Simple Harmonic Motion Calculator

Simple harmonic motion is periodic motion in which the restoring force is proportional to displacement and points toward equilibrium. An ideal mass attached to a linear spring is the standard example. Pulling the mass away from equilibrium stores elastic potential energy; releasing it converts that energy into kinetic energy and back again. With no damping or external drive, the motion repeats indefinitely with constant amplitude and a sinusoidal position function. For a mass-spring system, angular frequency equals the square root of spring constant divided by mass. Period is two pi divided by angular frequency, and ordinary frequency is the reciprocal of period. A stiffer spring therefore produces faster oscillation, while a larger mass produces slower oscillation. In this ideal model, amplitude does not affect the period because the restoring force remains linear at every displacement. Amplitude is the maximum distance from equilibrium. The instantaneous displacement must lie between negative amplitude and positive amplitude. At equilibrium, speed reaches its maximum value of angular frequency times amplitude and acceleration is zero. At either endpoint, speed is zero while acceleration magnitude is greatest. The calculator obtains speed from energy conservation as angular frequency times the square root of amplitude squared minus displacement squared. It obtains signed acceleration from negative angular frequency squared times displacement. The reported speed is a magnitude, not a direction. At most positions the oscillator can be moving toward either side, so displacement alone cannot determine the sign of velocity. Acceleration does have a definite sign because it always points back toward equilibrium. A positive displacement produces negative acceleration, and a negative displacement produces positive acceleration. SI inputs yield seconds, hertz, radians per second, meters per second, and meters per second squared. Real oscillators lose energy to friction and air resistance, may have springs with mass, and can become nonlinear at large extension. Damping gradually reduces amplitude and can alter the observed period. An applied periodic force can create resonance and a phase shift not represented by this undriven model. The calculator is therefore best used for textbook systems, preliminary engineering estimates, and checks where the spring follows Hooke's law and mass is concentrated in the moving body. For experimental work, compare the result with measured timing over several cycles and account for uncertainty in both mass and spring constant.