# Potential energy and equilibrium

It is said that a negative potential energy gives stable equilibrium but a positive potential energy gives an unstable equilibrium. why is it so?

Who says so? It doesn't even make sense since there is no such thing as negative potential energy.

then what about the potential energy of a electric dipole in an external electric field.its negative if the charge is positive and the angle subtended between the field and dipole is between 0 and 90

cnh1995
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then what about the potential energy of a electric dipole in an external electric field.its negative if the charge is positive and the angle subtended between the field and dipole is between 0 and 90
What do you mean by "if the charge is positive"? There are both positive and negative charges in a dipole.
It is said that a negative potential energy gives stable equilibrium but a positive potential energy gives an unstable equilibrium. why is it so?
Do you have any reference to this?

What do you mean by stability of equilibrium?

there are three kinds of equilibrium,namely stable, unstable and neutral. the potential energy of a dipole is negative of the dot product of electic dipole and electric field.
i was wrong when i said charge in my previous reply. instead it will b electric dipole.

It doesn't even make sense since there is no such thing as negative potential energy.

You say that -GMm/r is positive?

• sophiecentaur
sophiecentaur
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It is said that a negative potential energy gives stable equilibrium but a positive potential energy gives an unstable equilibrium. why is it so?
Good try but wrong wording. A situation where Potential energy is at a Minimum is stable and where it is at a Maximum it is unstable. All around a stable point, the points have higher PE and vice versa for an unstable point. This is true, whatever forces (gravitational or E/M forces are involved. Whether the forces are repulsive (positive PE) or attractive (negative PE) in the immediate vicinity, you can still be in a stable or unstable point.
You should read around this in Wiki or elsewhere and you will find many examples which don't appear to totally fit this general rule. A local minimum on a slope can 'detain' a ball in a stable way but a slightly larger displacement can make it roll down hill. Neutral Equilibrium is where a displacement makes no change in PE (A ball on a snooker table)
since there is no such thing as negative potential energy.
You should get your facts straight before leaping on someone. The Potential Gravitational Potential of Earth is
-MG/x, which is always negative and indicates attraction to the Earth.

• cnh1995
Nugatory
Mentor
Any time that you hear people talking (at least in classical physics) about positive or negative potential energy, it will be relative to some arbitrarily chosen zero point and often you are expected to know where that zero point is from the context. Thus, the debate in this thread about whether potential energies are positive or negative makes no sense unless and until the debaters have agree on a zero point. For example, ##E_P=-GMm/r## is negative at the earth's surface with the zero point at infinity; but if I choose to put the zero point at the earth's surface I can use (for values of ##h## that are small compared to the radius of the earth) ##E_P=mgh## which is zero at the surface of the earth and only goes negative if I've dug a hole.

But all of this is irrelevant to the original question, which is pretty much answered by the first part of SophieCentaur's post in #7 above.

I stand corrected, I apologize. As nugatory points out, it's a question of reference point, and I guess in the case of gravity the usual reference point is taken as infinitely far apart, and thus the potential energy can only decrease from the zero point.

Nugatory
Mentor
The Potential Gravitational Potential of Earth is
-MG/x, which is always negative and indicates attraction to the Earth.
Strictly speaking, what indicates attraction towards the earth is not the potential is negative, but rather that the gradient of the potential negative.

• rumborak
Sophie centaur i didnt really understand what u explained. can u please give me the link where its explained?

sophiecentaur
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2020 Award
Strictly speaking, what indicates attraction towards the earth is not the potential is negative, but rather that the gradient of the potential negative.
You gave my confidence a bit of a rattle, there.
-MG/x is still a negative quantity and I was (reasonably enough) using my x origin as the Earth's Centre. I guess you are pointing to situations within a region of negative potential where there could be a repulsive force . That could be, perhaps somewhere near the Moon, on the Earth's side. There would be a force away from Earth - a local 'trap' in an overall negative potential well.

olgerm
Gold Member
It is said that a negative potential energy gives stable equilibrium but a positive potential energy gives an unstable equilibrium.
I don't think it is true. Maybe they said that stable equilibrium points are points, where spatial derivative of potential energy is 0 (##\frac{\partial U}{\partial x}=0##) and second spatial derivative of potential energy is positive (##\frac{\partial^2 U}{\partial x^2}>0##). And unstable equilibrium points are points, where spatial derivative of potential energy is 0 (##\frac{\partial U}{\partial x}=0##) and second spatial derivative of potential energy is negative (##\frac{\partial^2 U}{\partial x^2}<0##).

https://en.wikipedia.org/wiki/Mechanical_equilibrium#Potential_energy_stability_test

sophiecentaur
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2020 Award
Sophie centaur i didnt really understand what u explained. can u please give me the link where its explained?
Try reading this link and see the diagram with the red dots. I was a bit disappointed that I didn't find a large number of links but that's something to start with.

i saw the Wikipedia link and i have made some conclusions. can u please correct me if I am wrong?
we know that any work is done to move from a region of higher potential to a region of lower potential. so when a body aquires stable equilibrium it is in the lowest possible potential energy state. Hence any diaturbance from that state brings it back to the initial one. In case of unstable equilibrium the neighbouring regions have lower potential. So any disturbance causes it to change its configuration altogether. But my question is then why does the body acquire equilibrium at a region of highest potential energy? i understand now what makes it unstable but in the first place what causes the equilibrium.

cnh1995
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i understand now what makes it unstable but in the first place what causes the equilibrium.
The force balance between the driving force and the restraining force.
When a stone is on the ground, it's in stable equilibrium. But when you lift it to a height h and hold it there, at that height, you are providing an equal and opposite force to the gravity (restraining force). Hence, you get an equilibrium which is unstable because the moment you release the stone, it will be back to its original position.

• esha
thanks for explaining it. i think i now understand it thoroughly

sophiecentaur
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2020 Award
i understand now what makes it unstable but in the first place what causes the equilibrium.
That can be achieved in many ways. If you balance a pencil on its point, the energy state is highest. You supplied 'just enough' energy to raise it to that point and to be stationary. There is no net force, one way or another, so it's in equilibrium but it's Unstable because any displacement at all will cause the Centre of Mass to drop and Bang.
You have to go over this in your mind, if it doesn't seem quite right to you, and apply the 'rules' strictly.

• esha
ok.. i think i got it clearly now