This is the "home page" for PHY385F - Vibrations, Waves and Thermal Physics. It was written by David Harrison, August, 1996. This is version 1.23, date (m/d/y) 11/26/96.
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Monday 2 - 3PM Wednesday 11AM - noon. Thursday 10AM - 11AM
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D. Halliday, R. Resnick and J. Walker, Fundamentals of Physics - Extended, 4th edition, John Wiley.
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This course is a rather broad survey of a number of topics in Physics that are of importance in a large number of fields of engineering.
Here we list the material to be covered this term, and the textbook references.
Topic Textbook Reference Elasticity §13.6 Oscillations Chapter 14 Waves I Chapter 17 Waves II Chapter 18 Electromagnetic Waves Chapter 38 Geometrical Optics Chapter 39 Interference Chapter 40 Diffraction Chapter 41 Temperature Chapter 19 Heat, 1st Law of Thermodynamics Chapter 20 Kinetic Theory of Gases Chapter 21 Entropy, 2nd Law of Thermodynamics Chapter 22
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Lectures begin on Friday, September 6. The lectures are on:
There are two tutorial sections, both meeting on Friday, 1300-1400. Tutorials begin on Friday, September 13. The two locations are:
The assignment of students to tutorial section will occur in lecture on September 6.
The Class Test will be on Friday, October 18, from 1200-1300.
The date of the Final Exam is to be announced.
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The marking scheme was ratified by the class on Tuesday, September 10.
The marking scheme has two components:
Then, the final grade in the course will be these two components weighted 50 Final/50 Term or 60 Final/40 Term, whichever is higher for each individual student.
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Obligatory reminder: the "heart" of learning in Physics (and much of Engineering) occurs in doing the problem sets. Thus, we urge you to do (at least) all suggested and assigned problem sets, whether they are collected or not.
We will be regularly suggesting problems from the textbook related to the course material. These will not be collected or marked, although they may be discussed in tutorial.
In addition, every two weeks a problem set will be handed out which will be collected and marked. The schedule for these problem sets is:
Handed out Due Sept 13 Sept 20 Sept 27 Oct 4 Oct 11 Oct 18 Oct 25 Nov 1 Nov 8 Nov 15 Nov 22 Nov 29
Another obligatory reminder: copying a solution to a problem you have not done teaches you nothing. This will certainly be reflected in your marks on the Test and Final Exam in the course.
All six problem sets will be weighted equally. For each marked problem, approximately 60% of the mark will be based on the method of solution, with the remaining mark determined by showing the ability to get to the correct answer.
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The two main purposes to the class test are:
The class test will be of 50 minute duration, and will take place on Friday, October 18, 1200-1300.
The format will be closed book with calculators allowed. Students may bring in a single 8 1/2 by 11 inch sheet of paper on which is written anything they like.
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The Final Exam will have the same format as the Class Test, except of course it will be longer and cover a wider range of material.
The format will be closed book with calculators allowed. Students may bring in a single 8 1/2 by 11 inch sheet of paper on which is written anything they like.
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Every week, I provide the tutors with a summary of what occured in class. Since it only touches the "high points" (or maybe "low points"!) of the classes, it may be useful for students to see those summaries. They are included below. They were typically written on Wednesday, i.e. after Tuesday's class.
On Friday I talked about elasticity, textbook section 13.6. I did not mention bulk modulus. I pointed out that in the linear region, the equation's similarity to a spring-mass means that the object will oscillate with a characteristic frequency. I also talk a bit about the Wilberforce spring.
Yesterday, I took a Wilberforce spring to class, and demonstrated the resonance. I mentioned that when E > 2G for the material of the spring, the spring will coil up as extended, when E < 2G it will uncoil as extended, and when E = 2G it will oscillate without either coiling or uncoiling.
Then I began Chapter 14. I followed the order and notation of the textbook. I got through section 14.7. Below I only talk about things I did somewhat differently from the text.
I tried to stress that we get SHM whenever the restoring "force" is proportional to displacement (where I later in the class showed why I put the word force in quotation marks). I also stated (and later proved) that any small amplitude oscillation is approximately SHM.
I tried to help them build intuition about SHM by asking questions like: "If the speed of a body in SHM is exactly 0 at certain times, what is x at those times?" Some students had a surprising amount of difficulty with this.
I showed that the energy of SHM is 1/2 k (xm)^2 two ways, once by evaluting the kinetic energy at the maximum speed, and once by integrating F.dx to get the potential energy, and evaluating that at its maximum value.
I did a Taylor expansion for an arbitrary potential about a local minimum and showed that for small maximum amplitudes it is approximately like the potential of SHM. I said something like "If it looks like a duck, walks like a duck, and quacks like a duck, then it is a duck." Similarly if the math of the motion is the same, the motion is the same.
When we did rotational motion in section 14.5, I said that here the restoring *torque* was proportional to angle. Thus the quotation marks about "force" earlier in the class. I also tried to stress that since the functional form of the equation of motion was the same as for the spring mass (although with different symbols) then the solution must have the same form also.
In the discussion of the pendulum, I tried to stress that for the pendulum with arbitrary maximum amplitude the equations are *not* solvable analytically. I did not discuss the physical pendulum.
Finally, I introduced the mathematical analogy between SHM and uniform circular motion as a visualisation aid, which would maybe help them with questions like those I had asked earlier in the class.
In class on Friday I began by doing a more careful review of the previous class than usual: this was to compensate a bit for the absolutely awful room in which that class occured.
Then I finished Chapter 14 by discussing damped SHM and forced damped SHM. I ended up talking about shock absorbers in cars, and one of the members of the class provided information about air filled shocks (which I didn't know anything about).
I followed the text pretty closely except that for over-damped motion I pointed out that the solution is the same as for the under-damped case, once we realize that cos(ix) = cosh(x). One of the students piped up "But I don't know how to use the cosh button on my calculator!" Funny guy.
Yesterday I began Chapter 17. I tried to stress that we get the wave function by:
I claimed that the text's derivation of the speed of a wave in Section 17.5 is unnecessarily complex. Instead I asked the class: "Consider watching a wave go by you at some fixed position. After a time t = the period, how far has the wave travelled?" They eventually got it, so the speed is:
v = x/t = wavelength / period
Similarly, the text's derivation of Eqns 17-15 and 17-16 is unncessary: just watch the wave going by you at some fixed position to get the phase of the w*t term.
I went somewhat beyond the textbook in Section 17.6. I began by stating the "it can be shown" that for sound waves in a long solid rod the speed = Sqrt[ E / rho ], where E is Young's modulus and rho is the density. I tried to show that this is a reasonable sort of result. This, then, sort of justified the argument the text uses in doing the 'Dimensional Analysis' part of getting v for a wave in a string. Also, after doing the full derivation, I talked about water waves, where "it can be shown" that the speed in shallow water is proporational to the square root of the depth of the water. This is the point at which I stressed that the frequency of a wave is always equal to the frequency of the source. So, as a wave approaches a shore and the depth becomes less, the speed decreases and so does the wavelength, so that v / wavelength = f = constant. I also showed how when the amplitude of the wave is not small compared to the depth, the speed of the crest of the wave is significantly greater than the speed of the trough: this explains breaking surf.
In Section 17.9, I extended the discussion by talking about the light from two flashlights, where the beams intersect. I also mentioned that the principle of linear superposition is only true from "small" amplitude oscillations.
I finished with interference (Sec. 17.11), done just about exactly like the text. I did *not* do any of the application of trig identities, just talked about what Eqn 17-44 means.
On Friday I finished Chapter 17. We did standing waves (Sec 17.12). I talked about reflections on a string with a fixed end and from a free end. We extended this to standing waves for a string fixed at both ends (Sec 17.13).
It isn't totally clear on first reading, but Chapter 17 concentrates on waves on a string, while Chapter 18 concentrates on sound waves.
I talked a little bit about Pythagoras' realisation that the notes of the musical scale correspond string lengths that are ratios of integers. We discusses how a complicated vibration on a string is just the Fourier series of the fundamental (the note the string is tuned to) plus the overtones. The relative amount of the overtones determines the 'timber' of the sound, and is why a guitar sounds different than a violin. I also showed how the overtones are related to the tonic as the third and fifth of the scale, i.e. they form the three notes of the chord (plus some of the higher overtones form the minor seventh).
Suggested problems from Chapter 17: 13P 19E 46E 53E and 61P
Tuesday we began Chapter 18. I had to go back to Section 13.6 to define the bulk modulus, which I had dropped when we talked about elasticity. I then just wrote down the relation between the speed of sound, the bulk modulus and the density. The students are not responsible for the derivation in the text, which I did not do.
For Section 18.3, I tried to indicate the plausability of Eqns 18-13 and 18-14, but did not do the derivation. The students are not responsible for the derivation either.
For Section 18.4, again the students are not responsible for the derivation of Eqn 18-26. I instead stressed that since the air molecules are executing simple harmonic motion, and we had shown that the energy in SHM is proportional the the square of the maximum amplitude, the energy in the wave is proportional to the square of the maximum amplitude.
I stated that this result can be generalised: the energy of any wave is proportional to the square of its maximum amplitude.
I used this to show how conservation of energy requires that:
The above is not in the text.
I extended Fig 18-8 by drawing in the lines of subjective equal loudness. Also, I indicated that our ears are most sensitive to sounds at 3 KHz, which is almost exactly the central frequency of a baby's cries.
I discussed that the ear is a logarithmic detector (which is not mentioned in the text) and that a 3 dB increase in sound levels sounds about twice as loud to us. I derived that this corresponds to and increase in intensity of 1.995 approximately 2.
Somehow, the students got me talking about why we can use a single sub-woofer in a stereo (because we can't tell where a very low frequency sound is coming from). This led me to talk about how we localise the source of sound (our two ears plus we swivel our heads slightly back and forth).
Then I talked about how the basilar membrane in the ear is actually a Fourier analyzer, because each part of the membrane is resonant with a sound wave of a particular frequency.
Much of Section 18.5 I had already done on Friday. I showed the standing waves for a string with one end fixed and the other free.
The students seemed to have a lot of trouble with figuring out what a reflected wave from the free end would look like, and how it would combine with the incident wave. Maybe you can help them out with this in tuturial.
On Friday I finished Sec. 18.5, talking about standing waves where there is an anti-node at one end.
I mentioned the exact analog between electrical impedance for a pulse travelling down a cable connected to a diffferent impedance cable and a sound wave travelling down a tube with an open end. The analogy is also true for microwaves travelling down a wave guide into another wave guide of different dimensions.
For a sound wave, the position of the antinode is Delta L beyond the end of the tube. The magnitude of Delta L is on the order of the radius of the tube.
I nearly finished Sec. 18.6 - Beats on Friday. The students seemed interested in discovering why one can tune a musical instrument such as a guitar to a reference note by eliminating the beats.
On Tuesday I finished beats by pointing out that the frequency of the beats is | f1 - f2 |, not 1/2 | f1 - f2 | as might be thought at first glance.
I then did Sec. 18.7 - The Doppler Effect. I followed the text pretty closely, and did not bother to set up the case with both the source and detector moving. I also did not do the relativistic Doppler effect. I mentioned that the first experimental test was done in the 19th century with a brass band on a flatbed car of a train. I asked the class why it took until the 19th century to test something that we hear every day when, for example, a car playing a loud radio passes us. They quickly got that it is because until the 19th century everyday speeds were very small compared to the speed of sound.
I discussed the sonic boom shock wave sort of heuristically. I mentioned that the same phenomenon leads to the bow wave from a boat travelling faster than the water wave speed. I also mentioned Cerenkov radiation as another example.
Suggested Problems Chapter 18: 1E, 11P, 23P, 37P, 67P, 87P.
I believe that with the exception of one section, the material in Chapter 38 is way beyond the level of this course, and not very appropriate to boot. I spent about 45 minutes trying to justify that:
1. A charge executing SHM will generate a "far field" travelling wave. Much of the justification hinged on Faraday's Law (which I reviewed by talking about a transformer) and the displacement current. 2. The E field is oriented in the plane of the oscillation of the charge.
We talked a bit about how when the field impinges on a wire, it will cause the electrons in the wire to oscillate at the frequency of the wave (which is of course the frequency of the source), and thus generate an oscillating voltage. So this explains how broadcast radio and TV works.
The only section of Chapter 38 that I did at all like the text is 38.7 - Polarisation. I followed the text pretty closely.
Suggested Problem Chapter 38: 55P.
I then began Chapter 39 - Ray Optics. I essentially finished Sec. 39.2 - Reflection & Refraction. I pointed out that we had seen a first sign of the refraction law when we talked about how a water wave incident on a shore will tend to bend so that the wave fronts end up parallel to the shore line.
On Friday I did Total Internal Reflection (Sec. 39.3), and discussed fiber optics as an example. Next was Polarisation by Reflection (Sec. 39.4). Both of these followed the textbook fairly closely.
Questions in class indicate that many students are having trouble with the whole concept of polarisation.
Next I discussed how when we see point on an object, what we see is all the light rays being emitted from that point that enter our eye. This apparantly trivial point escapes many students, who then get confused about images.
Next was Plane Mirrors (Sec. 39.5) I did sample problem 39-4.
I ended with a supplement by asking the class "Why do mirrors invert left and right, and not up and down?" They were pretty well stumped. The answer I claim is that mirrors invert neither left/right or up/down. Mirrors reverse front and back.
On Tuesday we began with Spherical Mirrors (Sec. 39.6). I followed the text fairly closely, and did a table of all the sign conventions for p, i and f. I introduced the phrase "virtual object" for when p is negative, and promised to show what that means later.
The students had asked me why phenomena like the Brewster angle occur. I stated that the classical model is a 19th century vestage that is both complex and ugly. I also mentioned that it involves modelling how the light is absorbed and re-emitted. By way of example I stated that if I have two crossed polarisers so no light passed the second, inserting a polariser at an intermediate angle allows light to pass through all three. I said that this was sort of mysterious classically, but that I thought it was a good example of the Heisenberg Uncertainty Principle. I don't know how many students had any idea what I was talking about.
I followed the text fairly closely Spherical Refracting Surfaces (Sec 39.8).
I did Thin Lenses (Sec. 39.9) by stating that the image formed by surface 1 becomes the object for surface 2. This is how a "virtual object" can occur. I did a table of all the sign conventions for p, i, and f for lenses.
I used ray tracing for a thin lens to show what the thin lens approximation means, in particular that for a thin lens we approximate that the ray through the center is not deflected.
I discussed how the optics we were doing in this Chapter is OK for simple systems, but in the real world for anything beyond a trivial system, ray tracing computer software is how it is done now. I showed a transparancy of such software solving a problem involving a bi-convex lens and a mirror at an angle. This was just to give the students some flavor of what this software can do. I also followed the text closely on Ray Tracing (Sec. 39.7).
I did a lens example of a sound wave in water encounter a lens made of a thin rubber lens-shaped baloon filled with air.
I briefly discussed combinations of lenses as just that the image of the first lens becomes the object of the second lens, etc. I anticipate doing this more carefully on Friday.
Finally I discussed aberrations, which the textbook only briefly mentions at the end of Sec. 39.10.
After introducing chromatic aberration for a lens, I showed how an "achromatic prism" can elimate dispersion. Similarly we can construct an "achromatic lens" of two different glasses with different amounts of dispersion to reduce if not cancel spherical aberration. It is difficult to describe this in words and it is not in the text: if you want further information drop by to discuss.
I also discussed spherical aberration, and that to reduce it one must eliminate the rays far from the lens axis by something like an iris.
Finally, I showed a transparancy of four semi-modern camera lens which are designed to eliminate both of these aberrations plus some others such as astigmatism and curvature of field.
After class some students stuck around and we talked about f-stops, depth of field and other topics camera-esque.
Friday's class was dominated by questions. The only things I did was to finish Chapt. 39 by:
1. Discussing how the magnification for all the optics cases we have done is m = -i/p, and what the significance of the minus sign is.
2. Going through a 2 lens system in detail, calculating magnifications, showing how the image from the first lens is the object for the second lens, and showing how virtual objects can arise.
Suggested problems from Chapter 39: 6P, 17E, 29E, 49P and 63P
On Tuesday I began by discussing how physics always builds on previous physics. Thus, it is important to understand the basics or one will be hopelessly lost later on.
I then began Chapter 40. I introduced the material somewhat different from the text, talking about when the ray approximation is valid. We also talked about how when we don't want a ray approximation, such as producing a high frequency note from a loudspeaker such that the sound is radiated out in all directions, we must reduce the size of the speaker to on the order of the wavelength of the sound wave: thus tweeters must be small.
I emphasised Huygens principle. I derived Snell's Law. I also pointed out that much of the material in the Chapter we had also seen in Sec 17.11.
In Sec. 40.4 - Young's Expt, we spent a fair amount of time talking about how to get a point source by putting light from an extended source through a pinhole, and why this is necessary to get the interference pattern.
Some students continue to be confused about how when we have an incoherent source, this procedure allows the interference pattern anyhow. I have stated why in as many ways as I can think of: perhaps you can think of some additional ways to say it.
I did a numerical example which showed that if the distance between the slits is too large the interference pattern becomes essentially unobservable because the distance between the fringes becomes too small. Thus we would see the average of the maxima and minima, which is just the sum of the intensities from the two slits individually.
I finished with 40.6 by introducing phasors. Some students had seen phasors in some electrical course option they had taken. I kept trying to emphasise that the phase difference phi is just a way of describing the difference in path length | r1 - r2 |, with only mixed success I think. I also did the general case where the two fields do not have the same magnitude, which isn't done explicitly by the text.
Friday's class was, of course, the test.
I began Tuesday's class by doing a phasor analysis of 4 equally spaced coherent light sources and asked the students for the phase angles leading to minima in the intensity. With some struggle they got the answers: 90 deg, 180 deg, 270 deg, NOT 360 deg, ...
I followed the text closely for thin film interference, Sec. 40.7.
I did a "once over lightly" treatment of the Michelson Interferometer, Sec. 40.8. I did discuss the Michelson-Morley experiment a bit, and also spent a while talking about Michelson's 1896 pronouncement that physics was a dead field because everything was known. I also talked about how similar claims have been surfacing in the past few months from people who should know better!
Suggested Problems Chapt 40: 9P, 31P, 41P, 47E, 65P
I then began Chapter 41 - Diffraction. I covered sections 41.2, 41.3 and 41.4, and followed the textbook fairly closely with the following exceptions:
In Sec 41.2, I considered dividing the slit in half. Then rays for which a/2 sin(theta) = wavelength will constructively interfere. But, when we divided the slit into quarters, we derived that when a sin(theta) = 2 wavelength we get destructive interference. What is wrong?
The answer is that although the two rays for whuch a/2 sin(theta) = wavelength will constructively interfere, there is another set of rays coming from the 1/4 and 3/4 position of the slit for which a/2 sin (theta) also = wavelength and will constructively interfere. But the phase of the resultant from this second pair is 180 degrees of out phase with the resultant from the first pair, so we get an overall cancellation.
In Secs 41.3 and 41.4, I introduced the notation that when we divide the slit into widths delta(x), we end up with n phasors, and that n * delta(x) = a.
Also, I reserved the symbol phi for the relative phase angle between two adjacent phasors, consistent with the notation we have been using. Thus, what the text calls phi I called phi-prime.
On Friday, for the very few who showed up I spent the class going over some problems from Jearl Walker's "The Flying Circus of Physics" which were related to our topic of waves. This led into a general discussion of problem solving strategies and technques.
On Tuesday I finished Chapter 41. I hadn't anticipated getting that far, so didn't have any suggested problems from the textbook: I'll do that on Friday.
I extended the discussion of the Rayleigh criterion (Sec. 41.5) by talking about electron microscopes.
In two-slit diffraction (Sec. 41.6) I saw no need to introduce the notation using the variable "beta", so kept the variable "phi" which the text used in Sec. 40.6.
For multiple slits (Sec 41.7) I followed the text fairly closely.
For X-ray diffraction (Sec. 41.9) I briefly mentioned "Miller indices", although I couldn't remember their name in class. Other than that this too followed the text fairly closely.
As I mentioned last week, I have finished Chapter 41 but had not suggested any textbook problems. On Friday I rectified that:
Suggested Problems Chapter 41: 15P, 25E, 41P, 43E, 77P
I then began Chapter 19, which focuses on temperature. Throughout both Friday's class and yesterday's I often diverged from the textbook. This this summary is somewhat longer than usual; I hope it is helpful to you.
I *did* follow the text fairly closely in Sec. 19.2, which discusses the Kelvin scale. I claim the text is wrong when it states that when the universe began the temperature was 10^39 K: it was infinite.
I stressed that as we seek a good definition of temperature, we will spiral towards it. I led the class through a discussion that ended with the the realisation that the temperature measured the heat density; I stressed that this was close but far from the final word on the topic.
I next introduced temperature as a measure of the tendency of a body to give up heat.
I spent some discussing the history of thermodynamics. We talked about the 18th century "caloric" theory. In the 19th century people, especially Joule, realised that heat is a form of energy. They lacked a good atomic theory at that time, so built thermodynamics as an axiomatic system.
I discussed Euclidean geometry as an example of an axiomatic system. In classical thermodynamics the Laws of Thermodynamics are the axioms.
It turns out that the axiomatic basis for Thermodynamics was not quite complete, and so a "housekeeping" additional axiom, the 'Zeroth Law of Thermodynamics' was introduced. I did not state the law the way the textbook does. Instead I stated it:
One has three bodies. If T1 = T2 And T2 = T3 Then T1 = T3
For section 19.4 I spent some time discussion what an "operational definition" is. Then I introduced yet another definition of temperature, an operational one: temperature is what is measured with a good thermometer.
I followed the text in defining the triple point of water, but also mentioned that the pressure at the triple point is 0.0006 atmospheres.
I then discussed where the number 273.16 in the definition of the temperature of the triple point came from. I discussed a constant volume gas thermometer *not* as a standard for the operational definition of temperature but as a way of measuring absolute zero. We measure the temperature with a thermometer that measures in Celsius, and extrapolate to zero pressure. Until recently the accepted value of that extrapolation was -273.16 Celsius, now the accepted value is -273.15 Celsius.
I then talked about the constant volume gas thermometer similar to the textbook: a measurement standard for temperature. I added volume expansion terms as one of the corrections necessary, and tried to stress how difficult these experiments are to perform.
I followed the text fairly closely in Sec 19.6, discussing the Celsius and Fahrenheit scales.
For Sec 19.7 - Thermal Expansion, I did it "right" by starting with:
dL = alpha L dT
If alpha is independent of T, this integrates to:
ln( L/L0 ) = alpha ( T - T0 )
or:
L = L0 exp( alpha Delta(T) )
Expanding in a Taylor series and throwing our higher order terms we finally got to textbook equation 19-9.
For the remainder of the section I followed the text with one exception: for Figure 19-12 I made the vertical axis the volume of 1 gram of water, not the specific volume.
On Friday I finished Chapter 19 by doing a detailed example of the Mercury bulb thermometer. We used "realistic" numbers and derived that term for expansion of the Hg up the stem. We then used realistic numbers for the glass the thermometer is made of and calculated the expansion of the stem itself. I showed that provided the expansion of the mercury is greater than that of the glass, since both are linear the glass expansion makes no difference: one marks a "0" on the stem at 0 degrees C and a "100" on the stem at 100 degrees C, and then the thermomater will be correct at all temperatures. I stated but did not prove that the volume expansion of the glass similarly does not have affect the accuracy of the thermometer. I sketched briefly how to do a stem correction, which *does* matter for the most accurate possible result.
Suggested Problems Chapter 19: 1E, 5E, 16P, 43P.
I then began Chapter 20. I gave a historical overview of Count Rumford's contribution to the destruction of the caloric theory (and also some biographical information on what a strange combination of philanthropist and scoundrel he was).
On Tuesday I began by mentioning that Rumford did an experiment to show that the mass of caloric in 1 calorie of heat is less than 0.000013 mg. Of course, according to classical physics since heat is just internal energy the mass of 1 calorie is zero. According to E = m c^2, the mass is 4 x 10^(-11) mg.
I followed the text fairly closely for Sec 20.3 - Absorption of Heat by Solids and Liquids.
I posed a question to the class. Given hot coffee and cool cream, when should the cream be added to get the coffee to drinking temperature as quickly as possible. Some students knew immediately the answer: add it right away. We talked a bit about how recent experiments indicate that super-cooling is very important in how long it takes something to freeze, and that often the system that starts at the higher temperature will freeze first because the added turbulence makes it less likely to become super-cooled.
I did say that what the text calls "heat of fusion" is often called the "latent heat of fusion"; similarly the "heat of vaporization" is often called "latent heat of vaporization".
I followed the text fairly closely in Sec. 20.4, going through all the pV graphs.
In Sec. 20.5 - The First Law, I didn't exactly folow the text. I personally find the form that is most intuitive to be:
Q = Delta( Eint ) + W
so that is the form I presented. I tried to stress that all we are saying is that (1) heat is internal energy Eint, and (2) energy is conserved.
I tried to be a little more careful than the text's brief footnote about what the "dbar" means in the differential form of the First Law. I talked explicitly about how Eint is a function fE of pressure and volume:
Eint = fE(p,V)
and so the derivative d Eint is the partial of fE wrt p times dp plus the partial of fE wrt V times dV.
There is no function fQ such that:
Q = fQ(p,V)
and similarly no function fW such that:
W = fW(p,v)
Thus dQ and dW don't exist in a mathematical sense although we can certainly consider infinitesimal changes in Q and W, which is what "dbar" means.
In Sec 20.6, I more or less followed the text with one exception:
For free adiabatic expansion of a gas, the text is slightly wrong when they say the final temperature equals the initial temperature. The final temperature is slightly less than the initial, and the difference increases as the initial density of the gas increases. The reason is that as the gas expands, some if its internal energy goes into the increasing potential energy of the molecule-molecule bonding so less is available for kinetic energy of vibration.
For Sec. 20.7 - Heat Transfer, I followed the text fairly closely.
In discussing conduction, I did point out the analogy between the Thermal Resistance and Electrical Resistance.
In the discussion of convection, I mentioned that a convection process is the adiabatic expansion of air rising above the earth, and that when the air cools below the dew point clouds are formed. I also discussed how in a storm window, convection is the major heat loss since the R value for air is so high.
In the discussion of radiation, I mentioned that the rate is proportional to T^4. We also spent some time talking about thermal equilibrium, using a person in a room as the example. Such a person must be dumping about 100 Watts more heat due to radiation from their skin than they receive from the radiation from the walls of the room since the basal metabolism is on the order of 100 Watts.
Suggested Problems Chapter 20: 7E, 27P, 41P, 49E and 59P.
In Sec. 21.3 - Ideal Gases, I mentioned that the approximations involve ignoring the volume of the molecules and the attraction between molecules.
I then presented the Van der Waals equation, which is not in the text, and used some "real life" numbers to indicate the size of the corrections introduced by this equation over the ideal gas one. I also discussed how the equation tries to account for the two factors ignored in the ideal gas one.
I followed the text fairly closely in Secs. 21.4 and 21.5 with one small exception: I discussed how Einstein's work in 1905 on Brownian motion allowed the first good calculation of Avogadro's number.
On Friday, I discussed diffusion (Sec. 21.6) and also justified the result with a discussion of the drunkard's walk. From that we simply stated the equation for the mean free path.
I went through Sample Problem 21-7 sort of carefully. I also derived the statement in that problem that the molecules are on average 11 d apart (except I got 11.9 d).
I then discussed plasmas, which are not in the text at all (except tangentially in Problem 31.62). I began with a discussion of a gas discharge tube, relating the mean free path to the density of molecules. Thus, when the pressure is reduced, the mean free path increases, so the kinetic energy the molecules acquire between collisions increases. When that energy approaches the binding energy of the electrons, a plasma is formed. Recombination leads to the emission of light. I sort of waved my hands about why some people call plasmas the fourth state of matter.
On Tuesday, I discussed the magnetic bottle for plasma confinement, and related to the spiralling of protons and electrons about the Earth's magnetic field in the Van Allen belt.
I followed Sec. 21.8 of text fairly closely, deriving relations between Eint, Cv, Cp, etc.
In Sec. 21.9, I introduced degrees of freedom and equipartition of energy with monatomic and polyatomic gases only. After we got Cv = f/2 R, I then justified why f = 5 for a diatomic gas.
In Sec. 21.10 - A Hint of Quantum Theory, I also put Fig 20.2 up on the board again and pointed out that the limiting value for C for metals at high temperature is 6 R, the value for a polyatomic ideal gas.
I then departed from the text entirely, to point out that Fig 20.2 is only true for *some* metals, and that this was difficult to understand classically. I also showed how arguing that diatomic molecules don't have energy corresponding to rotation about their axis doesn't really make any sense, since the atoms are not really point particles. Thus the usual statement of equipartition of energy, such as the text's on page 591, involves a fair amount of heuristic hand-waving to actually apply. I mentioned that these heuristics are very useful since doing it right, i.e. with Quantum Mechanics, is very time consuming.
That said, I told the class that we would skip Sec. 21.11 of the text.
Suggested Problems Chapter 21: 14P, 30P, 43P, 63P, 69P.
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