New Guidance for Free Body Diagrams on the AP Exams!

Some families argue politics or religion.  AP Physics readers have been arguing free body diagrams.

Many good teachers have differing opinions about the purpose and proper construction of the free body diagram.  The grading of free bodies on the AP exam has been the subject of extremely lively debate for many years.  Everyone, fortunately, has common goals:

* Award credit to students for correct physics

* Do NOT award credit to students for vague or incorrect physics

* Have enough flexibility in the rubric to accomodate alternative but reasonable interpretations of the correct construction of the free body diagram.

Perhaps the loudest of the shouting involves the issue of vector components.  Many of us -- including me -- take it as a matter of faith that a free body should never include components of a force.  Others reasonably point out that since the whole point of the diagram is to determine the magnitudes of the forces, and since one must break down angled vectors into components at some point, such components represent a necessary and important part of problem solving technique.  I see the legitimacy to both arguments. 

In order to quiet the arguments and send a clear message about free body components, the AP development has decided to change the language used to ask students to draw free body diagrams.  The test questions now will SPECIFICALLY forbid components on free body diagrams, and will even remind students to use a separate space to draw components where necessary.  You can see the white paper issued by the development committee here within the College Board's website.

Moral of the story:  Argument settled by fiat.  DO NOT put components on free body diagrams, or you will lose points.  Break angled vectors into components on a diagram separate from the initial free body.

First Law of Thermodynamics and the Sign of Each Term

The first law of thermodynamics is a statement of energy conservation.  In equation form, it states that

ΔU = Q+W

where each variable has the following meaning:

ΔU is the change in the gas's internal energy
Q is the heat ADDED to the gas
W is the work done ON the gas.

A typical test question will show a pressure-vs-volume graph, and expect the student to use the graph to determine a value, or at least a sign, for each of the quantities.  Therefore, it's worth memorizing the method of determining each variable from a PV diagram:

ΔU: Read the axes of the graph, and ΔU = (3/2)PV.
Q: CANNOT BE FOUND FROM THE GRAPH!
W: Look at the area under the graph.

The quiz/poll thingie I had up last week is shown to the right.  This question only asks for the signs of the three variables. 

Based on the definition ΔU = (3/2)PV, ΔU is positive when the gas's temperature increases, and negative when the temperature drops.  (Why?  Look at the equation, and remember that PV = nRT.) 

The sign of W can be trickier.  Work is done ON the gas when the gas's volume DECREASES. Think of a piston compressing the gas... a force is applied ON the gas over some distance.  So we consider W to be a positive quantity when a gas compresses, and a negative quantity when a gas expands. 

And finaly, the sign of Q can only be determined from the first law equation.  One must find ΔU and W and plug in.

And now, the answer to this week's quiz:

Start with ΔU.  This process starts and ends at the same spot on the PV diagram -- the product of PV does not change. Therefore, ΔU is zero -- there is no net change in internal energy.


Now look at W.  Consider each sub-process individually.  In process A-B, the gas expands, so W is negative.  In process B-C, there is no volume change, so no work is done on or by the gas.  And in process C-A, the gas's volume decreases, so W is positive.  But what is the sign of W for the overall process A-B-C-A?  There is more area under the graph in the expansion from A-B than in the compression from C-A.  So, the net value of W is negative -- net work is done BY the gas.
 
And finally, use the first law to find the sign of Q.  ΔU = Q+W.  Solving for Q, we find that Q = ΔU - W.  Since ΔU was zero and W was negative, then algebraically, Q must be positive.  This means that, in net, heat is added to the gas.
 
As a side note -- this process represents a cycle of a heat engine.  Despite what happens in each individual process, IN NET, some amount of added heat is converted into some amount of work done by a gas.  That's what is meant by a heat engine.
 
More polls soon...
 
GCJ
 

A different use of a clicker quiz -- snell's law

Springtime for seniors brings a competition as to who can do the least amount of work.  That's certainly not true for all seniors, but if you listen to any high school faculty this time of year, you'd think that this year's crop of 17 year olds were the passive-aggressive scourge of Satan.  In a previous note, I explain my use of the "exemption" as one prong in my defense against the senior slide.

The gist of the exemption:  Students who put in particularly strong effort on out-of-class assignments earn the right to skip a future assignment of their choice.  The first exemption of the year in general physics is awarded to the student with the highest homework average from the previous trimester.  From here on, exemptions will be given for things like maintaining an A homework average over the course of a full week, writing a perfect quiz when that quiz is based on a homework assignment, or a perfect fundamentals quiz.

The homework problem for today involved refraction in a triangular block of glass.  I've introduced Snell's Law already, and we've done several basic problems with simple geometry.  Today's problem was more complicated, because the normal was NOT straight up and down the page.  Take a look at the diagram to the right, which is slightly editied from a problem in the Glencoe text.  The homework problem did not label the 49 degree angle; instead, it gave θ_i as 45 degrees, and asked to find angles A and B, and θ_exit

I warned everyone ahead of time that angles A and B are NOT 60 degree angles.  That doesn't prevent half the class from making that assumption anyway, but it sets up for success those students who pay attention.

For today's quiz, I gave the diagram shown, with the 49 degree angle.  I asked the six questions at the end of this post, to be answered on the classroom response system (the clickers).

Now, I am careful NOT to use the clickers for quiz purposes through most of the year.  At first, I absolutely do not want students seeing how their peers did.  I don't want boasting about good scores, I don't want the sour-grapes rationalization that inevitably follows when someone gets a 25%.  Over the course of the year, 1/4 on a multiple choice quiz is a drop in the bucket.  My guys can deal with that.  What they CAN'T deal with at first is the idea that the average score on the quiz might only be 50%. 

In the spring, though, I use the clickers because I WANT score distributions to be public.  Everyone needs to see that yes, people are getting 6 correct answers on this quiz.  The clickers' instant feedback allows me to pinpoint the folks who missed a straightforward question, and make sure they know why they got it wrong.  And most importantly... when the quiz is over, I can IMMEDIATELY and publicly award an exemption to everyone who got a perfect score. 

GCJ



1. What is the angle of incidence at the left edge, labeled θ_i in the diagram?

2. What is the angle of refraction at the left edge, labeled r in the diagram?

3. What is angle A?

4. What is angle B?

5. What is the angle of incidence at the right edge?

6. What is θ_exit?

 

Double Slit -- convert all distance quantities to meters

We're beginning double-slit problems right now. This is not a difficult topic, especially for AP physics B students who have spent all year learning how to learn physics. They generally pick up quickly what the variables in dsinθ=mλ mean.  They have a bit of trouble with a deep understanding of m.  At first, I just get them to see that m = 0 at the central maximum, m = 0.5 at the first dark spot, m = 1 at the first bright spot, etc.  Once they have facility with getting physically reasonable answers out of the relevant equation, we talk about m as the number of wavelengths in the path difference between two waves.  The meaning of m is the only true conceptual challenge in double slits.

Getting correct answers out of dsinθ=mλ can sometimes be a chore, though.  It's easy to think that such a straightforward calculation doesn't require careful thought; and students hopefully have been trained by now that plugging into the calculator is of minor importance compared to a conceptual understanding of the topic at hand. Nevertheless, I usually need to remind the class that all distance quantities in this equation, or in its small-θ companion x = mλL/d, must be in METERS.

Here's today's quiz.  Note that I ask for no problem solving or calculation; I just make sure everyone knows what the variables mean, and how to convert to meters.  The most common mistake on this quiz:  620 nm can be written as 620 x 10^-9 m, or as 6.20 x 10^-7 m, but NOT as 6.20 x 10^-9 m.

GCJ

A red laser with wavelength 620 nm in air shines through two slits which are separated by 0.50 mm. On a screen 2.0 m away from the slits, the laser makes an interference pattern. The brightest spot is located directly in front of the two slits. You are asked to find the location of the nearest bright spot to the central maximum.

1. What is the relevant equation?

2. Assign a value to the following variables. USE UNITS OF METERS FOR ALL DISTANCE QUANTITIES!!! Indicate the variable that is not given in the problem statement; don’t bother to solve for it.

d =


θ =


m =


λ =

Poll Answer: Electric Field Created by a Charge

The poll question stem, which was posted throughout the week, is shown to the right.  Here's my answer.

Let's start with direction.  The electric field produced by a positive charge points away from the charge; the electric field produced by a negative charge points toward the charge.  Since a negative charge produces this electric field, the direction is "toward the charge."

Now for the magnitude of the electric field.  First of all, our students must be taught to use the variables given in the problem.  I know that the textbook says that the electric field due to a charge is E = kq/r^2.  This problem defined the distance from the charge as "L".  Thus, we must use "L" and not "r" in the answer.

Secondly, our students must understand what the word "magnitude" means.  All vector quantities, including electric field, include an amount of something (the magnitude) and a direction.  For example, a velocity might be 30 m/s toward the south.  In this case, "30 m/s" is the velocity's magnitude, and "south" is the direction. 

Now, when a vector is constrained to one direction (or when we're dealing with components of a 2-d or 3-d vector), it is mathematically convenient to define a negative and a positive direction.  If I'm working a kinematics problem with this car, I might call north the positive direction, in which case the velocity vector can be plugged in simply as "-30 m/s".  But, importantly, the magnitude of this vector is still "30 m/s."  The magnitude of a vector can not be negative!

Electric field is a vector.  When using equations regarding electric fields, never plug in the sign of the charge!  use the equation to find the magnitude of an electric field or force.  Then, use memorized facts to determine the direction of an electric field or force. 

In this case, then, the negativeness of the charge producing the electric field does not affect the field's magnitude.  The field has magnitude kq/L^2.  The direction of the electric field is toward the charge.

I like the poll... New question coming soon!

GCJ

Using Red AND Green Lasers

As soon as we get back from spring break, I will be discussing "physical optics" -- double slits, diffraction gratings, thin films, and so on.  I always start with the double slit -- why light through a double slit produces a diffraction pattern, and the equation for the location of the minima and maxima. 

Here the best demonstration, I think, is to shine a red laser through a diffraction grating.  (No, I don't start with a real double slit... a diffraction grating produces sharp maxima which don't invite arguments about whether we're really seeing bright and dark spots; and, the mathematics for diffraction gratings are identical to those of double slits.  Later, after the class is comfortable predicting the location of the bright spots, I explain the qualitative differences between diffraction gratings, double slits, and single slits.)

This demonstration can start off quantitatively.  Using the known slit spacing of the diffraction grating, the wavelength of the laser (printed on there somewhere), and the distance to the screen, predict the distance between the first two bright spots using dsinθ = mλ. 

Next, ask the qualitative question: if the red laser is replaced by a green laser, what happens to the distance between bright spots?  Let the class argue it out amongst themselves:  the wavelength of green light is smaller than the wavelength of red light.  (Yes, you have to know that fact for the AP exam.)  By the above equation, reducing the wavelength while keeping d and m constant reduces sin θ as well.  As the sine of an angle decreases, the angle itself decreases.  So, the angle to the bright spot decreases, and the bright spots will be closer together. 

Note that it is legitimate to argue using the small angle approximation equation x = mλL/d.  You'd find that reducing the wavelength reduces x, the distance from the central maximum to the mth spot.

Then, if you have different diffraction gratings, you can ask how the distance between spots will change using these other gratings.

"But I don't have a green laser," you say.  I bought one a few years ago for $200.  But now both green and red lasers have come way down in price.  Search around.  I just bought a pack of twelve keychain red lasers for about $2 each; I found green lasers for about $10 each.  Considering its other uses in geometric optics (for ray tracing in an aquarium, say) or in an astronomy unit (to point out stars in a nighttime observing session), the $10 is worth it.

Magnetism Introduction: Assigning reading, a reading quiz, and personally handwritten notes

I ask students to read their textbooks very, very rarely.  Textbooks can be excellent references, they can provide sources of problems, examples of problem solving processes in action... but it is rare that a first-time physics student can simply read and understand a section in a textbook.

Well into the school year, after I have hammered the class about the difference between physics and math, when the students themselves are demanding demonstrations and physical (rather than mathematical) explanations of new topics... then and only then will I consider assigning some reading.  And even then, that reading must have a clear and useful purpose.

One reading assignment I've used repeatedly is to introduce the magnetic force on a charge and the first right hand rule.  Serway's presentation is pretty good, though equation-heavy and technically written.

If I were to present this topic in class, I'd be stopped with questions at every turn:  "Is a magnetic field the same thing as an electric field?  What does the θ mean in F=qvBsinθ   ?  Does a negative charge get forced the opposite way as a positive charge?  Why doesn't a charge experience a force if it moves along the magnetic field lines?  Why do we use B for magnetic field, not M?"  I'm quite proud of my class for their inquisitiveness, I'm pleased that they expect these sorts of questions to be answered.  There have been numerous times in class when I've encouraged, nay, DEMANDED such questioning.  It's difficult for me to communicate "Shut up right now and listen for ten minutes while I just show you the fundamentals.  You'll have time to play with these new ideas on homework and in class tomorrow, but for now, I just have to feed you this information.  So be patient and quit buggin' me."

I assign the brief section of text in which the equation F=qvBsinθ and the right hand rule is introduced.  Sure, this is pretty confusing, even to my experienced AP students.  But now they know what questions to ask.  And the text answers many of these questions, so I can go straight to a presentation on the right hand rule, followed by several demonstrations in which I use a manget to deflect a beam of electrons in the direction predicted by the right hand rule.

Here's how I set up the reading assignment.  Instead of two homework problems on, say, Thursday night, one of the problems is replaced by the reading assignment.  Here's what I say:

Problem 2: READ Serway chapter 19-3, about magnetic fields. Magnetic fields are different from electric fields. I expect you to know equation 19.1, including what each term means; and, you should look at how to figure out the direction of a magnetic force. Friday's quiz will be basic questions about this reading.  You will be allowed to use your notes on this quiz if the notes were personally handwritten by you.

Note that I've referenced the only relevant equation in the section, though Serway provides others that are not particularly relevant.  Also note that by promising a quiz I ensure that a student who has difficulty doesn't just throw up his hands and say "I don't get it."  He's welcome to say that, but he'll fail the quiz.

(I'm particularly fond of the "you may use personally handwritten notes" proviso.  I credit Haverford history professor Roger Lane for this idea.  Half of the course grade in US History was based on periodic pop quizzes based on the assigned reading.  We were always allowed to use our notes, but only if these notes were handwritten by us, not xeroxed or highlighted.  So we all took pretty good reading notes.  And, the quizzes weren't so hard, 'cause we had paid such good attention to the reading, 'cause we had taken such careful notes.  Insidious, that Roger Lane.)

Anyway, here's the quiz.  Note that questions 3-5 are as much about knowing when a magnetic field DOESN'T produce a force as about the right hand rule.

1. What are the units of a magnetic field?

2. State the equation for the magnetic force on a charged particle. Define each variable.

3. A positive charge moves to the right in a magnetic field that points toward the top of the page. State the direction of the force on this charge.

4. A negative charge moves to the left in a magnetic field that points to the right. State the direction of the force on this charge.

5. A positive charge is at rest in a magnetic field that points toward the bottom of the page. State the direction of the force on this charge.

Trying out the google "poll" function

This is rather primitive, but might be fun.  Take a look at the left sidebar, where a fundmentals multiple choice question masquerades as a poll.  Feel free to vote... your response is anonymous.  I can't figure out how to format equations properly in the poll.  The text editor won't allow html tags.  Any thoughts?

I'll leave the poll up until I get a few votes.  Then, if there's disagreement about the answer, I'll write a post about it.

UPDATE -- Okay, I screwed up the initial poll question.  It asks for magnitude and direction of the electric field, and I left off the direction choices.  Apologies.

GCJ

Mail Time: B field of a straight wire, and waves before magnetism?

It was good to hear from Fed Duay, a two-year veteran of my AP Summer Institute at Manhattan College*

* Which, for the uninitiated, is in the Bronx.  No, I don't get it, either.

Fed said:

Hello!

I have two questions. We are doing the "B field of a straight wire lab", where we can use a compass aligned to the earth's magnetic field and trigonometry to find B's value; then we graph "B vs. 1/r" and use the slope to find the "vacuum permeability" value [or should we find the current as compare to the ammeter reading?]. However I have come across two values for the earth's field: 2x10^-5 T and 5x10^-5 T (or 20 microT and 50 microT respectively). What do you use for the earth's field value?

I actually do the experiment the other way -- I use the ammeter reading and mu naught to find the magnetic field.  I like either of the two ways you suggested.  That's one of the beautiful aspects of the graphical approach to laboratory... Depending on what you measure or what you look up, a single experiment can be done in a wide variety of ways.

As for the value of Bearth:  You're only finding the HORIZONTAL COMPONENT of the earth's magnetic field.  Along the east coast, the magnetic field points more down than north, at a "dip angle" that can be close to 70 degrees off of horiontal. 

The site http://www.ngdc.noaa.gov/geomag/magfield.shtml will tell you the local magnetic field, including all components. I find at Woodberry Forest the northward magnetic field component is 2.0 x 10^-5 T.

(Fed continues with his second question:)

Also, I see that last year you covered part of waves before finishing magnetism; I am guessing that you needed to do the "standing wave lab on a string" before finishing EM. Is this the reason or is there something else I should be aware of?

Nothing other than personal preference is in play here.  The intricacies of electromagnetic waves aren't included on the AP physics B exam.  Certainly students are expected to know the EM spectrum, the visible wavelengths, which colors of light have higher frequencies, and so on; but the fact that electric and magnetic fields oscillate in accordance with Maxwell's equations is irrelevant at the physics B level.  Standing waves are in no way a prerequisite to magnetism.

I tried sticking in the wave section before magnetism because that breaks up the toughest parts of the AP course. Electricity and magnetism kick my students' butts, especially coming in the dead of winter when they're busy and in bad moods, anyway. I put waves in between, because waves have some cool demonstrations, are easily vizualizable, and are (comparatively) easy .

(You got a question you want answered in Mail Time?  Either post a comment, or email me at greg_jacobs@woodberry.org.  Those who include an astute and witty criticism of the Cincinnati Bengals or Reds impending disastrous seasons are most likely to see their questions answered.)

Translucent Grading

Woodberry Forest is on a trimester system, which means that I just yesterday finished my grades and comments for the fourth marking period.  (Comments?  Yes, I write a short missive about every student, 5-6 times a year.)  The prize for finishing the week-long process of writing exams, giving exams, grading exams, calculating grades, writing comments, and entering grades and comments into the database is... two weeks of spring break.  And I am thankful.

Now is as good a time as any to discuss the meaning of the grades that I assign.  Yeah, sure, many schools  have some sort of official policy stating what percentage represents what grade.  Such policies are as fungible -- and as useful -- as a steaming pile of dirt in a cow pasture.  It is my job to funge students' grades so that the letter I assign communicates meaningfully to students, parents, and other interested parties (read: colleges and scholarship committees).

The overriding philosophy of my grades is TRANSLUCENCY.  No, no, not "transparency"... I don't want students to know their grades every day.  Regular posting of grades leads to slackage from the top students, hopeless near the bottom of the class.  And I don't want the grading process to be so much of a mystery that rumors begin to circulate of Jacobs throwing lab reports down stairs -- if assignment of grades is opaque, if standards seem vague, why should anyone in the class work hard? 

No, I truly mean "translucent."  I want a grading system that allows a student to calculate his grade with reasonable precision any time during the marking period.  I also want the system to be intricate enough to discourage students from actually carrying out that calculation.

I personally use a weighted average of labs (10%), quizzes (15%), homework (20%), and tests (50%) to determine an overall course grade.  Each component is converted to the official school scale via the "square root curve" -- take the square root of the percentage score and multiply by 10.  This makes an 81% into an A, 64% into a B, 49% a C, and so on.  I do all these calculations in a rather intensly complicated spreadsheet that I have developed over the years.  Students can figure out their course grade, too -- if they keep all their assignments and then use the correct weighting without messing up data entry into the TI-84.  You see?  Translucent.

I have a sense of what the final grades should mean.  In AP physics, my grade reflects a probability of earning a score on the official May exam, with A translating into a 5 and B- translating into a 3 or 4.  In the general physics course, an A student has learned physics to the AP standard, but has been asked to master three times fewer topics than an AP student.  In general physics, a C is the old "gentleman's C" -- the student worked hard, did what I asked, but couldn't quite put together more than a cursory understanding of basic problem solving.  (Or, a C student is someone capable of As if he could be bothered to study.)

Now, I don't include in my calculations any sort of official "fudge factor" like a class participation grade or the like.  There's nothing inherently wrong with evaluating class participation, I just don't feel like subjecting my students to such a subjective evaluation, nor do I feel like dealing with the fallout from students who aren't given perfect class participation scores.  So, what do I do when my calculated grades don't seem to exactly match student performance?

I do my fudging by dropping some number of homework or quiz scores each marking period.  I look at each student's calculated grade.  I never merely change one person's grade -- bumping individuals up from, say, a B+ to an A- actually got me in considerable trouble my first years teaching.  So, I look at the overall grades.  If they seem generally low, I'll drop one more quiz.  If they seem high, I'll drop one fewer homework.  This way, I can't be accused of playing favorites, since the entire class gets bumped slightly up or down.  But, I can determine what happens to students right on the border between grade cutoffs.

Grades are necessary because they motivate students and because they communicate about performance.  Grades must be assigned fairly, and must be based as much as possible on objective criteria.  Nevertheless, I don't ever want a student focused on his grade to the detriment of the development of his physics skills.  That's why I make my grading translucent.

General Physics exam review, a "clicker" exercise, and trying out SCRIBD

In addition to AP physics, I teach a general high school physics course for juniors and seniors.  This course is intended to be accessable to all college bound students, even those not gifted in or particularly interested in math and science.  In fact, a significant number of students who didn't take our freshman course are required to take and pass this General Physics course.

Now, everyone has a different perspective and philosophy about general high school physics.  My approach -- which isn't necessarily the RIGHT approach, but has been successful for me -- is to teach a limited number of topics nearly to the depth demanded by the AP physics B exam.  By March 1, we have covered most of physics B mechanics.  (How come I claim to call this "general" and not "AP" physics?  Because my AP class covered this same set of topics by November 1.)

My trimester exam, coming up on Thursday, is 100 points worth of AP-inspired free response problems.  Some are directly off of old AP exams.  Others are modified slightly, but still maintain the spirit of AP problem solving.  Students of all abilities usually do quite well on this exam.  I always say that if this class could take an AP exam limited only to the topics that we cover, they would be getting mainly 5s, with no one below a 3.

If I am going to give such a comprehensive and rigorous exam, I must prepare my students for it the same way I prepare my AP class for their national exam.  Of course, most of this preparation comes via the coursework we've done all year.  But I spend the last week before the test in review mode.

The primary review method I've used this week involves a multiple step process:  a homework problem, sometimes a review quiz over that problem, and an in-class "clicker" exercise.  The "clickers," or "classroom response system," allow me to present a series of questions as an extra credit competition.  Class members are divided randomly into groups of two.  Each group submits a single answer to each question.  They earn one point for a correct answer, plus another point for every group who gets the answer wrong.  This scoring system encourages collaboration with the groups, while discouraging reliance on one or two students who might carry the whole class along.

I plan on posting a couple of these clicker exercises over the next week.  Thing is, it is often difficult to post quizzes or class handouts on this site.  Equations and diagrams don't copy well into google's "blogger."  But I think I've found a place where I can very simply upload my word documents, and where you can easily download them as well.  Check out "scribd.com".  This site looks to me like the text equivalent of flickr.  I was asked to create a username and password -- I'm "gregcjacobs," though I won't tell you my password.  The site is free, and I haven't received any spam from them yet.

I've posted two documents on scribd that I'd like people to try to download, just to see if this works. 

The first is a homework assignment:  it's an elevator problem, based very loosely on 2005 AP physics form B problem 1.  (This problem could also be given as a 10 minute quiz, or as a test problem.)  Click on the link, and see if it works.

The second document is a multiple choice exercise for the clickers that is based on the elevator problem. 

Please post a comment, firstly saying whether you could access these files an a format that you can use; also, feel free to comment on the utility of the exercises, or any thoughts related to this general exam review strategy.

GCJ