(adsbygoogle = window.adsbygoogle || []).push({}); I'm trying to find the work done by a harmonic oscillator when it moves from [tex]x_{0} = 0 m[/tex] to [tex]x_{max} = 1 m[/tex].

The oscillator has initial velocity [tex]v_{0}[/tex], a maximum height of [tex]x_{max} = 1 m[/tex], initial height of [tex]x_{0} = 0 m[/tex], a spring constant of [tex]k[/tex], a mass of [tex]m = 1 kg[/tex], and a damping factor of [tex]b[/tex].

It can be represented by the following differential equation:

[tex]m x'' + b x' + k x = 0 [/tex]

Solving for [tex]x(t)[/tex] will result in this equation:

[tex]x = A e^{-\gamma t} sin(\omega_1 t) [/tex]

where [tex]\gamma = b / (2 m ) = b / 2 [/tex]

and [tex]\omega_1 = ( \omega_0^2 + \gamma^2 )^{1/2} [/tex]

and [tex]\omega_0 = ( k / m )^{1/2}[/tex]

I think that the potential energy at [tex]x_{max} = 1 m[/tex] will equal the initial kinetic energy minus the work required to move the mass to the peak. Work, in general is

[tex]W = \int F dx[/tex]

I have a spring force [tex]F_{spring} = - k x[/tex], but I also have a damping force [tex]F_{damping} = - b \frac{dx}{dt}[/tex]. I tried working out [tex]W = \int ( -kx - b\frac{dx}{dt} ) dx[/tex] but it didn't seem to add up right.

Any ideas on how I can solve [tex]x_{max} = A e^{-\gamma t} = 1 m[/tex] in terms of the initial velocity that will be required to get it there?

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# Finding the Work of a Damped Harmonic Oscillator

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