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Protons stay together in the nucleus due to the strong nuclear force, otherwise known as binding energy. This force is the fundamental glue, so to speak, in everything. It overshadows the electromagnetic force by several orders of magnitude, so that the protons do not fly apart due to like charges repelling each other.

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Q: The most abundent isotope of lead contains 82 protons and 124 neutrons packed closely together in the nucleus why do the protons stay together in the nucleus rather than fly apart?
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