Why is the mean density the same for all nuclei? Tell me if this is a correct theory? So the radius $R$ of the nucleus is directly proportional to $A^{1/3}$ (the nucleon number).
As $$V = \frac 43 \pi r^3,$$ this makes $V$ directly proportional to $R^2$. Also, as the nucleon number increases, the mass also increases and as the masses of protons and neutrons are similar you could say that the mass of the nucleon is directly proportional to the nucleon number.
If you put all of this together, you get the mass of the nucleon being directly proportional to the volume where the constant is the density.
Thus, that is why the density is constant for all nuclei?
 A: Indeed, this is a non-trivial fact about the nuclei, as opposed to atoms. The atoms are held together by the Coulomb forces, and their density increases (i.e. they become more compact) as the charge of the nucleus and the number of electrons increase.
A: Density is $$\rho = \frac mV$$
Expressing for nuclei mass and volume, gives :
$$ \rho = \frac{A\,\mu}{4/3 \,\pi \left(r_o A^{1/3}\right)^3} $$
Simplifying gives :
$$ \rho = \frac{\mu}{4/3 \,\pi~ r_o^3 } $$
Where $r_o = 1.25 ~\text{fm}$ and $\mu$ is typical nucleon mass. Thus nuclei density is constant.
A: Yes, the nuclear density is approximately constant with nucleon number. Expanding on Vadim's answer, perhaps I can explain why this is somewhat intuitive.
Nucleons are held together by the residual strong force, which is attractive between nucleons and mediated by massive pions. As as result, it has limited range such that nucleons only feel the pull of their nearest neighbours (and to a much lesser extent their next-to-nearest neightbours). Consequently, we expect that we can keep adding nucleons to the nucleus without it increasing in density, as the inner nucleons do not attract the outer nucleons, which would in turn lead to a more compact (and hence more dense) nucleus.
Now in practice, one should expect that the nuclear density will in fact decrease slightly with radius, as the repulsive (electromagnetic) force between protons in the nucleus has an infinite range.
If instead the nucleus were made of nucleons held together by an infinite range, attractive force, then we would expect the density to increase with nucleon number.
