Relationship between force and potential energy

[ZapperZ]

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.

 

[sophiecentaur]

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 but it doesn't make my Physics wrong.

 

[ZapperZ]

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 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.

 

[sophiecentaur]

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 realize 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.

 

[ZapperZ]

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 realize 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.

 

[sophiecentaur]

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.