There is a close connection between simple harmonic motion and uniform circular motion. This connection is widely taught and included in standard textbooks. Here, we exploit this connection to simultaneously derive two results from introductory mechanics: the period of a mass-spring system and the centripetal acceleration formula. Previously published derivations of the centripetal acceleration formula often rely on calculus, either via explicit computation or arguing geometrically about limits. One approach relies entirely on the kinematics of vectors, without invoking limits. Our approach likewise does not use limits, but is rooted in dynamics.