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A State Equation Approach to Harmonic Oscillation

We discuss the motion of a harmonic oscillator using a state equation approach commonly applied in a modern control theory (MCT) of engineering. Instead of a second-order differential equation learnt by students in classes on mechanics, the state equation is written in a matrix form of a first-order differential equation of the state vector [Formula: see text]. We present the motion of a mass-spring-damper (MSD) system on an inclined plane and a simple pendulum in a viscous fluid. In the former case, we treat the gravitational force along the inclined plane as an input signal to the system. In the latter case, we treat the buoyant force and the drag force from the fluid as feedback to the system, controlling the final state of the system. The state equation approach presented in this article will help university students bridge the gap between physics and engineering.