# Higer potential energy->more mass?

## Main Question or Discussion Point

Higer potential energy-->more mass?

If I go from the cellar to the loft, does my mass increase? If not, then where is the potential energy stored?

Thanks for helping an unsure student! =)

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CompuChip
Homework Helper

In theory, yes. However, it is a nice exercise to calculate how much percent of your total body weight that would be (calculate the extra potential energy and use $\Delta m = \Delta E / c^2$). You will find that it is in fact so small that it is even immeasurable with the best instruments we have... probably your calculator will just even give zero.

I get a weight increase of a factor of order 10-14 when climbing a 100 meter building from the surface of the earth... that's about one millionth of a millionth of a percent!

jtbell
Mentor

Potential energy is a property of a system, not of the individual objects that make up the system. So any mass associated with potential energy contributes to the mass of the system, and you can't consider it as being localized or distributed among the individual objects in the system. In your case, the "system" is you plus the Earth, and it's kind of difficult to test this by "weighing" you and the Earth together.

A better example is an atomic nucleus, whose mass is less than the sum of the masses of the individual protons and neutrons in it, because of the (negative) potential energy that they have when they are bound together.

I read this thread earlier and found it interesting. It raised an interesting problem to me. As you said the increase in mass for a short height is extremely small. However there is no bound to how "high" you can raise an object. You could increase the distance forever, which should also mean the energy stored in and thus mass would increase forever. I don't think this is the case. I may be mistaken but I thought that since gravity has no distance limit every bit of matter should have gravitational potential energy relative to every other bit. Two very distant stars should have a significant increase in mass from the high potential energy. This also presents a problem since that increase in mass should increase the gravity between them, and create an endless cycle. Each increase in mass would increase the gravity by a small but non zero amount.

I tried to figure this out for myself, I realize you can't use the simply formula for potential energy at great distances, as g doesn't remain constant, so I used the more complex:
$$J = -G \frac{m_1 m_2}{R}$$
However I was confused by it's usage. The result is a negative number, which got higher (closer to 0) as the distance increased. By doing one calculation with the radius of the Earth, and another with radius + 1, and taking the difference I found that the result agreed with the simpler formula. So I could find the difference in energies based on a change in distance. But I was confused as to how to get the energy for a given distance. The obvious solution was to compare it to a 0 distance, but then I had to divide by 0. So I settled on using 1, since that would effectively remove the distance variable from the formula. Giving me J = -G * m1 * m2 to find the maximum energy two masses could have, then using the formula to find out how much they had for the given distance, and then finding the difference.

Anyway, I want to know if J = G * m1 * m2 gives the maximum amount of gravitational potential energy two masses could have relative to each other. If it does it would help explain why everything doesn't have an infinite mass from potential energy, since there would be a finite limit on the increase. To see if I understand all this I did the calculation for the potential energy Jupiter has relative to the Sun.
m1 = 1.98892 * 1030
m2 = 1.8987 * 1027
R = 7.5 * 1011
J = G * m1 * m2 gave me 2.51996663 * 1047 for a maximum energy, and the formula for the distance gave me 3.35995551 * 1035, and their difference was so close to the max that I my calculator didn't even bother. So this means that Jupiter's potential gravity relative to the Sun is 2.51996663 * 1047J, which is very close to the maximum possible, correct? If that is correct that is 2.80384101 * 1030kg, which is about double the combined mass of the Sun and Jupiter alone. That doesn't seem like it can be true. And if it is shouldn't this mass then have to be taken into account?

QuantumPion
Gold Member

G is in the units of m^3/kg/s^2. So what you calculated with r=1 is the gravitational force of the Sun and Jupiter if their center of masses were 1 meter apart, which is impossible since the surfaces of both bodies is obviously much larger.

Yes, but we are talking about potential energy here, just because you'd have to collapse the Sun and Jupiter into singularities to utilize it doesn't mean it's not there. If the energy is stored in them, no matter how difficult it may be to release it, there should be an increase in mass due to it.

jtbell I didn't see your reply while I was typing mine. It did seem like it would help explain why this added mass doesn't influence our calculations of the orbit of Jupiter. However, even if it wouldn't affect the orbit of Jupiter, I don't see how it wouldn't change the orbits of the more outer (dwarf) planets, for example Pluto. If my above calculation for the potential energy is correct then the Sun Jupiter system should have an 140% increase in mass from the added energy.

Yeah increasing the potential energy will increase the mass but it will be negligible. It's also interesting to note that an increase in mass is not the only unusual aspect of an increasing the gravitational potential energy, time dilation is another.

rcgldr
Homework Helper

Take the case of a 2 body system in space with elliptical orbits. Wouldn't the total energy remain constant while cycling between kinetic energy (bodies close at high speed) and gravitational potential energy (bodies far apart a slow speed)? Wouldn't the net total mass of this system remain constant?

jtbell
Mentor

Yes. Ignoring gravitational radiation, of course.

Given the example of the Earth and the Moon G * m1 * m2 = 2.93 * 1037J, and the difference at average orbital distance is negligible. This gives an increase in mass of 3.26 * 1020kg, which is about a 0.0054% increase in mass. Shouldn't this mass affect the orbit of the Earth and Moon system around the Sun?

QuantumPion
Gold Member

Given the example of the Earth and the Moon G * m1 * m2 = 2.93 * 1037J, and the difference at average orbital distance is negligible. This gives an increase in mass of 3.26 * 1020kg, which is about a 0.0054% increase in mass. Shouldn't this mass affect the orbit of the Earth and Moon system around the Sun?
I only get a mass defect of 9.4*10^10 kg, calculating the difference in GPE of the earth-moon system between apogee and perigee of orbits. This is only one trillionth of a percent difference in mass of the system. Although this does not take into account the energy of orbital mechanics which I am not well versed in.