Harmonicmotion is considered simple if it is undamped, i.e. if it keep on to oscillate uniformly over time. Of particular interest are the frequency f or also period T of the oscillations. Consider an object subject to merely two forces: one because of the acceleration of the object and the other due to a restoring force The total force is
F = ma + kx,
Where k is the spring constant or restoring force, this is able to be expressed as a second order linear differential equation as follows,
It is known that solutions to such equations involve trig functions. We define a generalized cosine as x = Acos(ωt + φ). Then, dx/dt = -Aωsin(ωt +φ) and d2/dt2= -Aω2cos(ωt + φ).
Substituting
-Aω^{2}cos(ωt + φ) + A( k/m) cos(ωt + φ) = 0
So,
ω2=k/m
ω =√k/m.
But ω = 2πf = 2π/T. Hence,
f =1/2π√k/m and
T =2π√m/k