# Significance of negative potential energy

#### UMath1

Using the potential energy formula -GMm/R^2 I understand that the maximum potential energy is 0. But what is the significance of the potential energy as it approaches negative infinity? I am not sure how exactly to word this, but I cannot seem to grasp the what negative potential energy tells us about an object.

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#### mfb

Mentor
The potential energy is lower than the (arbitrary!) reference point: you have to put energy in to move the objects apart. The closer the objects are, the more energy you need to separate them.

#### Nugatory

Mentor
Using the potential energy formula -GMm/R^2 I understand that the maximum potential energy is 0. But what is the significance of the potential energy as it approaches negative infinity? I am not sure how exactly to word this, but I cannot seem to grasp the what negative potential energy tells us about an object.
You can put the zero point anywhere you want. For example, I could say that the potential energy of a mass sitting on the floor of my living room is zero ($mgh$, and $h$ is zero because it's sitting on the floor). I pick it up and put it on a table one meter high, and it's potential energy becomes $mg\times{1m}$ ($mgh$, and $h$ is one meter). But what's the potential energy of the same object if it's resting on the basement floor, two meters below the living room floor? The potential energy is less than when the object is in living room, which means less than zero, so it has to be negative (in fact, it's $mg\times{-2m}$ because $h$ is -2).

If that negative potential energy bothers me, I can make it go away by taking advantage of the fact that my living room floor is 100 meters above sea level... Now the potential energy of the object in the basement $mg\times{98m}$, on the living room floor is $mg\times{100m}$, and on the table is $mg\times{101m}$. Of course the differences in the energy are still the same, and it only those differences that are physically significant.

When you're dealing with gravity, or any other force that falls to zero at infinity, it turns out to be very convenient to set the zero point to be the potential energy at infinity. But it's just a convention, and you could put it somewhere else if you wanted.

#### jtbell

Mentor
Using the potential energy formula -GMm/R^2
Correction: gravitational potential energy (with the reference point at infinity) is -GMm/r (not r^2).

#### UMath1

So basically the negative potential energy tells the amount of work required to move the object to its maximum potential energy?

#### HallsofIvy

Science Advisor
So basically the negative potential energy tells the amount of work required to move the object to its maximum potential energy?
The crucial point is that the value of the potential energy, whether positive or negative, has no "physical significance". The only thing that has physical significance is the difference in values of the potential energy at different points. If the potential energy at one point is -30 and at another point is -20, that is exactly the same, physically, as if the potential energy at the first point were +1000 and at the second point +1010.

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#### nasu

So basically the negative potential energy tells the amount of work required to move the object to its maximum potential energy?
No, the maximum has no relevance. The work is to move the object to the reference point for potential. As the reference point is arbitrary, it can be anywhere. Including the maximum, but not necessarily there.

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