The conservation of mass says, if the initial state has a mass M_init and this system undergoes some changes, then the sum of the masses of all products MUST be equal to the same amount M_init. For example chemists could not leave without this law, they have initial chemicals which undergo a chemical reaction, then the sum of the masses of all products must be exactly the same as the sum of the masses of the initial components.
Also, imagine that you are sitting in a car with your dog. You drive the car to a weight station ( you know, these exits on interstates when you cross a state line, and all trucks must go and be weighted) and sitting in the car with your doggie you measure the weight of this system ( you+car+doggie). You measured let us say 2120 pounds. Then you take your dog and you get out of the car, and you measure your car and let us say you get 2000 pounds. Then you go home, find your mom scale and measure yourself, and suppose you found 100 pounds. Then it is obvious that your dog has a weight of 20 pounds. ( you could not put him onto the scale, because he is always running and would not stay :-))
This approach however would not work with elementary particles such as neutrons and protons and electrons and .... .
The problem there is that the mass of a moving object depends on the speed it moves with. This effect is noticeable only if the object moves very very fast, namely with speeds very close to the speed of light. So in the world of elementary particles ( they tend to move very fast), the conservation of mass as you applied DOES NOT HOLD! The rest masses are not conserved. That is why for rest masses you CANNOT say that the
neutron mass = proton mass + electron mass + neutrino mass
as you did in the case of your car, when you had
the mass of the car with you and your dog = mass of the car + your mass + mass of the dog.
There are other quantities which ARE CONSERVED, such as the energy , and impulse, and those are the ones we physicists use to determine masses and verify theoretical predictions, but that is another story. Hope this explanation helped you to understand where the problem with measuring the masses is.
Cheers Arnold Pompos
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