green-beans said:
Homework Statement
During class, we were considering position dependent forces and we used the following differential equation to describe what is happening to the object in this case:
m*(d2x/dt2) = F(x) (i.e. ma=F(x))
Then, we were introduced the potential V(x) which we defined as:
F(x) = -(dV/dx) = -mg
I am a bit confused where did the negative sign come from and why it is equal to -mg as I though F(x) is the force dependent only on position (and not g).
Also, I do not quite understand what physical meaning does potential V have. Does it just show the energy level of an object at a particular position in space?
I hope this is not confusing and thank you for your help in advance!
Homework Equations
As stated before the following equation was introduced, the meaning of which I cannot really understand:
F(x) = -(dV/dx) = -mg
The Attempt at a Solution
I suspect the sign has something to do with the direction in which the position changes.
You would understand the negative sign better as soon as you think of a concrete example, a body being lifted or falling under gravity, a spring compressed or stretched, a gas compressed adiabatically,... Then you can see with the convention of the negative sign, higher potential is visualisable as a 'store of energy', of ability to move, or more precisely to accelerate, matter.
You ask 'where does the negative sign come from?'. In a sense it doesn't come from anywhere, it was not derived from anything nor logically forced on us. It was given you just as a
definition. In theory you could equally well have defined potential with a positive sign instead of negative. We would then have had a law that in a purely position-dependent system, the difference between potential and kinetic energy Is constant. We then would have had books with all the same physics but half of the signs in the equations would have been opposite to what we have. However you maybe can see that it would have been a lot less suggestive and convenient than the definition adopted (which you are now obliged to keep always in mind and stick to), refer again to my examples above. We can then say we have a 'conserved quantity', total energy of such a system is conserved, is constant.
For your question you thought the force depended only on position. Well firstly I take it you have no problem with the fact that m
should be there - there is more gravity force on a cannonball than on a penny. In a system of a lever, a penny could lift up the cannonball with no change in total potential, or without change of total potential the cannonball could lift up the penny, the cannonball descending a short distance and the penny arising a much largeer distance. Maybe somebody will explain this better or with a better example.
H'mm well saying the force depends only on position is maybe not quite accurate, it would be more accurate to say the force
on that given body varies only with its position. The force can depend on other factors too. For example if the body is electrically charged, the force on by another charge depends on charges. But when they are fixed, it varies according only to position and you still have conservation of energy. Then when you go to into it you find that really this conservation of energy only works because other things, mass and charge are conserved too. If the body was made out of dry ice which is evaporating, then the force on it will vary with time and not according its position only.. So then you would have to ask or define what exactly is a 'body', and say that a lump of dry ice is not one, It's doesn't have the requisite properties of permanence. Quite philosophical. I think you are in fact forced to say there is an underlying scientific law needed to make a sensible system, that says '
there exist bodies of constant mass'.
You ask what is the physical significance of potential or potential energy? Really it's all in your first paragraph, that's it. Plus it is very helpful concept for thinking about all dynamics of bodies. The conservation law means that it can be thought of like a
thing, as
stuff, that can be divided up, reapportioned, in physical processes just like stuff, spoonfuls of sugar or something.
Stuff Is something very concrete isn't it? We seem to call it 'mass', and as such it is a very valuable possession of physicists (and engineers and chemists) that
helps them think. They will tell you that they think
physically and concretely. Not airy-fairy abstract like those mathematicians and philosophers. Though how we know about this stuff is because it has dynamics. We know it through these properties of motion. It's not a thing that can be independently observed, we just have to introduce it to make dynamics sensible. As we see, this concrete stuff is really quite abstract. And you have to be quite careful to avoid circular reasoning. So as well as physicists liking the energy concept for helping them think, perhaps they like it also for helping them not to.
But then I'm not a physicist, or mathematician.
You did not need to apologise for being too confusing - the thing is quite subtle really. Maybe I have been worse-
"Does it just show the energy level of an object at a particular position in space?" I think is circular in this context and you could forget that.
