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We investigate the occurrence and nature of naked singularity for theinhomogeneous gravitational collapse of Tolman-Bondi dust clouds.It is shownthat the naked singularities form at the center of the collapsing cloud in awide class of collapse models which includes the earlier cases considered byEardley and Smarr and Christodoulou. This class also contains self-similar aswell as non-self-similar models. The structure and strength of this singularityis examined and the question is investigated as to when a non-zero measure setof non-spacelike trajectories could be emitted from the singularity as opposedto isolated trajectories coming out. It is seen that the weak energy conditionand positivity of energy density ensures that the families of non-spaceliketrajectories come out of the singularity. The curvature strength of the nakedsingularity is examined which provides an important test for its physicalsignificance and powerful curvature growth near the naked singularity ispointed out for several subclasses considered. The conditions are discussed forthe naked singularity to be globally naked. Implications for the basic issue ofthe final fate of gravitational collapse are considered once theinhomogeneities in the matter distribution are taken into account. It is arguedthat a physical formulation for the cosmic censorship may be evolved whichavoids the features above. Possibilities in this direction are discussed whileindicating that the analysis presented here should be useful for any possiblerigorous formulation of the cosmic censorship hypothesis.Comment: 41 pages, TIFR preprint TAP 9/9

Naked singularities in spherically symmetric inhomogeneous Tolman-Bondi dust cloud collapse