am not confused about that. I really think that bringing in SHM was not helpful.

Talking to the wrong person is a common problem :smile: but it doesn't make my Physics wrong.
 
  • #33
ZapperZ
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If time is not an issue then how would a student be confused in the context of a harmonic oscillator? The maximum displacement in SHM is arbitrary and depends on where it's been let go.- That's my second category (2) of Maximum and it doesn't involve differentiation with respect to displacement. I am not confused about that. I really think that bringing in SHM was not helpful.

Talking to the wrong person is a common problem :smile: but it doesn't make my Physics wrong.
No, but it shows that you are equally confused in many of these conversations.

1. The harmonic oscillator has the potential form of U = 1/2 kx2. Notice, I'm not bring in ANY time factor here.

2. Post #14 says that all one has to do to find the "maximum force" is to find where the 2nd derivative (with respect to "x", if there is any confusion here) is zero.

3. Student performs the 2nd derivative of U = 1/2 kx2. He/she gets "k" as an answer, and wonders why he/she can't find the "zero" for this result. Does this mean that there are no values of "x" for the force to be a maximum?

The rule doesn't work when there are no local maxima/minima in the 1st derivative.

Now, where exactly in what I said above is wrong?

Zz.
 
  • #34
sophiecentaur
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The harmonic oscillator has the potential form of U = 1/2 kx2.
Just one turning value. A minimum at x=0 and, of course, a finite maximum displacement but the derivative increases all the time. Would you call that "a maximum" at the arbitrary extremes of displacement?
Does this mean that there are no values of "x" for the force to be a maximum?
In the displacement / force law there is no 'maximum' because there is no turning value up there. I think your confused student would just sit and think about it for a minute and realise the difference. Just because the oscillator only explores a part of the Hooke's law doesn't mean that it's found a maximum. If he/she is interested enough to find this a problem all that's necessary is to increase the displacement and prove that the force still increases. Hooke's Law doesn't predict a turning value anywhere until the spring breaks. I guess that would be a valid maximum.

No, but it shows that you are equally confused in many of these conversations.
Now now! That should be in a PM, I think. :wink:
 
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  • #35
ZapperZ
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Just one turning value. A minimum at x=0 and, of course, a finite maximum displacement but the derivative increases all the time. Would you call that "a maximum" at the arbitrary extremes of displacement?
There's no "time". The force is linear with displacement. There is no "maximum" as in a "peak" in the curve, and thus, no zeros in the second derivative with x. But there is a maximum force that is applied by the spring, and that occurs at maximum displacement. This can't be found simply by blind mathematics. It can only be found by understanding the physics, i.e. the system has a maximum displacement, and at that maximum displacement, one gets maximum force.

In the displacement / force law there is no 'maximum' because there is no turning value up there. I think your confused student would just sit and think about it for a minute and realise the difference. Just because the oscillator only explores a part of the Hooke's law doesn't mean that it's found a maximum. If he/she is interested enough to find this a problem all that's necessary is to increase the displacement and prove that the force still increases. Hooke's Law doesn't predict a turning value anywhere until the spring breaks. I guess that would be a valid maximum.
But that is exactly what I've been arguing about! This can't be done simply by blind mathematics of "find the zeroes in the 2nd derivative"! It requires the PHYSICS and knowing that the force is linear with displacement, and that there is a range of values for the displacement. The mathematics does NOT tell you this.

Early on in this thread, I gave the simplest answer that I believe matched this thread level, which is simply to look at the slope of the U(x) curve, and find where the slope is maximum. That corresponds to the maximum absolute value of the force. This works no matter if there are local maximum in the force curve or not. If it is a straight line, then the maximum slope is the same everywhere. This simplest approach works and it is what WE ALL know here, and was clearly described in the Hyperphysics link. I do not know why we just did not stick with this and added on the complications of the 2nd derivative that doesn't even work all the time in picking up the location of maximum force. To me, this thread has been sufficiently answered PRIOR to Post #14.

Zz.
 
  • #36
sophiecentaur
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The mathematics does NOT tell you this.
Without using Maths, how would you know the Potential Energy for a given displacement?
picking up the location of maximum force.
That is entirely up to the choice of the experimenter and has nothing do do with the Physics of the system. And I think this is the nub of our disagreement. If one was interested in the Physics of the setup, one would very soon come to the conclusion (experimentally) that there is no 'Maximum' except either when the experimenter chickens out or the spring departs from the simple law and would you seriously call that a maximum? (Apart from it being the biggest number used).
To me, this thread has been sufficiently answered PRIOR to Post #14.
I totally agree and have made this same point several times higher up. And slope of PE against distance, as we all agree, tells you the Force. (Ye Gods - that's more than thirty posts.
 

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