Serway & Jewett — Oscillatory Motion: Problem 24

Show that the time rate of change of mechanical energy for a damped, undriven oscillator is given by \( dE/dt = -bv^2 \) and hence is always negative. To do so, differentiate the expression for the mechanical energy of an oscillator, \( E = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 \), and use Equation 15.31.