Support recovery of sparse signals from noisy measurements with orthogonal matching pursuit (OMP) has been extensively studied. In this paper, we show that for any K-sparse signal x, if a sensing matrix A satisfies the restricted isometry property (RIP) with restricted isometry constant δ K+ 1 <; 1/√K + 1, then under some constraints on the minimum magnitude of nonzero elements of x, OMP exactly recovers the support of x from its measurements y = Ax + v in K iterations, where v is a noise vector that is ℓ 2 or ℓ ∞ bounded. This sufficient condition is sharp in terms of δ K+ 1 since for any given positive integer K and any 1/√K + 1 ≤ δ <; 1, there always exists a matrix A satisfying the RIP with δ K+ 1 = δ for which OMP fails to recover a K-sparse signal x in K iterations. Also, our constraints on the minimum magnitude of nonzero elements of x are weaker than existing ones. Moreover, we propose worst case necessary conditions for the exact support recovery of x, characterized by the minimum magnitude of the nonzero elements of x.