Simple harmonic motion (SHM) is the simplest form of oscillatory motion. A particle is said to execute a SHM when it oscillates under the action of a (restoring) force that is always directed opposite to its instantaneous displacement (from its mean position) and whose instantaneous magnitude varies in direct proportion to the magnitude of its instantaneous displacement.
In mathematical terms, it implies that
Fx = -kx
Now, Fx = Max , where M is the mass of the oscillating particle and ax is the magnitude of its instantaneous acceleration when its displacement is x.
∴ For a particle executing SHM, we have
We may, therefore, also say;
“An oscillating particle is said to execute a S.H.M if its acceleration is always directed towards its mean (or central) position and the instantaneous magnitude, of this acceleration, varies in direct proportion to the magnitude of the instantaneous displacement of the particle (from its mean (or central) position).
We know that
∴ For a particle in SHM, we have
We speak of this equation as the ‘differential equation’ for a simple harmonic motion.
