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## Homework Statement

A forced damped oscilator of mass ##m## has a displacement varying with time of ##x=Asin(\omega t) ## The restive force is ## -bv##. For a driving frequency ##\omega## that is less than the natural frequency ## \omega_{0}##, sketch graphs of potential energy, kinetic energy and total energy of the oscillator over the cycle.

## Homework Equations

## The Attempt at a Solution

From the equation given, and the definitions of kinetic and potential energy for an oscillator:

(1) Kinetic Energy:

$$ E_{K}=\frac{1}{2}m\dot{x}^{2} = \frac{1}{2}mA^{2}\omega^{2}cos(\omega t) $$

(2) Potential Energy,

$$ E_{P}=\frac{1}{2}kx^{2}=\frac{1}{2}mA^{2}sin^{2}(\omega t) $$

These are simple and easy to plot. My issue comes when thinking about the total energy:

(3) Total Energy:

$$ E_{T}= E_{K}+E_{P} +E_{D} $$

##E_{D} ## is the energy lost due to the damping force, ##F_{D}##, where ##F_{D}=-b\dot{x}##.

At some time ##t##, then :

$$ E_{D} = \int_{0}^{x(t)} F_{D} \ dx = \int^_{0}^{x(t)} \omega \sqrt{1-x^{2}} \ dx $$

This can be evaluated by making the substitution ##x=sin(u) \implies dx=cos(u) du ##.

So:

$$ E_{D} = \int_{0}^{sin(x(t))} \omega cos^{2}u \ du = \bigg|_{0}^{sin(x(t)} \frac{sin(2u)}{4} +\frac{u}{2} $$

So:

$$E_{T}=\frac{1}{2}mA^{2}\bigg(sin^{2}(\omega t)+\omega^{2}cos(\omega t)\bigg)-bAsin(\omega t) $$

Is this the right approach though? The reason I am confused is that there will be points in the motion where the damping force will do work with the system - whereas it should be at all times against it?

Thanks!

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