Chapter 14, Problem 61P

(III) (a) Show that the total mechanical energy, $E=\frac{1}{2}m{v}^2+\frac{1}{2}k{x}^2$, as a function of lime for a lightly damped harmonic oscillator is

$E=\frac{1}{2}k{A}^2{e}^{-(b/m)t}=E_0{e}^{-(b/m)t}$

where E₀ is the total mechanical energy at t = 0. (Assume ${\omega }^{\prime }\gg b/2m$.) (b) Show that the fractional energy lost per period is

$\frac{\Delta E}{E}=\frac{2\pi b}{m{\omega }_0}=\frac{2\pi }{Q}.$

where ${\omega }_0=\sqrt{k/m}$ and Q = mω₀/b is called the quality factor or Q value of the system. A larger Q value means the system can undergo oscillations for a longer time.