Consider a mass that is connected to a spring on a frictionless horizontal surface. To understand the oscillatory motion of the system, apply DID TASC . This gives:
ΣF = ma → -kx = ma
The acceleration is the second time derivative of the position:
The solutions to this equation of motion are periodic functions of time and have the generic form:
x(t) = A cos ω t or   x(t) = B sin ωt
Here A is the amplitude of the oscillation, and ω is the angular frequency.
Taking two time derivatives of x(t) gives:
|= -Aω sin ωt|
|= -Aω2 cos ωt →   a = -ω2 x|
Comparing with the original equation of motion a = - k x/m, the angular frequency is: ω2 = k/m.
The frequency ω is related to the period T of the oscillation by:   T = 2 π/ ω
Simple harmonic motion (SHM) is described by x(t) = A cos ωt (or x(t) = A sin ωt).
The governing equation of SHM is: a = - ω2 x.