A system executing simple harmonic motion is called a simple harmonic oscillator. In simple harmonic motion, the force acting on the system at any instant, is directly proportional to the displacement from a fixed point in its path and the direction of this force is towards that fixed point. Thus, the system executes the motion under a linear restoring force.

If the displacement of the system from a fixed point is x, the linear restoring force is -Kx, where K is a constant which is called the force constant. Thus no other force except the linear restoring force acts on a simple harmonic oscillator. As a result, the oscillator executes vibrations of constant amplitude and with a constant frequency. These oscillations are called the free oscillations.

Let a particle of mass m be executing simple harmonic oscillations. The acceleration of the particle at displacement x from a fixed point will be d²x/ dt²  .

m(d²x/dt²) ∝ -x

or, m(d²x/dt²) = -Kx

where K is a constant, which is called force constant of the particle. Here the negative sign tells that the direction of force acting on the particle (or acceleration) is opposite to the direction of increase in displacement

Acce1eration of the particle  d²x /dt² =  Kx /m

Let k/m =ω²

 then,

Acceleration of the particle  d^{2 x} /dt²   = –ω² x

This is the differential equation of S.H.M.