I do understand this from a energy point of view. However, let's consider a system with two small mass point in a classical case. The total mass of the system should be $M=m_{1}+m_{2}-\frac{E}{c^{2}}$ where $E$ is the gravitational potential energy between $m_{1}$ and $m_{2}$. I know this effect is very small and can be ignored in classical mechanics, but it should be okay if we insist to consider it. If so, how this effect works when measure the mass of the system. Why is the mass of the system larger when the mass points are far apart?
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4$\begingroup$ The question in the title seems unrelated to the question in the body. $\endgroup$– probably_someoneCommented May 25, 2020 at 23:20
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$\begingroup$ If you understand it from an energy point of view, what else is there to ask about? $\endgroup$– Jon CusterCommented May 25, 2020 at 23:23
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$\begingroup$ @probably_someone suppose you are in high school then such question is very reasonable. This may be why SR is tagged too here. $\endgroup$– jw_Commented May 26, 2020 at 1:54
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$\begingroup$ Not only the potential but also the kinetic energy in the rest frame of the two masses contributes to the rest frame energy, hence the mass, of the system. $\endgroup$– my2ctsCommented May 26, 2020 at 9:29
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2 Answers
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The change in mass of a composite system due to the kinetic or potential energy of its components is very real. The amount of kinetic and potential energy of the components of the system affects how the composite system accelerates when you apply a force to the system as a whole, and it also affects the amount of gravitational force it exerts on other masses.
Nuclei are a great example of this - their binding energy actually ends up decreasing their mass, and we can detect this quite easily in, for example, mass spectrometers (which measure how charged particles accelerate in an electromagnetic field, behavior which depends on both their charge and their mass). You can actually change this mass deficit, as it is called, by putting the nucleus in an excited state. In the excited state, the number of protons and neutrons in the nucleus are the same, but its mass is different! This is because a nucleus in an excited state has a different binding energy.
answered May 26, 2020 at 3:00
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$\begingroup$ Thanks for the answer. I suppose my question is how to derive this from the force analysis. We should be able to obtain the same result when we analyse the force and the energy. $\endgroup$ Commented May 26, 2020 at 10:38
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$\begingroup$ And also in special relativity, we have $E^{2}=(pc)^{2}+(mc^{2})^{2}$. Why there is no term relating to the potential energy? $\endgroup$ Commented May 26, 2020 at 10:48
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$\begingroup$ @Yukuncao At sufficiently low speeds, we can still say $F=ma$, so the total mass of the composite system can be measured by dividing a known force by the acceleration produced by that force. The tricky part is exerting a force such that the whole composite system accelerates, without changing its internal structure. This is fairly straightforward for nuclei, since there's such a large energy gap between their ground state and the first excited state, but for gravitationally-bound systems it will be difficult to do this without changing the kinetic and potential energy of the components. $\endgroup$ Commented May 26, 2020 at 15:08
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1$\begingroup$ @Yukuncao The potential energy term is part of $m$. $\endgroup$ Commented May 26, 2020 at 15:11
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Why is the mass of the system larger when the mass points are far apart?
The mass of the system is not larger when the mass points are far apart or are close together. Since there is no interchange between mass and energy during this process.
During nuclear binding process, as the process you described, some energy is stored as gravitational potential energy and also EM potential energy, these stored energy are very large since the particles are brought to very close to each other. These are based on gravitational and EM interaction.
During this process, nuclear reaction also take place, and mass is converted to energy in this reaction, that is the mass is decreased. This is based on strong interaction. This reaction only happen if the two particles are brought to close enough to each other, or only the process you described will happen during which there is no mass change since there is no process in which mass is converted to energy.
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1$\begingroup$ There is definitely an interchange between mass and energy in the example in the question. For any composite system in special relativity, the kinetic energy of the individual masses and the potential energy of the gravitational field both contribute to the total mass of the system. The energy contained in the gravitational field increases as you increase the separation between the masses, so the mass of the composite system increases as a result. $\endgroup$ Commented May 26, 2020 at 2:24
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$\begingroup$ This answer is incorrect. It is based on the pop-sci idea that E=mc^2 describes a conversion between mass and energy. I.e. that mass can change into energy by ceasing to be mass or vice versa. This is wrong. Mass at all times has energy and energy at all times has mass. If a system at rest loses energy then it loses mass because the energy it lost has mass and took that mass away. $\endgroup$– DaleCommented May 26, 2020 at 11:13
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$\begingroup$ @probably_someone Yes if you do the exact calculation there will be mass change, but that is not the point of the question right? The question talks about the much larger mass change during the reaction. $\endgroup$– jw_Commented May 30, 2020 at 4:10
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$\begingroup$ @Dale But shouldn'd a pop question be answered in a pop way? This incorrectness seems doesn't prevent the answer to answer the point of the question right? Or say does such details need be made clear for such question? $\endgroup$– jw_Commented May 30, 2020 at 4:12
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$\begingroup$ @jw_ not when the pop science answer is wrong. Your whole first paragraph is factually wrong. As is the last half of your last paragraph $\endgroup$– DaleCommented May 30, 2020 at 12:24