Hogg's Teaching

what's the right answer?

2011-11-12T18:17:00.001-05:00

In her K/1 class (ages five to seven), the 'fuzz was doing "the weather" when an argument broke out between those who thought the clouds "move the Sun around" and those who thought the clouds "block out the Sun". She let the discussion proceed, encouraging contributions. In the end, the "move the Sun around" group got the consensus. What to do? You can correct them all, and then they learn that scientific truths are handed down by more knowledgeable authorities ("How do you know the Universe is expanding?" "I read it in a book."). Or you can let it lie, in which case they go home thinking they know something that in fact is wrong. Or (best, but extremely time-consuming), you can go through the process of having them turn their pseudo-scientific explanations into predictions about other phenomena, or have them extrapolate their model into other domains, and then see why or where it breaks. That's (to my mind) the only solution you could possibly call science, but it would require an absolutely radical replacement of the current curriculum and structure of school. The 'fuzz didn't have the right (it was someone else's classroom) to blow the schedule, and she didn't want to be a priestess, so she let it lie and moved on to the next activity.

vectors and their derivatives

2011-09-14T23:27:00.000-04:00

The time derivative of velocity is acceleration, both vectors of course. But I was reminded in office hours today of just how hard it is to get across the idea that the velocity vector and the acceleration vector can point in totally different directions. And some students have trouble seeing this when a ballistic stone is going upwards along some (parabolic) arc, some have trouble seeing it when it is going down, and some have trouble seeing it at the top. That is, different students have very different problems visualizing the differences of the vectors over time.

I said in lecture that this issue was deep but I didn't emphasize it enough. I feel like it is so big it almost needs its own week!

Stokes vs ram pressure

2011-09-05T06:30:00.002-04:00

Macroscopically, air resistance is ram pressure (proportional to cross-sectional area times velocity squared). Microscopically, drag is Stokes-like (proportional to radius times velocity). Where does the cross-over happen? I didn't have the guts to put that on problem set one of my course for pre-health students, but it will be in around problem set eight. It could be on problem set one, because the transition can be obtained purely by dimensional analysis.

In transport processes, there are often qualitatively different effects working at small scales than at large. Another good example is diffusion vs convection.

what does a future doctor not need to know?

2011-09-03T06:18:00.000-04:00

My big challenge in preparing my General Physics I syllabus is to figure out what to cut, when the majority of the students are pre-health. I cut thermodynamics, because we have learned that it is taught also in chemistry (and other places). I then wanted to add more material about fluids and elastic solids (pretty relevant to medicine, it seems), so what to cut? I ended up cutting most of rotation, spinning, and angular momentum. Why? To understand the body, you do need to know about torques (how does your arm work, static structures, and so on) but you don't really need to conserve angular momentum. Or do you? The centrifuge spins, but it doesn't have angular dynamics.

(I will be doing the centrifuge.)

parallax distance

2011-07-15T03:08:00.002-04:00

The parallax distance to an object is the distance you get by moving your (single) eye's position by a transverse distance x, measuring the angular displacement θ of the object given that move, and dividing the transverse distance by the angle.

Problem-set problem: You are outside after the rain late in the day and you see a rainbow. What is the parallax distance to the rainbow?

(Thanks to Andrei Gruzinov at NYU.)

memorizing capitals

2011-06-07T09:08:00.004-04:00

Cheryn (9 years old): What's the capital of Kansas?

Hogg (40 years old): I don't know; it is a useless piece of knowledge, because I can look it up in a few seconds on the internet.

Dustin (30 years old): In other words "I don't care, because I can just ask my iPhone." but then your teacher says "Ah, but what will you do if your iPhone isn't working?" answer: "I am in New York City; if the iPhones aren't working, I don't need to know the capital of Kansas, I need to know how to shoot, skin, and dress a squirrel!"

don't make students hate math!

2011-04-28T19:23:00.003-04:00

In my previous post I implied, inadvertently, that we should only teach useful math. That was not my point. My point was that we should not teach math if the effect of that teaching is to cause most students to hate it. Of course if we could teach it such that the effect was to cause most students to love it, I would be all for teaching it!

don't teach math!

2011-04-26T08:37:00.004-04:00

In a nice conversation about writing for education, Adam Gidwitz (the author of A Tale Dark and Grimm) pointed me to The Mathematician's Lament. The book makes (much more clearly than I) a point I have been making informally for years: If you want students to know and love math, you definitely should not teach it in school! (Same for literature.) The mathematics requirements in school empty the subject of its meaning and point, and are useless to boot. How many non-scientists use the quadratic formula, ever? Discovering the formula would be fun, using it is a drag (and exceedingly rare).

visualization of high dynamic-range data

2011-04-02T09:11:00.003-04:00

college: what's the point?

2011-02-06T10:05:00.006-05:00

In the State of the Union address of 2011 January 25, Obama raised one of his standard themes, which is the great importance of college education:

If we take these steps—if we raise expectations for every child, and give them the best possible chance at an education, from the day they are born until the last job they take—we will reach the goal that I set two years ago: By the end of the decade, America will once again have the highest proportion of college graduates in the world.

Though I don't disagree that we need to give all Americans great opportunities, I disagree that the best way to achieve that is through universal college attendance.

First of all, as pointed out here and many other places, though employers want to see college degrees, they do not really look for many of the things in a potential employee that college brings. That is, it is not clear that college is delivering useful things to employers. A good example is math skills; even high-tech employers do not look for math skills among potential employees.

Second, since most of the important skills for employers are communication skills, flexibility, problem-solving, and ability to learn (rather than specific content knowledge), employees ought to be well prepared by the end of high school. How can (more than) full-time work for twelve years not be enough to teach 18-year-olds the communication and common-sense skills they need to do most jobs? I realize that schools face huge challenges, but why not make helping the schools the top priority?

Third, and related to the second, the reason Obama and other politicians focus on sending students to college instead of improving K–12 is that college is private. Fixing schools takes public money; sending kids to college takes private money. Even state schools in America charge tens of thousands in tuition, and they don't have the capacity to take up Obama's slack. So requiring all kids to go to college is like levying an enormous quasi-voluntary tax on parents, relieving society of the responsibility of providing opportunity to the less fortunate. If a kid has trouble, in this vision, it is because his parents didn't do right by him.

Indeed, this ties it all together: Abandon the public schools, because improvements there would reduce disparity and improve the public sphere. Require (effectively) all potential employees to go to expensive colleges, because that is tax-neutral and increases disparity! What happened to the common good?

Apollo 11

2010-12-27T09:26:00.006-05:00

Pete Mao (Caltech) sent me a note for Newton's birthday, pointing out that all sensible transfer orbits to and from the Moon ought to have half-periods (transfer times) of about 5 days. And yet, as he also pointed out, Apollo 11 came back in 2.5 days. What gives? Did NASA waste fuel to improve the filmic value of the mission, or does the enormous tidal effect of the Sun change the sensible set of transfer orbits for some reason I don't understand?

(His note on this subject also had a nice discussion about what a 5-year-old wants when he or she asks a scientific question, and why just answering it is the wrong response.)

diagnosis

2010-09-03T08:41:00.002-04:00

I posted about diagnosis on my research blog and that made me realize that I should post here too. Almost all of science is diagnosis: You must diagnose your code, you must diagnose disagreements between theory and observation, you must diagnose inconsistencies in results. And yet diagnosis is so far outside any formal curriculum, no-one knows what you would even think about teaching there! It is taught to doctors and automobile mechanics, but not astronomers, who need it just as much, or perhaps even more.

Seymour Papert makes this point very clearly—and much more generally—in his book Mindstorms. One of his main arguments that computers must play an important role in education, or in thinking about thinking, is that programming a computer forces us to confront bugs. These require diagnosis (and also lead to interesting insights, a consequence he discusses extensively).

astronomical detectors

2010-05-17T10:21:00.003-04:00

Another problem-set problem, inspired by an email today:

Imagine measuring the brightness of two stars, one of which is a 100,000 K blackbody, and one of which is a 3,000 K blackbody. You are interested in the flux ratio in the V band. In one experiment, you perform this pair of measurements with a (flatfielded, calibrated, etc) CCD with a standard V-band filter on it. In another, you perform this pair of measurements with a bolometer with the same filter on it. Do you expect the flux ratio you measure to be the same in the two cases, and, if not, what is the expected difference? Imagine that both detectors are very high in quantum efficiency over the wavelength range of interest, and that you are capable of making accurate, calibrated, sky-subtracted measurements in both cases.

perspective

2010-04-12T12:15:00.004-04:00

Future problem-set problem: These two images were taken from the same three-space camera position, with the same digital camera. What, physically, was changed about the camera, and why did it have the effect that it did?

infinite jerk

2008-09-23T20:54:00.004-04:00

Last week I worked through some kinematics, and it led to a discussion in lecture about the jerk, that is, the time derivative of the acceleration. I was emphasizing that the velocity must be a continuous function of time (lest we have infinite accelerations and hence infinite forces), but then I realized that most people in the class (physics majors) felt that the acceleration must also be a continuous function of time.

After a bit of thought I agreed with them, but not for the reasons they wanted; indeed several were unsatisfied with my discussion: As far as I can tell, the only thing that limits the jerk is the propagation of information. When you slam on the brakes on your car, it takes a finite time for the brake shoe to come in contact with the wheel. This is not really a fundamental problem with infinite jerk (the jerk appears in no physical law), but it any real situation because information propagation (and other kinds of changes) happen at finite speeds, the jerk can never really be infinite.

vector subleties

2008-09-13T10:46:00.007-04:00

It is vectors all week this week, in my class, and in two classes I have taught for others. It is understandable that they confuse students, even physics majors with good backgrounds. Here are some subleties that I like to point out:

Vectors have a magnitude and a direction, but that is not sufficient. They also have a coordinate-free existence or description, and they form a linear space (with the usual linear operators). In this sense, despite what every textbook says, the unit vectors that define the coordinate system are not vectors!
Although vectors carry around all this geometric baggage, they have a magnitude and a direction and nothing else. I can still confuse the physics majors by sliding around vectors on the board. There is no position associated with a velocity vector, and we confuse the students by always drawing the velocity as coming from the object that is moving.
Multiplication of a vector by a scalar is usually conceived as changing the magnitude of the vector, which it does, but it also changes the units, in many cases of interest (for example when a displacement is multiplied by an inverse time to make a velocity). So it often produces a new vector that is not longer than the original vector, nor shorter, but really incomparable.
There is a perfect symmetry between the relationship between velocity and position and the relationship between acceleration and velocity. However, it is far harder for students to understand that the acceleration vector can point perpendicular to the velocity vector than it is to understand that the velocity vector can point perpendicular to the position vector. No amount of class time spent on this point is wasted, in my experience.

everything is an approximation

2008-09-10T20:23:00.004-04:00

One of the main things I emphasized in today's class (computing a trajectory in gravity near the surface of the Earth with no air forces) is that every calculation in physics is an approximation. The parabolic trajectory near the surface of the Earth is an approximation to the tip of a very eccentric ellipse, and the eccentric ellipse comes out only in the Newtonian approximation to GR, and even that only holds if there are no other forces acting (and there always are). There are also small adjustments for reduced mass, and if the object has non-trivial extension. Crazy! And in high school this is all taught like it is exact: Just plug numbers into the equations!

dimensions and symmetry

2008-09-05T07:58:00.002-04:00

In class, I spend a lot of time on the dimensions of physical quantities. If the left-hand side of an equation is an energy, then the right-hand side must be also. Or if the left-hand side is a quantity measured in J, then the right-hand side must be also. Is it right to call this property a symmetry of physical law? It acts very much like a symmetry, because it selects out of all the things you might write down a very small fraction that could conceivably be physical laws or physical results. On the other hand, these issues are so fundamental, they almost transcend that.

is our children learning?

2008-08-21T07:20:00.004-04:00

Teaching style / technique came up at the MPIA mess hall lunch table last week. I said that I use the highly interactive style. Some at the table opined that it probably only helps the best students and is bad for the worst. I differed, but realized that I have essentially no objective evidence. There are certainly studies on the subject, many of which do limited forms of controlled experiments, and they tend to support my view. But it is truly impossible to do a proper differential experiment.

You can't teach two classes at identical quality in two different styles to two identical sets of incoming students. It gets worse the more you think about it. For just one example of the biases: Even if you get two teachers to agree, the one that agrees to do the newer style will tend to be younger; that is, very different when it comes to interacting with the students. And for another: There is no agreed-upon evaluation of student knowledge or aptitude, before or after, and the ones that exist (FCI, for example) favor certain kinds of knowledge.

This relates to a bigger issue I sometimes expound upon from my soap box: When it comes to human or important things, like teaching, hiring, promotion, and supervision, scientists tend to become non-scientists, and throw repeatable empiricism out the window. They are convinced that what they are doing is right despite being able to muster no piece of objective evidence (indeed, evidence is often concealed behind confidentiality or human subjects rules).

For all these reasons, I am a big fan of those who are trying to actually make measurements of the effectiveness of various educational strategies. More power to them, and I will continue to take their advice.

OLPC: the leading-order term

2008-04-30T10:41:00.005-04:00

There was a meltdown on the One Laptop Per Child developer list because Negroponte (OLPC chief) said that they would work on a Windows version. In the ensuing discussion (which was, in fact, very enlightening and constructive), many issues came up, about code development and open-source and the educational value of having computers in the hands of children.

One point made by one of the developers, with which I strongly disagree, is that the leading-order term for the educational impact of the OLPC is that it gives children access to the web. Although I love the web (as my non-existent readers know), this is not the leading educational impact of OLPC, if OLPC is successful. If the main point is the web access, then give the students all ASUS or Nokia or Classmate low-cost computers and be done with it!

The leading-order term in the OLPC project is that the computer is a device that can be modified, programmed, altered, and made to do new things. The project de-mystifies computers and electronics and technology and software and the web. It is not access to the worlds information, but an introduction to the world's modifiability and opportunity for innovation. Unfortunately, I don't think everyone on the project agrees, and I don't think that the countries that are investing in OLPC understand. This may bode ill for what might be right now a marriage of convenience between constructivist educators and countries hungry for development (of the economic kind, not the code kind).

teaching physics teachers

2008-04-11T23:48:00.004-04:00

I took a break from my no-teaching, all-research sabbatical to make a guest appearance this week in Jhumki Basu's course Recent Advances in Physics in NYU's education program. Her students are building new science units with help and ideas from current researchers. I presented not really my research, but some of my research techniques: estimation and approximation. No surprise there!

I showed on dimensional grounds that cars like the ones we currently drive will never do far better than 30 miles per gallon. 100 maybe. But never 1000. A nice result, with important implications, using only techniques that a high schooler could easily muster.

After I spoke, we discussed, and it was noted by one and all that despite the simplicity of the techniques, in fact estimation and approximation techniques are non-trivial and sophisticated. It is hard to incorporate them incrementally into the existing New York State middle- and high-school curricula. On the other hand, it is my (perhaps optimistic and/or utopian) view that if these things were the focus of quantitative education from day one, they would be easy to have mastered by the end of high school. Of course the teachers I was talking to are going into the system that exists; they can't start from scratch!

Many other interesting things came up, which I hope to blog about at some pont in the future, including students' lack of contact with machinery and hardware and electronics, and the idea (that I hold, but others don't) that education ought to give students skills and tools, rather than knowledge.

mean average rainfall

2008-03-15T11:23:00.002-04:00

I dropped in on Sanjoy Mahajan's course 6.055/2.038 Art of approximation in science and engineering at MIT yesterday. We learned about mean average rainfall; you can estimate it pretty well by considering the mean Solar flux, the specific heat of vaporization of water, and the density of water. If you assume all of the Solar flux goes into evaporating the oceans you get 5 m/yr of rainfall, but the true average on the earth is about 1 m/yr; the factor of 5 comes from things like the fact that much of the earth is land, much is covered by clouds, light is reflected, light is absorbed by other processes, and other messy details of the energy budget.

After class, Mahajan and I discussed the size of raindrops, which has a similarly simple calculation: They break up when the stresses exceed the surface tension stress; the main stress is air resistance, which, at terminal velocity, is balancing gravity. I haven't checked the calculation, but Mahajan says this gives you a few mm.

cosmology for kids

2007-12-29T22:03:00.001-05:00

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!

One Laptop per Enthusiast

2007-12-27T15:55:00.000-05:00

I am posting this content-free post from my brand-new OLPC, for which I am optimizing my planetarium and for which I hope to make an Astrometry.net client.

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.

assigning grades

2007-12-21T17:11:00.000-05:00

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.

weight, gravity, and contact force

2007-12-20T17:13:00.000-05:00

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.

material left out

2007-12-19T17:16:00.000-05:00

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.

last class

2007-12-12T22:19:00.000-05:00

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.

physics and society

2007-12-10T17:24:00.000-05:00

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).

reading memos and the arrow of time

2007-12-05T20:42:00.000-05:00

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.

rotation of solid objects

2007-12-04T16:53:00.000-05:00

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!

the OLPC

2007-11-27T21:47:00.000-05:00

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.

sheep–sheep collisions

2007-11-27T09:56:00.000-05:00

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.

space shuttle

2007-11-19T17:58:00.000-05:00

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.

accelerating a train

2007-11-12T21:57:00.000-05:00

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.

rolling down planes

2007-11-08T19:38:00.000-05:00

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.

complex numbers

2007-10-31T22:34:00.000-04:00

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!

midterm exam

2007-10-22T22:13:00.000-04:00

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.

energy misconception

2007-10-15T23:59:00.001-04:00

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-turned ski 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 to or 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.

numerical homework

2007-10-15T23:59:00.000-04:00

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.

golf and math

2007-10-10T21:52:00.000-04:00

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.

golf and air resistance

2007-10-08T23:31:00.000-04:00

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

2007-10-03T16:59:00.000-04:00

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.

blocks on planes

2007-10-02T20:46:00.001-04:00

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.

cars and energy

2007-09-24T16:32:00.000-04:00

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.

forces and formality

2007-09-20T11:20:00.001-04:00

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.