The Equations of Simple Harmonic Motion

Every simple oscillator, no matter what it's made of, obeys the same differential equation: acceleration proportional to negative position, a = -omega² x. The function x(t) = A cos(omega t + phi) solves that equation for any values of A and phi, which is why one formula covers springs, pendulums and anything else that feels a restoring force proportional to displacement. The three constants are not interchangeable: omega comes from the physics of the system itself (spring stiffness and mass, or gravity and length), while A and phi come from how you started the motion.

Amplitude A stretches the curve vertically: it's the maximum distance from equilibrium, fixed by how far you pulled or pushed the object before releasing it. Angular frequency omega stretches the curve horizontally: it sets how many radians of the cosine cycle pass per second, so period T = 2 π / omega and frequency f = omega / (2 π). The phase constant phi slides the whole curve left or right along the time axis without changing its height or its repeat rate. It just answers the question: where was the oscillator, and which way was it moving, at t = 0? Differentiating x(t) gives velocity v(t) = -A omega sin(omega t + phi), and differentiating again gives acceleration a(t) = -A omega² cos(omega t + phi) = -omega² x(t), tying the kinematics directly back to Newton's second law.

This equation is the hinge of the whole unit: earlier you found omega from force equations like F = -kx, and next you'll use x(t), v(t) and a(t) together to track how kinetic and potential energy trade off and to find velocity or acceleration at any position without needing time at all.