f) In this way one can (for weak damping) determine that:

where τ = m/b is the exponential damping time of the energy of the associated damped oscillator. This is one of the reasons defining dimensionless ζ is convenient. ζ is basically the inverse of Q, so that the larger Q (smaller ζ) is, the weaker the damping and the better the resonance will be!

It is worth mentioning in passing that one can easily reformulate the instantaneous or average power in terms of A, the driven amplitude, instead of F0, the driving force. The point then is that the power will be related to the amplitude of oscillation squared. This is intuitively reasonable, as the work removed per cycle is the damping force (proportional to A) integrated over the distance travelled (proportional to A). This is consistent with our developing rule of thumb that oscillator or wave energies or powers are (almost) always proportional to the oscillation or wave amplitude squared.

So what of this are you responsible for knowing? That depends on your interest and the level of your class. If you are a physics or math major or an engineer, you should probably work through the math in some detail. If you are a major in some other science or are premed, you probably don’t need to know all of the details, but you should still work to understand the general idea of resonance, as it is actually relevant to various aspects of biology and medicine and can occur in many other disciplines as well. At the very least, all students should be able to semi-quantitatively draw, and understand, Pavg(ω) for any given Q in the range from 3 to maybe 20 visibly to scale, specifically so that Pmax = F02/2b and Q ≈ ω0/ ω in your graph. Practice this on your homework! I nearly always ask this on one exam or quiz in addition to the homework.

The point of understanding Q pretty thoroughly is that oscillators with low Q quickly damp and don’t build up much amplitude even from a perfectly resonant ω = ω0 driving force. This is “good” when you are building bridges and skyscrapers. High Q means you get a large amplitude, eventually, even from a small but perfectly resonant driving force. This is “good” for jackhammers, musical instruments, understanding a good walking pace, and many other things.

9.5: Elastic Properties of Materials

It is now time to close a very important bootstrapping loop in your understanding of physics. From the beginning, we have used a number of force rules like “the normal force”, and “Hooke’s Law” because they were simple rules that we could directly observe and use to help us both understand ubiquitous phenomena and learn to use Newton’s Laws to quantitatively describe many of them.

We had to do this first, because until you understand force, work, energy, equations of motion and conservation principles – basic mechanics – you cannot start to understand the microscopic basis for the macroscopic “rules” that govern both everyday Newtonian physics and things like thermodynamics and chemistry and biology (all of which have rules at the macroscopic scale that follow from physics at the microscopic scale).

We’ve done a bit of this along the way – thought about microscopic causes of friction and drag forces, derived the ways in which a macroscopic object can be thought of as a pointlike microscopic object located at its center of mass (and ways it cannot, e.g. when describing its rotation). We’ve even thought a bit about things like compressibility of fluids, but we haven’t really thought about this enough.

Let’s fix that.

To do so, we need a microscopic model for a solid, in particular for the molecular bonds within a solid.

9.5.1: Simple Models for Molecular Bonds

Consider, then a very crude microscopic model for a solid. We know that this solid is made up of many elementary particles, and that those elementary particles interact to form nucleons, which bind together to form nuclei, which bind elementary electrons to form atoms and that the atoms in turn are bound together by short range (nearest neighbor) interatomic forces – “chemical bond” if you like – to actually form the solid.

From both the text and some of the homework problems you should have learned that a “generic” potential energy associated with the interaction of a pair of atoms with a chemical bond between them is given by a short range repulsion followed by a long range attraction. We saw a potential energy form like this as the “e ective potential energy” in gravitation (the form that contained an angular momentum barrier with L2 in it), but the physical origin of the terms is very di erent.

The repulsion in the molecular potential energy comes from first the Pauli exclusion principle189 in quantum mechanics (that makes the interpenetration of the electron clouds surrounding atoms energetically “expensive” as the underlying quantum states rearrange to satisfy it) plus the penetration of a screened Coulomb interaction. The former is truly beyond the scope of this course – it is quantum magic associated with electrons190 as fermions191 , where I’m inserting wikipedia links to lots of these terms so that interested students can use them as the starting points of wikipedia romps, as a lot of this is all absolutely fascinating and is one of the reasons physicists love physics, it is all just so very amazing.

Next semester you will learn about Coulomb’s Law192 , that describes the forces between two charged particles, and (using Gauss’s Law193 ) you will be able to understand how the electron cloud normally “screens” the nuclei as eventually the rearrangement brings increasingly “bare” nuclei close enough so that the atoms have a very strong net repulsion.

One thing that we won’t cover then, however, is how this simple/naive model, which leads to two atoms not interacting at all as soon as they are not “touching” (electron clouds interpenetrating) is replaced by one where two neutral atoms have a residual long range interaction due to dipoleinduced dipole forces, leading to what is called a London dispersion force194 . This force has the generic form of an attractive −C/r6 for a rather complicated C that parameterizes various details of the interatomic interaction.

Physicists and quantum chemists or engineers often idealize the exact/quantum theory with an approximate (semi)classical potential energy function that models these important generic features. For example, two very common models (one of which we already briefly explored in week 4, homework problem 4) is the Lennard-Jones potential195 (energy):

189Wikipedia: http://www.wikipedia.org/wiki/Pauli Exclusion Principle.

190Wikipedia: http://www.wikipedia.org/wiki/Electron.

191Wikipedia: http://www.wikipedia.org/wiki/Fermion.

192Wikipedia: http://www.wikipedia.org/wiki/Coulomb’s Law.

193Wikipedia: http://www.wikipedia.org/wiki/Gauss’ Law.

194Wikipedia: http://www.wikipedia.org/wiki/London dispersion force. This force is due to Fritz London who was

a Duke physicist of great reknown, although he derived this force from second-order perturbation theory long before he fled the rise of the Nazi party in Germany and eventually moved to the United States and took a position at Duke. London is honored with an special invited “London Lecture” at Duke every year. Just an interesting True Fact for my Duke students.

195Wikipedia: http://www.wikipedia.org/wiki/Lennard-Jones potential.