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If an object is sitting on a potential slop, why must there be a force to push it into the well?

Qmechanic
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frog_prog
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  • Surely the gradient of the potential is a force ? If the gradient of the potential is non-zero then it will act to move the object in the direction of decreasing potential. In other words $\vec F = - \nabla V$. – gandalf61 Aug 03 '21 at 11:57
  • A potential slop/non zero gradient is just another way of saying that there is a force acting. So what you are essentially asking is why force does what it does. – TheImperfectCrazy Aug 03 '21 at 11:57
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    Does this answer your question? Why are thermodynamic potentials minimised? – Chemomechanics Aug 03 '21 at 12:13
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    By the very definition of that which you measure to determine that you are on a slope!? I actually cannot grok your misunderstanding... It's like asking "if i punch a wall, why is the wall punched?" – PcMan Aug 03 '21 at 13:02
  • @PcMan Here is a possible way of interpreting the question: let's say we define potential energy as: energy that is released when in motion down a potential gradient. According to that definition, in order to release the potential energy the object needs to start moving first. As long as the object hasn't started moving no potential energy is released. It appears to me that there are always multiple ways to define concepts such that one paints oneself into a corner. – Cleonis Aug 03 '21 at 14:57

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Exactly, it is the force. If there is a conservative force, a potential can be defined as a function $U(\vec{r})$ that satisfies $$\vec{F}(\vec{r}) = -\nabla U(\vec{r})$$ Therefore the force pushes towards the minimum of the potential.

The common image of a potential "well" into which the objects "fall" is a great analogy because if you consider the gravitational force field here on Earth, $\vec{F} = -mg\hat{z}$ and so $U = mgz$. In this case a "physical well" (like a hole in the ground) is also a "potential well", because $U$ is proportional to $z$.

But one should be careful not to bring this analogy too far. In general a potential is just a way to express the fact that there is a force, and has nothing to do with height and falling.

Prallax
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It is because of the principle of minimum energy, which states that for a closed system the internal energy will decrease and approach a minimum value at equilibrium.

Dom Tesilbirth Shira
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  • Erm... this is circular reasoning. 'Minimum energy' looks as if it is saying something different, but it isn't; it is rephrasing. The standard example of saying the same thing with different words is in a play by Moliere. In the play he lampoons a group of physicians providing an explanation in macaronic Latin of the sleep-inducing properties of opium as stemming from its "virtus dormitiva". – Cleonis Aug 03 '21 at 14:32
  • 'Minimum energy' looks as if it is saying something different, but it isn't; it is rephrasing. I disagree. The principle is based on entropy maximization, which is new information - not merely rephrasing but providing the underlying framework for all real processes. Entropy maximization simply means that we more often observe situations that have a statistically higher likelihood of occurring, such as a collection of molecules at a lower potential but warmer. One derivation is shown here. – Chemomechanics Aug 03 '21 at 14:41
  • Let me expand on my previous comment about what counts as explanation, and what counts as circular reasoning. Example of explanation: the transition from Kepler's three laws of celestial motion to Newton's laws of celestial motion. The universal law of gravity unifies Kepler's 1st and 3rd law. Kepler's 2nd law (the area law) can be seen as an instance of conservation of angular momentum. Universal gravity gives the means to calculate gravitational influence of multiple sources of gravity. Newton's laws explain Kepler's laws because they describe the physics taking place at a deeper level. – Cleonis Aug 03 '21 at 14:46