Today I started writing a short piece on the early universe for kids; maybe eight year-olds? I am not sure I have what it takes, but I thought I would give it a shot. No wonder I never get anything done!
The tiny keyboard is going to take some practice! Here is a shot of me reflected in my office window, taken with the OLPC camera.
I finished assigning grades. I was nice. I would love to teach classes without grades; I think that grading only interferes with the educational channel between students and teachers. The only educational advantages (forget institutional advantages) of grading that I can see are the following:
(1) Graded assignments are much more likely to be taken seriously and completed. This consideration is paternalistic but it may have some force in the
real world. (2) Graded assignments give the students low-bandwidth but high-impact feedback on their knowledge and gaps in that knowledge. This consideration is also paternalistic, since verbal analysis of student performance contains far more information, but it is true in my experience that a low grade (on, say, a mid-term) is much more likely to inspire hard work than a verbal suggestion.
That said, grading makes my relationship with the students somewhat adversarial. It also discourages students (especially those with performance-based financial support) from taking risks with their course selection. This point, maybe the most important, is the one that clearly shows grading to be incompatible with the (important) mission of the University.
On the final exam, I asked the following:
Explain why the astronauts in the Space Shuttle are weightless.
I was lenient in grading. But my position is actually at odds with most of the textbooks. Here's why.
The standard textbook answer is something like
Actually, the astronauts on the Shuttle still have weight, since there are still gravitational forces acting on them. However, they feel like they are weightless because they are in an accelerating reference frame that is accelerating at the acceleration that the gravitational force is providing. This will be followed with various things about equivalence and plummeting elevators and non-inertial forces and so on.
My explanation is that the gravitational force on an object is not the weight of the object, when the word
weight is properly understood. The weight of an object is not the gravitational force but rather the contact force that holds the object up against gravity (and non-inertial forces). It is this contact force, after all, that a spring scale measures, because a spring scale does its job by providing a contact force. It is also this contact force, after all, that you perceive by having your feet pressed onto the floor or behind pressed into the seat of your chair. Indeed, gravitational forces can never be measured locally or internally (that's equivalence!), all you can measure is the stresses and strains required to oppose them in our non-inertial (by GR standards) frame.
My view makes the astronauts not misled but truly weightless. It also makes it true, not apparent, that one is lighter at the top of a hill and heavier at the bottom of a hill on a roller-coaster, and same for the related changes you experience in an elevator.
Since my last post I thought of many things I would like to have included but didn't. I guess the biggest is that I got two questions near the end that were effectively about Coriolis force. I did no non-inertial forces and/or accelerated reference frames. And this would have been fun to bring up in the context of Galileo, because there is a non-trivial Coriolis effect on his (apocryphal) experiment at the leaning tower of Pisa. It also is a nice use of cross products and pseudo-vectors.
I gave my last class today; still have to prepare a final. It has been a fun semester, and the students have performed well with a set of extremely difficult problems in lectures and recitations and problem sets, including ill-posed problems, approximate problems, and numerical problems. I ran out of time at the end, as I always do. Many subjects were not covered. Perhaps my biggest regret is not making it to the analysis of non-circular orbits using the effective one-dimensional radial potential. This analysis takes a long time to set up and then solve, but it is so damned beautiful.
In class today, I said that the biggest impacts of physics on society have been (1) the Brahe, Kepler, Galileo, Newton understanding of celestial mechanics, and (2) the development of the atomic bomb. The former was the beginning of precise, quantitative science (I think), undermining the authority of non-empirical scholarship in areas of natural philosophy, and establishing calculus and quantitative observation as the key tools of physics. The latter created an ability to (trivially and unilaterally) destroy ourselves and pushed all of world politics to global mutual destruction brinksmanship. After these two I might put the discoveries of AC and DC electricity (and batteries and the like), and the optimization of steam engines (and the discovery of thermodynamics and the like).
At the suggestion of my pedagogy mentor Sanjoy Mahajan (MIT), I assign
reading memos, to be turned in before the class in which I expect the reading to be done. This encourages the students to do the reading (I give a small fraction of the grade for doing the memos), and it also gives me some exceedingly insightful feedback about what works and what doesn't in the book I am using (Chabay & Sherwood).
In Chapter 11 (this week's chapter), the book makes the requisite notes about the increase in entropy possibly having something to do with the advance of time, a subject I avoid for its capability of generating enormous quantities of speculation. In the reading memos, one of my students (John Morrow) asked:
A closed system will tend toward maximum entropy. Is it possible for it to reach maximum entropy before the rest of the universe? And if so wouldn't that imply that time would stop for the closed system?
Beautiful question! This either throws doubt on that whole crazy idea, or else implies that all systems that have or perceive time must be out of equilibrium? Insane! But of course that would be some of that speculation I abhor.
I spoke yesterday in class about rotating solid bodies, in particular when the object is spinning around an axis not aligned with one of the principal axes of the moment of inertia tensor. The challenging point is that if you fix the axis of rotation then you get bearing forces or torques, but if you spin torque-free then the axis of rotation necessarily precesses. Of course the details are pretty hard for a student seeing this material for the first time; many of my colleagues would drop this from an intro course. But I love mechanical engineering, and this particular process comes up just about everywhere in our every-day experience. I can't help but talk about it!
Discussions with Yann LeCun (NYU) and Jon Barron (NYU) got me excited again about my project to produce planetarium software for the One Laptop Per Child project. My planetarium is a totally human-readable, child-hackable, ascii python executable. And it is fast to compensate for the slow hardware. That said, the OLPC is the best possible platform for field support for amateur and student observing.
A very nice paper appeared on the arXiv today on sheep–sheep collisions. It makes the point, which I stressed here, that collisions involve immense forces. It also makes some realistic estimates of the physical properties of the horns and skulls of bighorn sheep. But perhaps my favorite thing about the paper is that it begins by deconstructing a laughably wrong analysis in one of the many bad textbooks.
In class today I did
the rocket, the calculation of the trajectory of a device that is propelled by a continuous stream of momentum-carrying propellant. It accelerates as it sheds mass, and the final velocity is related to the initial velocity by a logarithm of the mass ratio (initial to final). I then worked out how big the Space Shuttle's fuel tanks need to be—relative to the orbiter—to get the orbiter to orbit, assuming that the propellant is expelled at around the speed of sound. Insane! It turns out that the space shuttle boosters expel propellant at around 4000 m/s, far, far higher than the speed of sound. If I know anything about hydrodynamics, this is non-trivial. Kudos to those NASA engineers.
There is a nice old problem of the force exerted by an engine pulling a set of train cars, all of which are initially at rest on a frictionless track, and all of which end up moving at speed v. You can think about it in terms of momentum (the change in momentum is due to a force acting over a time) or in terms of kinetic energy (the kinetic energy is produced by a force acting over a distance). The two ways of thinking about it get different answers by a factor of two!
The resolution is simple: The momentum of the train must be created by a force, and since it is a vector law in a one-dimensional problem, the total force times the total time must equal the total momentum. Any other solution fails to conserve momentum. The work done by the engine can go into kinetic energy, but it can also go into other forms of energy (like oscillation or dissipation in the train linkages). Energy is conserved as long as the kinetic energy is less than the work done. This resolves the discrepancy, and creates the nice result that when a train is accelerated, there must be dissipation, or else the train will be left vibrating or oscillating as it goes. I think this is an example of impedance matching.
Sanjoy Mahajan (MIT) and I have spent a lot of time talking about balls rolling down planes, in part because it is a very rich physics problem, and in part because it was the experiment that allowed Galileo to infer the constant acceleration behavior and galilean relativity. I started on this problem in class yesterday, but considering only the three energies: potential, linear kinetic, and rotational kinetic. When I asked the class to predict the outcome, I was surprised that I could get all three answers to the question "will the tube roll down the plane faster, slower, or at the same speed as a block sliding with little friction?" Even when we were done, not everyone got the inference, but I have to admit, the issue is subtle. Next time I will start to look at the problem from the point of view of forces; unfortunately, the class isn't quite ready for torques yet.
I showed the students complex numbers today, in the context of solving the damped harmonic oscillator. I took my time, and treated it as a cultural romp rather than a physics problem, so I enjoyed myself. What I was surprised to learn is that almost every student in the room knew that the square root of negative one is i. What is up with high school math that every student in the room learned what an imaginary number is, while not a single one learned how to program a computer? Which is easier, which is more relevant to them, and which is more natural given their interests and materials? I would say: Programming, programming, and programming!
My midterm exam was today, and I think I made it too hard. I haven't finished grading it, but I don't have any student who is likely to get perfect, and many who got very little completely right. It is my own damned fault, because most of the problems in the exam were multi-faceted and conceptually rich. Now I have the job of reassuring them that I won't kill them at grading time. Evaluation is very tricky, because it creates an adversarial relationship between student and teacher, exactly when cooperation and trust are required for learning.
Here's an energy example problem I discuss briefly every year, because it brings up a serious student misconception. I did it today in class, and it worked as usual, although about one fifth of the class got it right straight off the bat.
A block slides from rest down a long, slanted ramp that ends with a small, up-turnedski jump(I usually draw the end of the ski jump at about 45 degrees elevation above the horizontal). Air resistance and friction are negligible. After sliding down the ramp and leaving the jump, the block will fly on a parabolic trajectory. Will the peak of that parabolic trajectory come up above the vertical height of the starting point, exactly to the height of the starting point, very slightly below the height of the starting point, or well below the height of the starting point?
The students want to go with
slightly below. The correct answer is
well below, because the trajectory in gravity never brings the horizontal component of velocity to zero, and therefore never brings the kinetic energy to zero, or even close to zero. This leads to a nice discussion and an instructive comparison with the typical
roller-coaster problems out there.
Many students found it nearly impossible to complete the problem-set I gave with the golf shot with air resistance (PDF). I will have to analyze the problem sets to find out why. This was the third problem I have given on a problem set that involved making a numerical integration spreadsheet, so it wasn't integration per se that was hard for them. On the other hand, this was definitely the most physically challenging of the numerical integration problems I have given. Once again, I learned that the fact that I could do the problem in 20 minutes in Microsoft (tm) Excel (tm) does not mean that the students will find it easy! And I don't want to denigrate the students, many of whom clearly put in long hours on that problem. This is a level of enthusiasm I want to harness in this class!
After the mid-term, I will back off to a problem involving the numerical integration of sine and cosine, which is a straight-up math problem, but nonetheless very instructive.
Andrei Gruzinov (NYU) proved analytically my conjecture, made yesterday, that the distance a golf ball flies is dependent on the initial
muzzle velocity only logarithmically, in the air-resistance-dominated limit. Let's hear it for uninformed intuition! Actually, I have to admit that my intuition was highly informed by messing around with numerical integration spreadsheets. Gruzinov's analysis also confirms my conclusion that good golfers hit the ball about as far as it is physically possible to hit, given annoying limits like the speed of sound.
I just posted a problem set with a problem that involves numerical integration of a golf shot with air resistance, and comparison to the no-air case. With air, the golf ball must be hit far, far, far harder! In fact, if one assumes standard ram-pressure air resistance and a 45-degree elevation shot, it is essentially impossible to hit a ball 250 yards (as good golfers have no trouble doing). Of course, when air resistance comes in, it is better to reduce the elevation angle (as good golfers do!), which makes the shot possible again.
The amazing fact is that golf shots are enormously affected by air resistance, and no air-free calculation is in the least bit relevant. For shots of hundreds of yards, the with-air shot requires factors of ten more
muzzle velocity, or factors of hundred more initial kinetic energy, if the elevation angle is held constant. These factors reduce a bit if you have the freedom to drop the elevation angle. Sweet!
Hooke's Law is usually described as F=−kx, and thought only to apply to idealized springs, but really Hooke's Law is that stress is proportional to strain, and it applies not just to idealized springs but to every object in the entire universe (for small distortions), for extremely fundamental reasons. What an observation Hooke made! I tried to impress this upon my class today.
In class yesterday I compared two problems: the block on an inclined plane (sliding without friction) and the car sliding around an icy (frictionless), banked curve. In the latter, the challenge is to enter the banked turn at exactly the right speed that you make it out the other side without either sliding uphill or down. The nice thing about doing both these in one lecture is that they have nearly identical free-body diagrams, both of which have a massive particle acted upon by gravity and a normal force (at the same angle to the vertical if you set it all up correctly), and yet the physical situations are so different and the accelerations point in different directions. The comparison brings out a lot of conceptual material about contact forces and kinematic constraints.
The banked turn example is also hilarious, and has many nice details, such as that in general if you enter a conical banked curve and slide around to the other side without friction, you will come out with the car pointing in a strange (and non-trivial to calculate) direction.
I worked out a page of dimensional analysis and order-of-magnitude estimation to compare automobile energy expenditure in the form of acceleration with energy expenditure in the form of battling air resistance (ram pressure). After putting it together I realized the obvious: The air resistance losses exceed the acceleration/braking losses when the journey is long enough that the car has swept up its own mass of air! This means that for typical US cars, acceleration/braking dominates for journeys much less than 1 km (or city journeys in which there are stops much more frequently than once every km), and battling air resistance dominates for journeys that are uninterrupted by stops for distances much longer than 1 km.
Yesterday in class I worked through the problem of a bouncing ball, concentrating on estimating the magnitude of the force from the floor at bounce. Not a single student was even close to getting the magnitude of that force correct, even after many minutes of discussion, a few minutes of working in small groups, and more discussion. Eventually two students got it and understood after my
demonstration in which I prepare to drop a book on a student's hands (comparing with the case in which the student is just holding the book).
Before, during, and after the class, students asked me if the class is going to be more formal soon or ever. I said
yes. But what disturbs me is that if we go and do formal problems with vectors and calculus before the class can see even roughly the magnitude of the normal force on a bouncing ball, we are teaching math, not physics. I understand where the students are coming from: They like physics in part because it is formal. But there is no point in calculating forces when you don't understand what forces are.