Imagine a system made of 4 pieces, each of mass m. Einstein pointed out that the total mass of the system should be Mc2 = ℰ + 4mc2
Here ℰ is the total internal energy of the system, the sum of the kinetic and potential energies of the 4 particles. But for systems in nature, the size of an atomic nucleus or larger, the total internal energy of a bound system is negative. (The trick is that when the potential energy between constituents approaches a constant as r approaches infinity, we set that constant to zero. Energy is always arbitrary up to a constant.)
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For example, and as a
reminder,
here is the potential energy for the attractive force
between two unlike charges. It is
negative everywhere and goes to zero (by our choice)
at
infinite distance between the charges. In a
system bound by an attractive force, each pair of
interacting particles has a negative total potential
energy! |
Notice this means that for such
a bound system, where the sum of the
total kinetic energy plus the sum of the total potential
energies of the pairs of interacting particles will always be
negative, in our example, M is less than 4m!
Physicists define the binding energy of a system as B = -ℰ,
so that B is the work it would take to disassemble the
system completely. Example: suppose each
particle has a mass of 10 MeV/c2, and the mass of
the system is measured to be 30 MeV/c2. What
is the binding energy of the system? ℰ = Mc2 - 4mc2
= 30 MeV - 40 MeV = -10 MeV. It would take 10 MeV of
work to pull the system apart.