LECTURE NOTES FOR PHYSICS 307: CLASSICAL MECHANICS
TEXT: THORNTON AND MARION'S CLASSICAL MECHANICS OF PARTICLES AND SYSTEMS
SOME USEFUL REFERENCE STUFF: Greek alphabet, metric prefixes, conversion factors
ASIGNMENTS: (Subject to change: check back often.)
HW #6: Due Friday, Oct. 24: Problems 3.4b, 7, 10, 22.
HW #7: Due Friday, Oct. 31: Problems 4.4, 5, 6, plus spreadsheet exercises. (Download the spreadsheets
ODEDrivenSHO.xls, ODECubicPendulum.xls, and ODEPlanarPendulum.xls. You will need to enable the macros.)
"Dailies":
Tuesday, Oct.14: Show that the critical damping solution works.
Tuesday, Oct. 21: Show that the KE is maximum when    0 .
WEEK 8: GO TO LECTURE 13
LECTURE 13: SECTION 3.6-7 (Return to top.)
THE DRIVEN HARMONIC OSCILLATOR
While most harmonic oscillators tend to lose energy and their oscillations tend to die down, this need not be the
case if we provide some external source of energy. This is what happens in a driven harmonic oscillator. The
driven harmonic oscillator has a natural frequency of oscillation, and usually the driving force has its own
frequency, which may be quite different from the system's natural frequency. Which one wins out? Let's save that
question for later.
Consider a soda can hanging from a rubber band -- a simple example of a mass on a spring. You can find its
natural frequency by bobbing it up and down. Now "bob" it up and down with a 10 second period -- real, real
s..l..o..w. Does it move at its natural frequency or with the 10 second period?
The equation for the driven harmonic oscillator is the same as for the damped harmonic oscillator, except for a
new term, the driving term, which appears on the right hand side:
x  2x   02 x  A cos t
,
where A=Fo/m, the amplitude of the driving force divided by the mass of the oscillator. Solutions to equations like
this, called inhomogeneous equations, consist of two parts: the "complementary" or undriven solution, which is
just the solution we got for the damped system, plus the "particular" solution, which is the result of the driving
force. The first of these terms is a transient solution. That is, it dies away usually after a few periods of the object's
oscillation. For most purposes, the only part of the solution that we're interested in is the particular solution. To
find this solution, we will assume that the position, x, of the object varies with the same frequency as the driving
force. Then the particular solution has the form
x p (t ) 

A cost   
2
  02

2
 4 2  2
which we can derive by simply plugging a solution of the form x = Aeiωt into the differential equation. A couple of
observations: First, if there is no damping, β = 0. In this case, the oscillation gets a very large when ω, the driving
frequency, approaches ω0, the natural frequency of the oscillator. This is called resonance. In the real world, β
never goes to zero. (How many physical systems can you think of that don't have friction?) However, the smaller
we can make the damping, the more dramatic this effect becomes.
(p.s. You can use the same spreadsheet as above, ODEDrivenSHO.xls, to illustrate this. This is the same
spreadsheet as you will use in HW#7 above.)
A few comments about resonance. First of all, there are actually two resonant frequencies. If we calculate the
frequency for which the amplitude is largest, we get the resonant frequency of
 R  02  2 2
Notice that this is different than the frequency shift we get for the damped pendulum. The other resonance we can
consider it is the one at which the kinetic energy of the system is a maximum. In this case,
 R  0
QUALITY FACTOR, Q:
we can describe how dramatic our resonance is by defining a quantity that we call the quality factor, Q. One
interpretation of the quality factor is that it is proportional to the ratio of the energy of the system divided by the
energy lost each period.
Q = 2 π (Energy of system)/(Energy lost per period);
Another interpretation is that Q tells us how sharply the amplitude of oscillation increases as we near resonance.
The quality factor is dimensionless. Our standard definition of it will be
Q  R / 2
In the limit of light damping, that is, in the limit as Q greater than 1, this can be approximated by
Q  R / 
Δω can be defined graphically in terms of the maximum amplitude. If you sketch the amplitude and as a function
of frequency, ω, and if Amax is its value at resonance, ω=ωR, then Δω is the difference in frequency between the
two values at which this amplitude reduces by a factor of 1.414. I will show this graphically in class, using the
spreadsheet ODEDrivenSHO.xls.
PHYSICAL SYSTEMS:
There are a number of physical systems besides a simple mass spring system which exhibit resonant behavior.
One of these is a molecule in the atmosphere which scatters light, causing the sky to look blue and sunrises and
sunsets to look reddish. This is known as Rayleigh scattering. In this case, you can think of the electron cloud
surrounding an atom as being like a harmonic oscillator. It has an equilibrium position, and, if an electric field
passes by, that can shift the position of this electron cloud away from equilibrium. Light emitted by the sun
consists of electromagnetic waves, and as these pass this atom, they can perturb the position of the electron
cloud. The phenomenon we see takes place well below resonance, which occurs in the UV. In fact, much of the
ultraviolet spectrum does not get through the atmosphere because its energy is absorbed by the atoms near
resonance. However, for visible light, as the frequency increases, the shifting of the electron cloud increases, and
more of this light gets redirected ("scattered"). This is the scattering that we see which occurs mostly for violet and
other light of short wavelength.
FRIDAY: NO CLASSES, FALL BREAK
YSBATs
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