Answer to Question #165478 in Classical Mechanics for Nono

Question

Consider a system with a damping force undergoing forced oscillations at an angular frequency omega

Find the instantaneous kinetic and potential energy of the system.

Answer 1

Answer

Displacement of damped harmonic oscillator is given

$x=ae^{-bt}sin(wt+\phi)$

Where

$w=\sqrt{w_0^2-b^2}$

Now kinetic energy is given by

$K=\frac{m}{2}(\frac{dx}{dt}) ^2$

Putting displacement x and differentiate with respect to t

$K=\frac{ma^2e^{-2bt}}{2}(b^2sin^2(wt+\phi) +w^2cos^2(wt+\phi) -2bwsin(wt+\phi) cos(wt+\phi))$

Now potential energy

$P=\frac{mw_0^2x^2}{2}$

Now putting value of displacement

$P=\frac{a^2mw_0^2e^{-2bt}sin^2(wt+\phi) }{2}$