# Thread: Potential energy and E=mc˛

1. ## Potential energy and E=mc˛

Do objects gain more mass when you increase their potential energy (due to E=mc˛)? Either by lifting them, or rocketing them in to space?

I understand that if there is an increase it will be very small.

(like the way protons get more masive in a accelerator due to their kinetic energy.)

Quite a silly question I know, but you only learn by asking.

No need to bump this one up, so I thank you all in advance.

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Not by lifting them. Neither would magnets gain weight by pulling them apart. Those two store energy in the field.

However, charging a battery or heating something up will add a miniscule weight to it.

The answer is yes, but you need to be careful about where that extra energy went.

3. I must ask...what are we talking about here?....the OP says mass, the 1st answer speaks of weight...

Mass and weight are not the same thing.

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(like the way protons get more masive in a accelerator due to their kinetic energy.)
This is a question of what you mean, and I think for precisely that reason this interpretation is today resisted. All attempts to measure mass depend upon the technologies we have for measuring mass, which start by measuring weight, and proceeding based upon theories we have for converting weights, gravitational mass and inertial mass.

A particle at high speed still has the same rest mass. We can write the equations of its motion in terms of its rest mass. It responds to forces and gravity differently at high velocity, but whether its inertial mass or gravitational mass has changed is a matter of interpretation depending upon how you decide to define these things for a particle at high speed. Precisely because you can write their equations of motion in terms of a rest mass (unaffected by speed) and a relative velocity. Saying it has a relativistic mass is really saying it has both a rest mass and kinetic energy, and the force or gravitation acts interacts also with that kinetic energy.

Certainly if you carry out a chemical reaction, you can measure a difference in mass, which matches the energy of reaction, and which is a kind of potential energy. But if you carry something up a hill, you won't measure any mass change if you take your scales up the hill with it. As someone said, it depends where the energy is stored. In the former case, you are measuring the mass of the item that has stored the energy. In the case of carrying it up the hill, it is the field rather than the object that stores the energy.

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Yes, according to the De Pretto formula, E=mc^2, objects do get a larger mass as they are lifted up or if their potential energy is increased.

6. Order of Kilopi
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We should also note that potential energy is a property of a system, not an object. So where we can for example define the gravitational potential energy of the earth-moon system, it makes no sense to talk about the potential energy of the moon or the earth by itself (unless we're considering either as a system of many bound particles of course). But potential energy does relate to mass in the way you'd expect. If you put a huge black box around the earth-moon system it will appear to have a mass consisting of the mass of the earth, the moon, and the potential energy of the system combined.

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Originally Posted by Sean14
Yes, according to the De Pretto formula, E=mc^2, objects do get a larger mass as they are lifted up or if their potential energy is increased.
I'm pretty sure that applies to the object if the energy of the object itself is increased, for example, by increasing it's temperature.

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Seems like the answers are 50/50. At least there's only two possible answers.

I'm still saying that mass does not increase by elevating an object. Spinning it like a top however will add a ridiculously small amount of mass.

9. Caveman has it right. It's especially interesting to note that since gravitational potential energy is negative for objects a finite distance apart, if you measure the mass of the Moon and the Earth separately (take each one far away from any other masses temporarily, so as to avoid any influence), the mass of the huge black box containing them both will be slightly less than the total, just like the mass of a bound hydrogen atom is a very tiny bit less (13.6 eV) than a proton and an electron separately. Working it out, it's a bigger number than you'd think. The binding energy of the Earth-Moon system is about -8.3 x 1011 kilograms equivalent. Of course, we offset half of that loss with the kinetic energy of the Earth-Moon system, which has a positive value, and would also be counted in the mass of our black box. That represents about one part in 10 trillion of the total mass of the system, which I'm pretty sure is well below the accuracy with which we can measure the mass of things like planets. For the hydrogen atom, the binding energy is on the order of one part in 100 million of the total mass of the system.

Edit to add: Just out of curiosity, I went over to Wikpedia's Orders of magnitude (mass) page to see if there was a good comparison for that amount of mass. It turns out that right at 4 x 1011 kilograms is the total mass of all the people on Earth. I thought that was an amusing coincidence.

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