After 1911 most scientists accepted the theory that the nucleus of an atom was very dense and very small and has a positive charge. What led scientists to accept this theory?

The breakthrough was the alpha particle scattering experiments of Marsden, Geiger and Rutherford. Basically they fired a beam of alpha particles (which are small, positively charged particles emitted by some radioactive sources), at a thin metal foil. Unexpectedly, instead of the slight scattering of the beam they expected, they found a very few particles were scattered back towards the source. They could only explain this if the metal atoms had a tiny central nucleus with a positive charge to repel the alpha particles. If you search you should be able to find Rutherford's own account of those experiments.