- Show that the period between successive zeros of a damped harmonic oscillator is
constant, and is half the period between successive maxima.
- Show that the ratio of two successive maxima in the displacement of a damped
harmonic oscillator is constant. [From Fowles and Cassiday 2005.]
- If the amplitude of a damped harmonic oscillator decreases to
of its initial
value after
periods show that the ratio of the period of oscillation to the period
of the oscillation with no damping is approximately
- Many oscillatory systems are subject to damping effects that are
not exactly analogous to the frictional damping considered in Section 7.
Nevertheless, such systems typically exhibit an exponential decrease
in their average stored energy of the form
.
It is possible to define an effective quality factor for such oscillators as
, where
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
Hz. What
is the effective
of the system? [Modified from French 1971.]
- According to classical electromagnetic theory, an accelerated electron
radiates energy at the rate
, where
,
is the charge on an electron,
the instantaneous
acceleration, and
the velocity of light in vacuum. If an electron were oscillating
in a straight-line with displacement
how much energy
would it radiate away during a single cycle? What is the effective
of this oscillator?
How many periods of oscillation would elapse before the energy of the
oscillation was reduced to half of its initial value? Substituting a typical optical
frequency (e.g., for green light) for
, give numerical estimates
for the
and half-life of the radiating system.
- Show that, on average, the energy of a damped harmonic oscillator of quality factor
decays by a factor
during
oscillation cycles. By what factor does the amplitude decay in the same time interval?
- Demonstrate that in the limit
the solution to the damped
harmonic oscillator equation becomes
- What are the resonant angular frequency and
quality factor of the circuit shown in the preceding figure? What is the average power absorbed at resonance?
- The power input
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,
.
- Using
, show that the average rate that the damping force does work is
.
- Substitute the value of
at an arbitrary driving frequency and, hence, obtain an expression for
.
- Demonstrate that this expression yields (120) in the limit that the driving frequency is close to the resonant frequency.

- Using
, show that the average rate that the damping force does work is
.
- The equation
governs the motion of an undamped harmonic oscillator driven by a sinusoidal
force of angular frequency
. 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