An unusual formulation for the work-energy theorem

Most textbooks define work and kinetic energy, then show the connection: the net work done on an object is equal to the change in that object’s kinetic energy.

My students have always had difficulty with this principle. I spend a LOT of time explaining how to calculate the net work on an object, either by taking the scalar sum of the work done by each of several forces, or by calculating the work done by the net force. It doesn’t matter – students will continue to set the work done by any old force equal to a change in kinetic energy (or sometimes to just kinetic energy, without the “change” part.)

The most confusing type of problem asks something like, “How much work must you do to raise a 1 kg block 1 m high?” Students don’t know whether the answer should be +10 J, -10 J, or zero. And don’t get me started on “How much work must be done by an external force to move a charge from point P to infinity.”

These problems are tricky because of the implicit assumption of zero kinetic energy at each position. Using the standard textbook definitions, I can get students to understand that since there’s no change in kinetic energy in these cases zero net work was done. But that doesn’t answer the question! This idea of work done by a non-conservative force is difficult in the context of the work-energy theorem.

My answer this year: I didn’t technically teach the work-energy theorem, as described in textbooks, at all.

I defined work properly, including how to find the sign of work done by a force. (If the force has a component parallel to displacement, work done by the force is positive; if the force component is antiparallel to the displacement, work done by the force is negative.) I defined kinetic energy, and gravitational potential energy.

Next I jumped straight into the conservation of energy, but I wrote it in a weird way. Consider an object moving between two positions A and B. We know the total mechanical energy at position A must equal the total mechanical energy at position B, less the work done by any non-conservative force that dissipates mechanical energy to thermal or other energy. Instead of using one side of an equation for position A and one side for position B, I wrote the following:

W_NC = (KE_B – KE_A) + (PE_B – PE_A)

How did I define this W_NC term? I called that the work done by a non-conservative force. (How did I define a “non-conservative force?” At this stage, I said a non-conservative force was anything aside from the force of gravity. Bear with me.)

In the standard conservation of energy without friction problem, W_NC goes to zero. If a block slides with some friction or air resistance, then W_NCbecomes a negative value; if we’re talking, say, an airplane with a propeller, then W_NC is the (positive) work done by the propeller. The class had little trouble with this formulation, especially as they quickly recognized that most of these terms will usually go to zero.

As we discussed other forms of potential energy – elastic potential energy and electrical potential energy – we merely changed our equation for PE, and we called W_NC the work done by any force that isn’t gravity, a spring, or electricity. This definition is entirely correct and consistent as long as we stick to problems involving a single form of potential energy. (At higher levels of physics we’d have to discuss the deeper meaning of a “conservative” force and how the potential energy is defined… but we’re not at a higher level of physics right now.)

The main advantages of this formulation of what I now call the “work energy theorem:”

• I spend essentially no time explaining how to find the NET work on an object. The concept of net work is not particularly important in this formulation. Good – that used to be confusing to my class.

• We only have to learn and use one overriding equation for all energy conservation-type problems. I used to teach Wnet = ΔKE, plus conservation of energy without friction, plus energy conservation with friction. That’s all accounted for in my newly formulated work-energy theorem.

• The problem in green in the third paragraph above no longer causes trouble! Because I’ve explicitly included how work done on an object can change both the object’s kinetic AND potential energy, there’s no confusion. Both KE terms go to zero, one of the PE terms goes to zero, and we end up with W_NC = PE_B. When we lift the object, we have to do work equal to mgh; when we move the charge from point P to infinity, since potential energy at infinity is zero, we do work equal to qV at point P.

My colleague teaching honors freshman physics tried this formulation, and he loved it. I’m going to do this again next year. Try it! Tell me what you think.

GCJ

Mail Time: Is Color Determined by Wavelength or Frequency?

Visible spectrum from betesoft.com

Darren Tarshis, a physics teacher in Hayward, CA, has some physical optics questions:

Imagine that red light with a wavelength of 600 nm passes from air to a chunk of, say, diamond. In the diamond, I know the speed slows, which causes the wavelength to shorten (because the frequency remains constant). In the diamond, would the light have a different color because of its new wavelength?

I always teach my students that for a sound wave, pitch is determined by the wavelength/frequency, and for a light wave, color is determined by the wavelength/frequency. But I'm starting to think this may be incorrect, and the pitch is actually determined by frequency only, not wavelength, and color is determined by wavelength only, not frequency.

 

Yup. Frequency determines color and pitch.  The red light stays red even in diamond.

As a quick example: My voice is baritone. Imagine that you are in the pool with your ears just under water, and I am standing on deck talking to you. When the sound waves from my voice enter the water, they start moving about 4 times faster. The frequency doesn't change -- frequency of a wave NEVER changes when the wave changes materials -- so the wavelength increases by a factor of four as well. If pitch were determined by wavelength, then my voice would sound not only soprano, but squeaky soprano. 

Similarly, have you ever stood underwater and looked up at the trees overhanging the pool?  The leaves of the trees still look green, even though the light speed (and thus the wavelength) has decreased by 25%. 

I also propose a fanciful biological rationale for pitch being related to frequency only. The eardrum vibrates in response to incoming sound waves. It is the rate of vibration -- the frequency -- that can be measured by the ear and converted to a frequency. But how would an ear measure wavelength? With a meterstick? With a teeny weeny tape measure that an invisible goblin sticks out of the ear to measure the peak-to-peak distance of the incoming sound wave?

As long as the sound wave is in room temperature air, or as long as the light wave is in a vacuum (or air), then wavelength and frequency can be used to desribe color and pitch interchangably. That's why it's perfectly okay to say that red light is about 700 nm, and violet is about 400 nm. Those wavelength values must change when the light enters diamond, but the frequency of a given color will never change.

GCJ

Two awesome links -- useful for class!

Screen shot from This Too Shall Pass video by OK Go

Although I spent three straight days working on class and demo setup at the end of spring break, I don't have anything I'm ready to post about these. 

Instead, I recommend two links:

1.  This music video by the Illinois rock band OK Go reminds me of the halcyon days when MTV actually played clever music videos rather than girls dancing without their vests on.  The 4-minute video chronicles the world's most awesome rube goldberg device -- it puts that Honda commercial from a few years back to shame.  Physics content galore!  Thanks to Carrie Russell for making me aware.

2.  Randall Munroe, author of the xkcd comic strip that I linked a couple of weeks ago, frequently produces content that I want plastered on the walls of my classroom.  I've already ordered his visual presentation of character interactions in Star Wars and Lord of the Rings (click on the image to enlage; go to his store to order the poster).  This chart about metric units and orders of magnitude will be coming next, once I get around to ordering again.

But today's special awesome poster isn't yet available in Munroe's store. A friend of his who works in nuclear physics was becoming frustrated at fielding so may calls from ignorant journalists in the wake of the issues at the Japanese nuclear plant.  At the friend's request, Munroe created this Radiation Dose Chart.  I've linked here to the "blag"* post about the chart because he summarizes the chart's history and rationale there.  Click on the small image for a large image.  Look for the bananaphone reference!  This chart is public domain, so you can get it printed and hung yourself; I'm just waiting until it shows up in the xkcd store.

* Munroe writes a "blag," not a "blog."  He just likes being technologically different.

I'm generally not a fan of "link dump" posts, but (a) these were too good to pass up, and (b) I don't have anything else today.  Time for class -- I'll have lots more by the weekend.

Mail Time: How many atomic energy level transitions?

Joshua Beck of North Carolina writes:

When solving problems with energy levels, and a question asks them to draw all the possible transitions for an electron that is starting in say, the 3rd excited state. Should they only be drawing transitions that start there, or would they draw a transition from 3 down to 2, then also from 2 down to 1, etc. Does that question make sense?

It does make sense.  They should draw ALL transitions, including:

* from level 3 to level 1

* from level 3 to level 2

* from level 2 to level 1

Students should be able to calculate the energy, wavelength, and frequency of all three possible emitted photons -- one of those will likely be parts (b) and (c) of the question you pose.

GCJ

Why (or why not) teach physics first?

I hear several common arguments advanced about why physics-first is a good or bad idea. Thing is, having been in a successful physics-first school for 11 years, the common arguments don’t stand up to scrutiny. The true reasons that I think physics-first can work are rarely mentioned in literature or conversation. Read on…

Four arguments that I have frequently heard about freshman physics:

#1) Since physics concepts underlie those of chemistry, students must learn physics in order to better understand chemistry. Only the merest kernel of truth. Yes, electrical attraction along with quantum rules define the structure of the atom, as well as the chemical interactions that an atom may have. Not even the most advanced freshman physics student will develop a deep enough knowledge of electrical attraction or quantum rules to assist in understanding chemistry. Okay, sure, a first-year physics course can deliberately do some introduction to atomic structure as a prelude to chemistry. So what? Most everyone was supposed to have learned the structure of the atom in 7th or 8th grade. Who says we can better teach that under the auspices of physics rather than physical science or chemistry?

#2) A deep understanding of physics requires mathematics beyond the freshman level. Not true. It is easily possible to teach conceptual physics with no explicit algebra required. Such a course can still teach the “Big Three” skills. Yeah, I’ll grant you that alumni of a freshman conceptual physics course are not ready to do the advanced waves/optics/quantum sophomore undergraduate sequence, but that’s not the point of high school physics.

#3) Physics involves more equipment that students can see, feel, and touch… more concepts within the realm of students’ prior experiences. And, so, physics is a better stepping stone into more abstract sciences. True, assuming physics is being taught correctly. A freshman physics course that consists of a teacher solving math problems on the board is doomed to failure. However, a successful freshman physics class allows students to test their predictions with actual carts, springs, balls, boats, lenses, resistors, lasers… and that is a perfect introduction to rigorous science.

#4) Physics is simply too difficult for general population freshmen. A descriptive biology course is more likely to promote success in students who don’t have a future in the quantitative sciences. Possibly true, possibly defeatist. If a school does not have teachers who are willing and able to teach freshmen physics enthusiastically and appropriately, then students are better served by a descriptive biology course, taking a physics course in college or as a more intellectually mature senior. However, with appropriate teaching, even general freshmen can be successful in physics; and some of these folks might see that they’re NOT actually shut off from a future in the quantitative sciences. I know my school has seen a non-negligible number of initially pessimistic students discover an interest in physics through our freshman course.

Two arguments that I rarely hear, but that are the most important reasons that we teach 9th grade physics:

Many freshmen will at first struggle with or rebel against the “rigorous” problem solving required in physics. By the end of the year, though, the class is comfortable with the problem solving process. They thus have little trouble with the more abstract problems in sophomore chemistry. YES! Something like 20% of our sophomores are new, having not taken freshman physics. Except for the very top-end students, the new sophomores exhibit the same initial struggles with chemistry that our freshmen did with physics. Our conclusion: the first rigorous, quantitative high school science course, whatever the content, presents intellectual obstacles to the general population of students. Those obstacles can be overcome just as effectively by freshmen as sophomores. The question is, do we want to teach chemistry or physics to these freshmen; then we’re back to argument #3 above.

It is politically easier to sell a required physics course to bright, enthusiastic new freshmen than to grade- and college- focused upperclassmen. The most important truth of all. Because so many of their parents fear physics, so do our students often fear physics. Our freshmen, and their parents, just accept that physics is the required science course that everyone takes. There’s little drama – students try to do what the teacher asks, they see their grades improve if they work hard; it’s just another class. However, an upperclass physics course takes on heightened significance in students’ minds. A poor start to the year in 12th grade physics causes anger and resentment – “This course is keeping me from getting into a good college!” Even before the course starts, the gossip amongst the fear mongerers will prejudice students. The initial attitude of freshmen toward physics is generally open-minded neutrality – they’re so busy adapting to everything high school entails that they don’t think about physics beyond the next day’s homework. But to teach a class of general-level seniors who arrive with their jaws set, their eyes narrowed, prepared to game their grade with every trick they know – that’s a setup for disaster.

Please understand that I don’t believe that physics-first is the correct approach in all situations. A successful physics-first program requires top-rate, dedicated 9th grade physics teachers, as well as a supportive department and higher administration. If any of those pieces is not in place, 9th grade physics might not work, and the bio-chem-physics sequence might well be more appropriate. In today’s and yesterday’s post I merely deconstruct arguments so that the potential of a 9th grade program (or the necessity to maintain a physics last sequence) is not dismissed out of hand.

GCJ

Physics First Discussions: Reveal the Agendas

No, not the FIRST robotics competition, PHYSICS-FIRST.

I’m in the middle of a two-week spring break right now. Since I’m incapable of relaxing and ignoring physics teaching for that long – just ask Burrito Girl, my wife and sidekick – I’ve been working on a major redesign of my school’s overall physics curriculum. In the process of this revision, I’ve had occasion to reflect on all of the physics-first conversations I’ve had over the years. In tomorrow’s post, I will address some of the major arguments for and against a physics-first approach.

One of the most important realizations I’ve come to about a physics-first discussion is that the discussion itself is pointless until agendas are revealed and acknowledged. The decision about an overall school curriculum should be made on the merits of the proposed courses, and how well a curriculum does or doesn’t suit the school’s particular constituency of students and teachers. Too often, participants in the conversation steer consensus toward their own predetermined outcomes without listening to or acknowledging reasonable arguments.

The physics first supporters with agendas whom I’ve encountered generally fall into two categories: (1) Evangelical types, generally disciples of Leon Lederman, who are on a mission to grant physics its rightful place as the first, best, and most important of the three major sciences.* (2) Administrators who want change for the sake of self-promotion, so they can say “Look what I did, I ushered in a physics first curriculum! Now give me a promotion.”

* A line about physics first from Lederman’s Wikipedia entry, emphasis mine: “Also known as “Right-Side Up Science” and “Biology Last,” this movement seeks to rearrange the current high school science curriculum…” That’s, without question, evangelism.

The physics first detractors with an agenda whom I’ve known generally have a very simple point of view: “I don’t want to change, because I’m happy teaching whatever I’m teaching.  Doing something different would require a lot of work, and I might lose my monopoly on my special course or special students.”

Is a physics-first approach right? There can be no general yes-or-no answer. The question is ill-posed without substantial context. The better questions are, how can a physics-first program be successful at our school, what about physics first could define its success for us, and why use physics-first rather than a standard alternative.  Answering these questions neutrally, without promoting an agenda, is the way toward designing physics course offerings appropriate for your school.

Tomorrow, I’ll address some common physics-first pro- and con- arguments.

GCJ

Pole vaulting and the apparent weight at the equator

http://xkcd.com/852/

My first reaction when my colleague El Mole showed me the cartoon at the right was that 2 cm is way too significant a difference.

[Pause while you read the comic.  Good, ain't it?]

The fundamental principle is correct.  Because the linear speed of a point on the earth's surface is larger at the equator than at the poles, the "apparent weight" of a person, and thus the "apparent gravitational field g," will be smaller.  In advanced mechanics classes, Newton's Second Law is formulated in the rotating reference frame of the earth, and the effective g is reduced by a  centrifugal acceleration term equal to v²/r.

Why centrifugal and not centripetal?  In an INERTIAL reference frame, acceleration in circular motion is toward the center, i.e. centripetal.  "Inertial reference frame" means, in a sense, imagine that we observe the universe from a stationary camera placed above the rotating object.  Then the net force on the object is continually changing direction so as to push the object toward the circle's center.  However, if we instead observe the world from the eyes of the rotating object itself, then it seems like we are being pushed away from the center of the circle, i.e. in a centrifugal direction.  And if we consider a person rotating at the equator, it makes sense to consider the rotating reference frame; it's more interesting and useful to figure out what the rotating person feels than to figure out what would be observed by a stationary flying saucer over the north pole..

But to have the earth's rotation make a difference of nearly an inch?  An inch is significant in pole vaulting!  The last time the pole vaulting world record was broken, it was by Ukrainian Sergey Bubka over a ten year stretch from 1984-1994.  Each time he broke the previous record, he did so by just one or two centimeters.  The question that the comic begs is, should someone aspire to break records, should he compete exclusively in Ecuador rather than in, say, London?  Would the location make any difference at all?

I made my own order-of-magnitude estimate to check the comic.*  The gravitational field due to earth, without reference to rotation, is about 10 m/s². That term will be lessened by the "centrifugal" acceleration** v²/r. 

First, find v.  The radius of the earth is about 6000 km.  Multiply by 2π to get the circumference at the equator to be about 40,000 km, which is 40 million m.  We go around this circumference in 24 hrs = 80,000 s or so.  This gives a speed in the neighborhood of 500 m/s.

The "centrifugal" acceleration is then (500 m/s)²/(6,000,000 m) = 0.04 m/s². Compared to the gravitational g of 10 m/s², the centrifugal term is, say, four tenths of a percent.

Now, in the absolutlely simplest model, we might consider a pole vaulter as running at a fast horizontal speed, then launching himself as a projectile with that same speed.  The maximum vertical height the vaulter obtains is governed by vertical kinematics, with a known vertical launch velocity v_oy, final vertical velocity of zero, and acceleration of g downward.  This max height can be shown to be v_oy² / 2g.  Point is, the maximum height depends inversely on the first power of g.

So now we reach the end of the story:  what happens when we reduce g by a few tenths of a percent?  We increase the pole vaulter's maximum height by a few tenths of a percent as well. 

Bubka's record vault is 6.15 m.  Increasing that jump by four tenths of a percent would increase his vault height by... a couple of centimeters.  The comic is right. 

In practice, could Bubka have just gone to Indonesia to add two centimeters to his record?  Not exactly.  Four tenths of a percent is the difference in the apparent g between the pole and the equator.  If Bubka set his record at the 1994 Santa Claus's Merry Elves Invitational, then our analysis is sound.  But the farthest north city I can envision holding a major international track meet is, say, Oslo, Norway, at 60 degrees north latitude.  In Oslo, the effective g will be less than at the pole, but not an entire 0.4% less.  Since the linear speed of someone rotating on the globe drops off from equator to pole as the cosine of the latitude***, in Oslo the effective g is reduced by only 0.1%. 

The Oslo-Jakarta pole vault differential is more like 1.5 cm, not a full 2 cm.  Close enough.

Having read all this, my question for you is, who is the more complete nerd?  The xkcd author for carrying out this calculation and basing a comic strip on it, or me for checking the accuracy of the calculation?

GCJ
*This particular comic is generally quite good about its physics.  In fact, I'd be far more comfortable asking the xkcd writers to check me than vice versa.

** xkcd can explain this better than I ever could:  http://xkcd.com/123/

*** At the equator, cos (0) = 1, so his speed relative to Earth's center would be 500 m/s; at the pole, cos (90) = 0, so his speed is 0 m/s.  In Oslo, his speed is 500 m/s cos (60), or half his speed at the equator, and by the calculation above, the correction to g is one-fourth of the correction at the equator.

Get the vocabulary right in the first law of thermodynamics!

Q and U from a They Might Be
Giants wiki

W represents work done on a gas. W cannot increase or decrease. W can be positive or negative; negative W means work is done BY a gas.

Q represents heat added to a gas. Q cannot increase or decrease. Q can be positive or negative; negative Q means heat is REMOVED from a gas.

U represents internal energy of a gas. U can increase or decrease.

ΔU represents the change in internal energy. ΔU cannot increase or decrease.  ΔU can be positive or negative; negative ΔU means the internal energy DECREASES.
This was all posted to a class folder yesterday, after I graded a problem about a PV diagram and the first law of thermodynamics.  The 4 students who used illegal phrasology* such as "W decreases" lost credit.  I thought it was worth a classwide reminder with our exam coming up today...

* penalty: 5 yards and loss of down

Using publicly available external resources for your class's exam





A trimester exam -- or in your case, probably, a semester exam -- should be an authentic evaluation of what each student has learned or not learned in your course.  I think of the exam much like a playoff football game, or a state track meet.  It's the culminating experience for the season, showing in black-and-white how well your students have done, and how well you've done teaching them.

My friend the football coach points out to me why his job is more stressful than mine.  "Put yourself in my shoes," he asks.  "Have someone else, not you, administer your exam.  Have everyone in the community -- students, parents, alumni, administrators, EVERYONE -- watching, so they see every right answer and every ridiculous answer.  And, put up a real-time scoreboard so all can see how your students are doing compared to our rival schools' students.  That's my reality, every game, every season."

Although I would accept his challenge at the drop of a hat, I'm not recommending competitive examinations for all.  Don't worry. 

I'm putting forth this coach's point of view because for him and his team, every game is a test -- a fair, objective measure of his team's skill and preparation.  No one ever complains "that's not fair, the other team threw a pass!"  Everyone knows the rules up front.  The referee and his crew enforce the rules, but do not give advice on strategy.  In football in particular, a bit of scouting by watching film will even let the team know what "topics" will be on their "test" -- will it be "Their quarterback is fast and can throw on the run.  How do you defend him?"  Or could it be "Their offensive line is huge and mean, but their running back is small.  What do you do?"

I encourage introductory physics teachers to make the conditions of their exams clear and consistent.  The format should not be fundamentally different from the other tests that have been assigned throughout the year.  Just as a football team would be thrown for a loop if they were asked to play the playoffs under Canadian rules with a 120 yard field, your students will be less successful on an exam if they see question types that are brand new.  My AP tests and exams are all authentically in AP format -- 1.3-minute, no-calculator multiple choice, followed by free response questions of 10-15 minutes each.  In general physics this year, all tests and exams include an equal mix of 2-minute, calculator-okay multiple choice; 4-minute "justify your answer" items; and open response items of 2-4 minutes each.  No one will have to read the directions on the exam -- they will be able to dive into the physics without worrying about the format.

How would a football team react if the official were ambiguous with his decisions and instructions?  If he didn't tell anyone he'd started the clock?  If he called a penalty, but marked off yardage without an announcement?  If he didn't bother to tell anyone what down it was?  Perhaps he could say, with some justification, that a high school football team ought to know when they break the rules, and they should be able to keep track of downs.  But that official still puts the teams in a situation in which they cannot show off their football prowess.  The game becomes an argument about nebulous rules rather than a contest of skill.

Similarly, exam questions must be crystal clear.  It's not acceptable to ask, "In an ideal gas, what happens to temperature when P goes up?"  Sure, perhaps every problem you did in class involved a closed, rigid container, so perhaps you expected the class to assume a constant volume and to understand that P means pressure.  But why not write the question clearly?  Why make the students interpret? 

"That's the problem, Greg," you might say.  "I wrote what I thought was a perfectly clear and fair question, but still the students had questions about the question.  How am I to know what's clear and what's not?  How am I supposed to get my students to develop a sense for the level and difficulty of exam items, and then ensure that the exam matches their expectations?"

Here I strongly suggest using some sort of EXTERNAL evaluation.  Aim your class from day one toward some sort of test available from someone else.  The most common example of such a test is associated with the Advanced Placement program -- the AP exam's topics and difficulty are carefully controlled from year to year.  I have hundreds upon hundreds of authentic test items from which I can populate my tests and exams.  These questions have already been vetted for clarity, difficulty, and correctness.  You don't have to be teaching an AP course to use AP test items!  For years I taught a general physics course that covered only about 1/3 of the AP curriculum, but still used authentic AP items on tests and exams.

If you want a lower level than AP physics, there are other vetted test banks available.  The New York Regents test has been given since the 1930s, and pretty much all published exams are available online.  The SAT II physics likewise has released exams, prep books, and topic outlines available publicly.  Some colleges release their freshman exams year after year; pick one and follow their lead. 

No matter which publicly available exam you choose to follow, your testing becomes much more authentic through using that external source.  You become less the "bad guy" for writing questions that are unclear, or too hard.  Rather, you become like the beloved coach, the one who carefully prepares his team to meet the challenge of the playoffs.  "That was a hard exam," they might say.  "But it was exactly the kind of exam you had prepared us for.  I think I did well."  That's what I want to hear.

Greg