# Energy in damped harmonic motion

1. May 29, 2014

### PsychonautQQ

Hey PF,
my book either got sloppy in a derivation or I am not connecting two very obvious dots.
It gives the energy of the damped harmonic oscillator as
E = (1/2)mv^2 + (1/2)kx^2
then takes the derivative with respect to time to get dE/dt.

then it gives the differential equation of motion as
ma + kx = -cv

okay cool i'm following so far...
then it says with this equation of motion we know that
dE/dt = -cv^2

what am I missing here?

2. May 29, 2014

### nasu

Ff= -cv is the damping (friction) force.
dE/dt is the rate of dissipation of the mechanical energy.
Or the power dissipated due to the action of the friction force.

3. May 29, 2014

### D H

Staff Emeritus
Which your text should have as $\dot E = v(ma + kx)$.

That's just a rearrangement of Newton's second law, with the external forces being the spring, linearly directed against displacement, and drag, linearly directed against motion. Mathematically, $F=ma = -kx - cv$. Adding $kx$ to both sides yields $ma+kx=-cv$. Substituting that result back into the expression for the time derivative of energy yields $\dot E = -cv^2$.