Exercises

1. Show that the period between successive zeros of a damped harmonic oscillator is constant, and is half the period between successive maxima.

2. Show that the ratio of two successive maxima in the displacement of a damped harmonic oscillator is constant. [From Fowles and Cassiday 2005.]

3. If the amplitude of a damped harmonic oscillator decreases to $1/{\rm e}$ of its initial value after $n\gg 1$ periods show that the ratio of the period of oscillation to the period of the oscillation with no damping is approximately
   $$1 + \frac{1}{8\pi^{\,2}\,n^{\,2}}.$$
   [From Fowles and Cassiday 2005.]

4. Many oscillatory systems are subject to damping effects that are not exactly analogous to the linear frictional damping considered in Section 2.2. Nevertheless, such systems typically exhibit an exponential decrease in their average stored energy of the form $\langle E\rangle = E_0\,\exp(-\nu\,t)$. It is possible to define an effective quality factor for such oscillators as $Q_f=\omega_0/\nu$, where $\omega_0$ is the natural angular oscillation frequency.

   For example, when the note “middle C” on a piano is struck its oscillation energy decreases to one half of its initial value in about 1 second. The frequency of middle C is $256$ Hz. What is the effective $Q_f$ of the system? [Modified from French 1971.]

5. According to classical electromagnetic theory, an accelerated electron radiates energy at the rate $K\,e^{\,2}\,a^{\,2}/c^{\,3}$, where $K=6\times 10^{\,9}\,{\rm N\,m^{\,2}}/{\rm C^{\,2}}$, $e$ is the charge on an electron, $a$ the instantaneous acceleration, and $c$ the velocity of light in vacuum (Fitzpatrick 2008).
   1. If an electron were oscillating in a straight line with displacement $x(t)= A\,\sin(2\pi\,f\,t)$, how much energy would it radiate away during a single cycle?
   2. What is the effective $Q_f$ of this oscillator?
   3. How many periods of oscillation would elapse before the energy of the oscillation was reduced to half of its initial value?
   4. Substituting a typical optical frequency (e.g., for green light) for $f$, give numerical estimates for the $Q_f$ and half-life of the radiating system.

6. Show that, on average, the energy of a damped harmonic oscillator of quality factor $Q_f\gg 1$ decays by a factor ${\rm e}^{-2\pi}\simeq 1.9\times 10^{-3}$ during $Q_f$ oscillation cycles. By what factor does the amplitude decay in the same time interval?

7. Demonstrate that in the limit $\nu\rightarrow2\,\omega_0$ the solution to the damped harmonic oscillator equation becomes
   $$x(t) = \left(x_0 + \left[v_0+ \omega_0\,x_0\right]t\right){\rm e}^{-\omega_0\,t},$$
   where $x_0=x(0)$ and $v_0=\dot{x}(0)$.

   Figure 2.10: Figure for Exercise 8.

   

8. 1. What are the resonant angular frequency and quality factor of the circuit shown in Figure 2.10?

   2. What is the average power absorbed at resonance?

9. The power input $\langle P\rangle$ required to maintain a constant amplitude oscillation in a driven damped harmonic oscillator can be calculated by recognizing that this power is minus the average rate that work is done by the damping force, $-m\,\nu\,\dot{x}$.
   1. Using $x = \hat{x}\,\cos(\omega\,t - \varphi)$, show that the average rate that the damping force does work is $-m\,\nu\,\omega^{\,2}\,\hat{x}^{\,2}/2$.

   2. Substitute the value of $\hat{x}$ at an arbitrary driving frequency and, hence, obtain an expression for $\langle P\rangle$.

   3. Demonstrate that this expression yields Equation (2.62) in the limit that the driving frequency is close to the resonant frequency.

10. The equation $m\,\ddot{x} + k\,x = F_0\,\sin(\omega\,t)$ governs the motion of an undamped harmonic oscillator driven by a sinusoidal force of angular frequency $\omega$.
    1. Show that the “time asymptotic” solution (i.e., the solution that would be time asymptotic were a small amount of damping added to the system) is
       $$x = \frac{F_0\,\sin(\omega\,t)}{m\,(\omega_0^{\,2}-\omega^{\,2})},$$
       where $\omega_0=\sqrt{k/m}$. Sketch the behavior of $x$ versus $t$ for $\omega<\omega_0$ and $\omega>\omega_0$.
    2. Demonstrate that if $x=\dot{x}=0$ at $t=0$ then the general solution is
       $$x =\frac{ F_0}{m\,(\omega_0^{\,2}-\omega^{\,2})}\left[\sin(\omega\,t)-\frac{\omega}{\omega_0}\,\sin(\omega_0\,t)\right].$$

    3. Show, finally, that if $\omega$ is close to the resonant frequency $\omega_0$ then
       $$x\simeq \frac{F_0}{2\,m\,\omega_0^{\,2}}\left[\sin(\omega_0\,t)-\omega_0\,t\,\cos(\omega_0\,t)\right].$$
       Sketch the behavior of $x$ versus $t$.
    [Modified from Pain 1999.]