For a damped harmonic oscillator with equation
show that the work done against the damping force in an infinitestimal time dt is equal to the loss of energy of the mass m during the same time interval dt.
In an infinitesimal time dt during which the mass m traverses a distance dx, the work done δw against the damping force βx is given by
Total energy of the system is given by
Thus the loss of energy of the mass in time dt is
