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

-Aω²cos(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

Posted Date: 6/29/2013 6:19:03 AM | Location : United States

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