"How do the protons and nucleus know that they have to lose mass to produce energy...?"
Notwithstanding the comments that this is a silly philosophy question, I think it is a good question for precisely that reason: you know that protons cannot "know" things and therefore we must find an explanation not involving "knowledge". It sounds as though you're thinking something along the lines of everything's getting to the end of the reaction and saying "hey, we need to have less mass now", which of course you know is preposterous. So maybe it helps to think of fusion / fission as a process: he protons are in constant contact with the process, and it is their loss of energy that "drives" the process (I use the quotes because the energy loss is necessary, but not sufficient, to make the process happen). At the risk of being too colloquial, you can almost say the protons use a small piece of themselves up in completing the reaction.
(Rest) Mass is simply the property that something has if it has a nonzero energy content as measured from a frame that the particle is at rest relative to. As in the other answers, this notion is neatly expressed in the equation (for the "four-momentum's Minkowski norm"(see footnote)) $E^2 - p^2\, c^2 = m^2\,c^4$: if you are in a rest frame relative to a particle, then its momentum $p$ is nought and it has an energy content if and only if $m\neq0$. So yes, mass really does measure an energy content. So after having taken part in the process, the protons have a smaller energy content than before as measured from a relatively stationary frame, so that they have less rest mass.
But not only does the property rest mass measure energy. It also gives rise to the property of inertia, or resistance of state of motion change to external force, i.e. to the proportionality constant $m$ in Newton's second law. For instance, if we confine a quantity of light inside a perfect, lossless resonant cavity, we can show that the system's inertia increases by an amount $E/c^2$ when we confine the light, where $E$ is the light's energy. I talk about the thought experiment that shows this in my answer here. Indeed, in the same way, most of the mass in your body is owing to the massless, but confined, gluons in the nucleusses of your body's atoms.
For accuracy, I should say that particle physics thinks of many conversions as noncontinuous events: simply branches in a Feynman diagram and does not try to penetrate the "process" or think of it as a continuous evolution as I have implied. But the key idea is that everything is connected and interacting, with the transfer of energy. For example, the fusion reaction of four protons to yield helium in the Sun is thought of as three discrete events:
The proposal of this process as the source of energy in stars led to the award of the 1967 Nobel prize to Hans Bethe
Footnote: I appreciate this phrase is likely to be jargon to you at this stage - I'm not trying to be a bothersome git- I use the phrase because you might like to use it as a phrase to google on as your understanding builds and you want to know where it comes from. See here.
When the spring reaches maximum compression, the mass is instantaneously at rest but it is not in static equilibrium. The net force on it is not zero : $kx \ne mg$. Like a pendulum at the end of each swing, there is a net force on the mass causing it to accelerate towards the equilibrium position - at which the net force on it is then (for an instant) zero. Your calculation is correct. It is your interpretaion of the result which is at fault.
At the lowest point the compression force in the spring is $kx=2mg$, acting upwards on the mass, while gravity is still pulling down with force $mg$. There is a net force of $mg$ acting upwards.
Best Answer
The fact that you can't compute the weight of an isotope by adding up the mass of the neutrons, protons, and electrons is because the kinetic and potential energy of the interactions affects the total energy, and hence mass. But you object because that include nuclear interactions.
OK. Look at molecules. The reason a molecule's weight isn't the sum of the weights of the isotopes of the atoms involved is because the the kinetic and potential energy of the interactions affects the total energy, and hence mass. So that counts.
Has this been measured? Yes. Firstly, we can measure the weight of molecules, and we can measure the weight of a macroscopic amount of molecules. Secondly, the energy difference is available as produced heat or required heat. When we measure the heat released or absorbed, we are measuring the energy associated with the change in mass. Just like in a nuclear reactor, except smaller amounts of energy per gram are needed/released since electromagnetic potential energies and kinetic energies are so much smaller.