What is the root mean square energy of a nucleon bound within a nucleus? How does the root mean square energy of a nucleon in a  nucleus of atomic mass number A in the ground state  depend on A ?
 A: The energy of a nucleon in a nucleus is not an observable quantity (think about how you would measure it), and so is entirely dependent on the theoretical model used to describe the nucleus. Once nucleons are bound into a nucleus, they don't retain their individual identity. There is just "the nucleus". In principle, one can --with a suitably large computer--calculate properties of nuclei directly from quarks and gluons without ever making reference to protons and neutrons. 
With that said, a simple theoretical model is a fermi gas of nucleons. For a fermi gas, the kinetic energy goes like the density to the 2/3 power. As Lewis Miller pointed out, nuclei saturate and so the density is roughly constant and independent of mass number at 0.16 nucleons per cubic fm. This gives a kinetic energy on the order of 40 MeV, with a potential energy that roughly cancels it to give about 8 MeV per nucleon. Keep in mind that this is a model, and that all we can actually measure is the total energy of the system. 
A: http://hyperphysics.phy-astr.gsu.edu/hbase/nucene/imgnuk/bcurv.gif 
The answer to your question is contained in the curve linked above. 
Details:  As can be seen from the binding energy curve, the binding (per nucleon) increases sharply as more nucleons are added at the lower mass numbers; it reaches a maximum at the Iron group (mass number near 56); and declines very slowly as the mass number A increases further.  
This behavior is due to a phenomenon known as the Saturation of Nuclear Forces. Because the nuclear force between nucleons is short ranged, the addition of more nucleons to a large A nucleus does not result in tighter binding (which is the case for electrons in a large atom).  If the nuclear force were the only interaction felt by nucleons, then the binding per nucleon would become independent of A for large A. The long range electromagnetic repulsion between protons breaks this pattern and leads to the slow decrease in binding for large A. By the same mechanism, the binding at low A increases sharply in the low A range as the surface to volume ratio declines with increasing A.
Nuclei in their ground states are quantum in nature and like atoms exhibit a shell structure. At certain mass numbers, shell closures lead to enhanced binding in analogy to the noble gas atoms. For nuclei, these shell closure numbers are called magic numbers and their occurrence leads to the small scale variations in the binding energy curve.
