The equation of motion for simple harmonic motion (SHM) is given by

Simple harmonic motion is characterized as a type of periodic motion where the restoring force acting on an object is directly proportional to its displacement from the equilibrium position. In the equation above, $x$ represents the displacement from equilibrium, $A$ denotes the amplitude of the motion, $\omega$ is the angular frequency, $t$ is time, and $\phi$ is the phase angle.

The angular frequency $\omega$ is related to the period $T$ of the motion through the equation

Here, the period $T$ is defined as the time required for one complete oscillation. The frequency $f$, which indicates how many oscillations occur per unit time, is the reciprocal of the period, expressed as

In addition to displacement, the velocity and acceleration of an object in simple harmonic motion can also be described mathematically. The velocity $v$ is given by the equation

while the acceleration $a$ is expressed as

These equations illustrate that the velocity reaches its maximum value when the displacement is zero, whereas the acceleration is at its peak when the displacement is at its maximum.

The equation of motion for simple harmonic motion is instrumental in analyzing a broad spectrum of phenomena, ranging from the oscillations of a mass on a spring to the vibrations of atoms within a crystal lattice. A thorough understanding of this equation is fundamental for comprehending the behavior of these various systems.