How come the protons in the nucleus of an atom do not spin around like the electrons do?
If you mean why are all the protons in the middle with the electrons moving around them, the following explanation might help:
The protons and neutrons in the nucleus are made of elementary particles called quarks. Quarks feel a force that electrons do not - the strong nuclear force. It is this force which holds the protons and neutrons together in the nucleus. Electrons do not literally orbit the nucleus as is sometimes depicted. This is just the easiest way to picture it. Really they exist in a sort of cloud around the nucleus.
But perhaps you are talking about the "spin" property of electrons, sometimes pictured as the particle spinning on its own axis. If this is what you are referring to then the answer is that protons (and neutrons) do spin. The property of spin is the same for electrons and for protons. They are both "spin 1/2 particles". This is a property that tells us about the quantum mechanical behaviour of the particle and behaves mathematically in the same way as angular momentum.
Alys Morgan, M.S., Physics Grad Student, Oxford
'Science is facts; just as houses are made of stones, so is science made of facts; but a pile of stones is not a house and a collection of facts is not necessarily science.'