How to justify an answer: Equations, Calculations, or Facts

One mole of an ideal gas expands from a volume of 5 L to a volume of 10 L at atmospheric pressure.  Does the gas's temperature increase, decrease, or remain the the same?  Justify your answer.

Justifications are tough for novice physics students, especially students with weak verbal skills.  Look at some common unacceptable responses to the question above:

* Increase, because the volume increased at constant pressure.  [This answer is a tautology -- it simply restates information given in the problem.]

* Increase, because the temperature had to go up to increase the volume.  [Still a tautology.  "Had to" doesn't add any physics understanding.]

If we're not vigilant about scanning justifications for tautologies, we will get them all the time.  Our students do not have the same ability to reason logically as we do.*  But rather than merely kvetch about these dang kids who couldn't justify their own existence, the onus is on us to teach the skill of justifying an answer in physics.

*One student's impeccable logic:  "One time I saw some unexplained lights at night.  I called the local airport control tower, who confirmed that they knew of no aircraft operating in my area.  Therefore, I was visited by a space alien."  And no, sorry, I am NOT kidding.

My students seem to respond well to my demand for one of three possible elements to make a justification legit.  They must include either equations, calculations, or facts of physics.  In some problems, only one of these will be of any use; in the question I posed above, any one of them might be useful.  For example, some reasonable justifications:

(1) Using equations:  "Solving the ideal gas law for temperature, T = PV / nR .  Here n is constant because it's a sealed container, and pressure is constant because the problem stated atmospheric pressure the whole way.  The only variable is volume, which is in the numerator.  So, when volume increases, temperature must increase as well."

(2) Using calculations:  "Use the ideal gas law, PV = nRT.  In the initial state, plug in values:

(10⁵ Pa)(0.005 m³) = (1 mol)(8 J/mol K)(T), so T = 62 K initially

Now in the final state:

(10⁵ Pa)(0.010 m³) = (1 mol)(8 J/mol K)(T), so T = 125 K finally

Thus, the temperature has increased.

(3) Using facts of physics:  "An isobaric process looks like a horizontal line on a PV diagram.  Isotherms are hyperbolas assymptotic to the axes on a PV diagram.  So, a horizontal line with increasing volume must jump to an isotherm that is farther from the origin, and thus representing a higher temperature.

Once I've demanded these elements of justification enough times, the class gets the idea.  And when someone is lazy, forgetful, or simply wrong, I don't have to argue about the legitimacy of his answer.  I just ask, "Did you use equations, calculations, or facts?"  If the answer is "no," the student usually hangs his head in shame without any further prompting.

Common misconceptions -- parallel resistors

Parallel resistors each take the same voltage, which is equal to the total.

Now ask a student: "Two 100 ohm resistors are connected in parallel to a 12 V battery.  Determine the voltage across one of the two resistors."  What does the student say?

Generally, that student reasons, "Parallel resistors take the same voltage.  The battery provides 12 V to two resistors equally, so that's 6 V across each."  D'oh.

How do I attempt to remedy this misconception?  Give everyone the chance to predict and then MEASURE the voltage across several resistors, as in this laboratory exercise.  When a student comes to my desk for me to sign off on his correct measurement, I throw the common misconception in his face.  I say, "Hey, Will, that doesn't make sense.  Seems to me, you've got two parallel resistors here, you should only get half the battery's voltage across each."  Will generally has two points to his rebuttal:  (1) "That's not the correct rule, Mr. Jacobs, the voltage across parallel resistors is equal to the total." And, most importantly, (2) "That's not what I measured.  I get the same voltage across everything."

Just The Facts: circuits

Been busy busy here, preparing for the USIYPT and writing 6000 words worth of 3rd marking period comments.  Colleagues and readers have asked a few times, "what do I need to teach about foo?" where foo is some sort of physics topic in AP, honors, general, or conceptual physics.  I've wanted to be able to QUICKLY point to a text or handout that gives a good, clear answer.

Problem is, textbooks tend to cover way too much -- the whole point of the textbook is to be comprehensive so the professor can choose what portions of each topic to teach.  The College Board's or the Regents Exam's course descriptions either give too little detail by saying merely "teach circuits", or they go into impenetrable edujargon ("The student should be able to compute the equivalent resistance of a set of series resistors, parallel resistors, or combinations thereof including but not limited to up to 5 resistors connected or not connected to a DC power source, unless there are more than two men on base.")

My colleague Curtis has asked me to write some blog entries giving "just the facts" that I feel are appropriate for an honors or AP introductory physics course.  Without knowing it, I had given him a useful blueprint for teaching conservation of momentum in collisions with this post.  I gave more detail than he wanted in this fluids lesson plan I wrote for the College Board, but he still found the article a useful basis for answering the question "what do I need to cover in static fluids?"

So today, I will list briefly Just The Facts that I teach when covering circuits in my honors physics course.  When I cover this or any unit, I usually start by writing key facts and equations on the board, then quizzing the students the next days to make them remember those facts and equations.  Over the course of a week or two, the class develops their ability to reason with the equations and facts, and to solve problems in the topic area.

In circuits, we need to know the following:

Definitions:

* V represents voltage, measured in volts (V).  Voltage is measured across a circuit component using a voltmeter.

* I represents current, measured in amperes (A).  Current flows thru a circuit component, and is measured with an ammeter.

* R represents resistance, measured in ohms (Ω).

* The symbols for batteries, resistors, ammeters, and ohmmeters

Ohm's Law

Ohm's law relates voltage, current, and resistance by V = IR.  Ohm's law may only be used if the current I and the voltage V are experienced by the resistance R.  (That means we can't randomly pick a voltage and a resistance from the problem and divide to get current; we must be sure the voltage we plugged in is actually measured across the resistance we're considering.)

Power

Power is given by IV, subject to the same usage restrictions as ohm's law.

Series Resistors:

* The current through each is the same, and equal to the total current

* The voltage across each is different, and adds to the total voltage

* The equivalent resistance is given by straight addition of individual resistances.

Parallel Resistors:

* The current through each is different, and adds to the total

* The voltage across each is the same, and equal to the total

* The equivalent resistance is given by an inverse formula, 1/R = 1/R + 1/R

Solving DC circuits up to 4 or 5 resistors with a single battery

* Simplification of basic resistance networks

* Calculating current, voltage, resistance, and power

* qualitative questions: "when another resistance is added in series, what happens to the current from the battery?"

Light Bulbs and circuits in a laboratory setting

* A light bulb's brightness is determined by the power it dissipates.

* A bulb has a known resistance which doesn't change no matter what the bulb is hooked to.

* An ammeter is connected in series with a circuit element

* A voltmeter is connected in parallel with a circuit element

{The following are also required on the AP exam}

Kirchoff's rules

* Current entering a junction equals current leaving a junction, a statement of charge conservation

* Voltage changes around a closed loop equal zero, a statement of energy conservation
* Deal with multiple battery circuits, but NOT with three-equation three-variable problems

Resistance of a wire based on the wire's properties

* R = ρL/A

Please communicate with me

Is this sort of post useful?  I mean, this is everything I teach in circuits.  Sure, I haven't discussed the exact methods by which I teach how to solve for current through the 100 ohm resistor, or the lab activities that I do.  I actually discuss many of those things elsewhere on the blog; please search.  And you may agree or disagree with what I consider fundamental for a first-year physics course.  Conceptual physics would probably eliminate the calculational aspects; AP Physics C needs to include RC, RL, and RLC circuits.  My question is, is this list of facts and skills a useful starting point for you to figure out your own list of facts and skills for your students to understand?  Please comment.

GCJ

A quantitative demonstration with springs... one that DOESN'T work!

All year long, the majority of my in-class time is spent doing quantitative demonstrations, in which the solution to an example problem is verified live in class via measurement.  Each demonstration seems to end the same way: I perform the measurement, we all see that the measurement matches the prediction within 10% or so... everyone exhales, and I say "Physics Works."

It's not a problem, per se, but by November or so the novelty of my approach has worn off.  Instead of bright shiny faces anticipating whether or not the experiment will match the prediction, I start to see resignation:  "Yeah, physics works, I know, we've done this a million times."  I've got to throw a changeup.

Once we've introduced the force and potential energy of a spring, I set up what looks like a routine quantitative demonstration.  I hang a mass from a spring, as shown in the diagram.  I determine the spring constant via the 5-second method that I discuss in this post.  This particular spring (the one on the left in the picture) has a spring constant of about 7 N/m.

Next, I hang 100 g of mass from the spring, and allow the spring to come to rest.  Using F=kx, I derive that the mass should stretch the spring by 14 cm; sure enough, a ruler placed alongside the spring shows 14 cm of stretch.  Physics works.

Now, I use the ruler to measure an additional 5 cm of stretch beyond the equilibrium position.  The question: After I release the mass from rest, what will its speed be when it passes through the equilibrium position again?

[Note to readers before you flame me about these next couple of paragraphs:  Keep reading.  Trust me.]

I use the force on a spring equation, F=kx, to calculate the force on the mass to be (7 N/m)(0.05 m) = 0.35 N.  Then, Fnet = ma, so the acceleration of the mass is (0.35 N)/(0.10 kg) = 3.5 m/s².

I take a brief interlude, if someone asks, to (correctly) explain that with a vertical spring, it's completely correct and most simple to treat it as if it were a horizontal spring, but with the x = 0 position at the place where the mass would hang at rest.  In other words, it is acceptable to consider that the spring force kx is the net force, as long as we define x = 0 at the resting position.


Now we have enough information to make a kinematics chart.  The mass starts from rest, its acceleration is 3.5 m/s² and it will move a distance of 5 cm.  Using the kinematics equation v² = v_o² + 2ax, the speed at the equilibrium position is 0.59 m/s, or 59 cm/s.

"How do we verify this prediction?" I ask the class.  They quickly and accurately tell me to place a motion detector below the mass, and to look at the velocity-time graph.  I do so, I read the v-t graph, and I see that it says about 40 cm/s.  Physics wor------  oops.  Physics DOESN'T work.


All year, I've conditioned the class to expect measurements accurate to 10 or 15%.  I'm not even getting within 30% this time... that's not a matter of "experimental error," especially when I try again and get the same result; or even more especially when I try it with the other spring, and I'm still off by more than 30%.  So, why didn't physics work?

Despite the desperate grasping at straws from some students, someone usually comes up with the right answer:  I used kinematics when acceleration was not constant.  The force of the spring was only 0.35 N immediately after release.  As the spring compressed, the spring force got smaller by F = kx.  So the acceleration got smaller.  So the mass didn't speed up as rapidly, and ended up going less than 59 cm/s.

How do we solve this problem correctly?  Using conservation of energy, of course.  Spring potential energy is converted to kinetic energy, 1/2 kx² = 1/2 mv².  Solving, I get v = 41 cm/s.  Now, physics works.

Side note:  Sure, occasionally a student will try to stop me from using kinematics with changing acceleration.  My response is a quick "shhh!!!," Dr. Evil Style, but with a smile.

GCJ

Collisions -- how much detail?

In a typical college physics textbook, the end-of-chapter problems about collisions begin with the simple and move on to the unreasonably complex.  I'm frequently asked by AP teachers:  How far do I go before complexity becomes "unreasonable?"

In judging the depth necessary for this or any topic, first recognize the motivation of the textbook authors.  They're not making a considered, pedagogically sound choice about what material is important, or even about the best way to present said material.  No, they quite reasonably want the largest audience possible.  The publisher is far more likely to hear "I didn't choose your book because I like to derive the formulas for inelastic collisions in two dimensions with a coefficient of restitution e, which isn't covered" than "I rejected your book because it had too much information."  Thus, we get 10³ page* tomes that touch on every possible aspect of "introductory" physics.

* And 10²dollar

Don't ever use the textbook as a sole guide to what's important.  Of course that begs the question:  how do you figure out what's important when teaching an AP or college-prep high school course?  The simple answer is to look at AP exams since about 1996* for guidance.  Summarize to yourself the kinds of questions that are asked in each format (multiple choice or free response).  Then, certainly if you're teaching AP, be sure to cover the types of questions that showed up; and ignore anything else, even if it's in the textbook.

*1996 approximately marks the transition on AP Physics B exams between the "Shut up and Calculate" era to the "Justify your answer" era.  Sorta like when major league baseball lowered the pitcher's mound in 1969.

But what about those who DON'T teach AP, or who aren't particularly fond of the College Board's curriculum or their funny little ways?  What if we were to start from first principles, and decide philosophically how much detail SHOULD be included in an advanced high school physics course?  Fair question.

I want to cover the basics both conceptually and calculationally.  But we don't want to perform any calculations that are so complicated that the mathematical methods outweigh the physics approach.  In an algebra based course, I assume fluency in basic algebra, and in using sin, cos, tan.  A multi-variable system that cannot reasonably be solved in a couple of minutes is out of bounds; similarly with any trig identity beyond sin/cos = tan.  This limit on calculation is born of philosophy -- I never want to do math for math's sake, or I would have gone into teaching math -- and of practicality as well -- if I put a detailed calculation on a test, I can't ask more than one or two questions in a 45 minute period.  (Plus I'd be testing math ability, not physics ability.)  So here's what I teach about collisions:

1. The fundamental meaning of "conservation" of momentum.  Everyone's got to understand the idea of an unchanging quantity; but also how an individual object's momentum can change without violating conservation.  We would test this understanding with conceptual questions like "A ball bounces off a wall.  Did its momentum change?  How is that consistent with conservation of momentum?"

2. Basic computations with momentum conservation in one dimension.  A no brainer.  "A cart moving 30 cm/s collides with and sticks to an identical resting cart.  What is the speed of the carts after collision?"  Carts can bounce, stick, be moving in any direction.

3. Definition of "elastic" collision.  Although some books and teachers split hairs over the precise definition of elastic, most define an "elastic collision" as one in which kinetic energy is conserved.  While momentum is conserved in all collisions, kinetic energy is only conserved in elastic collisions.  Colliding objects may not stick together in an elastic collision, though a collision is not necessarily proved to be elastic just because objects bounce off one another.

4. Calculation to determine whether a collision was or was not elastic.  Note that this is NOT the two-equation, two-variable calculation to predict speeds of two objects after an elastic collision.  No, all I'm suggesting here is that we teach students how to add up the total KE of all objects before a collision, add up total KE of all objects after a collision, and compare.  The question generally takes the form "Was the collision elastic?  Justify your answer."

5. The vector nature of momentum, the scalar nature of kinetic energy.  "Two identical carts move toward each other at the same speed, stick together, and remain at rest.  Does this violate conservation of momentum?  Does this violate conservation of KE?"  Everyone has to recognize that momentum in opposite directions can "cancel out," but that the phrase "kinetic energy in opposite directions" is silly.

6. Ability to consider horizontal and vertical momentum separately in a 2-d collision.  Once again, I would not ask for anything that required multi-variable system analysis.  But we can arrange problems such that the horizontal conservation of momentum is simple to solve; and where vertical momentum was zero before the collision, so must be zero (in sum) after collision.  Usually, such questions will be limited in scope to very simple calculations, or to conceptual questions:  "Calculate the initial vertical momentum of the system before collision.  What is the system's vertical momentum after collision?"  Or, "Is magnitude of the red ball's vertical momentum greater than, less than, or equal to the magnitude of the green ball's vertical momentum?  Explain."

That's about it.  No coefficients of restitution.  No proof of why 2-d elastic collisions always produce final velocity vectors at a 90 degree angle to one another.  All of the types of questions above can be phrased in a straightforward manner, allowing for answers in a couple of minutes.  The list of six ideas allows for both conceptual and calculational questions.  Good.

Equal length spring set

I hate doing unpaid shilling for deep-pocketed companies.  However, when PASCO provides exactly the right tool for teaching a physics concept, as they so often do, I can do nothing but spread the good word.

Take a look at the three springs you see in the picture to the right.  All are clearly equal in length and diameter.  They seem identical in every respect except color.

But when I hang a 500 g mass from each spring, the springs stretch different amounts.  The spring constants can be ranked, calculated, used for prediction of extension under a new load, etc.  What a great lab tool.

PASCO sells a five-spring set via the link here.  The red, blue, and yellow springs have a book value spring constant of 25 N/m, 30 N/m, and 35 N/m, respectively.  In my picture, the yellow spring is slightly less stretched than the blue one, but only slightly.  Well, that's okay, because (a) the yellow spring is supposed to have a larger spring constant than the blue spring, and (b) PASCO only lists the accuracy of each spring constant as +/- 5%.  It could well be that the blue spring is at the top edge of the 5% tolerance, or about 1.5 N/m too high; the yellow spring might be at the bottom edge of its tolerance, putting the stretch close to, but not quite, identical for each.

Now, for years I've used cheap-o springs that I bought in bulk from random sites online.  But these springs get stretched past their elastic limit, intertwined with other springs, and generally destroyed very quickly.  PASCO's set of five springs comes in a storage box, and these springs are tough to tangle.  Plus the color codes allow you to determine very quickly how carefully a lab group has done an experiment.  I'd suggest getting at least a demonstration set of these, if not a class set, unless your budget is lilliputian.

I always argue that physics equipment should be bought over many years, a few pieces at a time.  That's not the reality of most schools' budgeting processes, which generally give more money than you know what to do with all at once, but then nearly nothing for years on end.  That's simply not practical.  You want to be able to buy new toys when you find out about them; don't buy too much at once, spread your purchases out over the years.  

Well, my colleague Curtis bought a set of these, and now I'm jealous.  Good thing we ordered another set for me for next year...  Thanks, Curtis, and thanks, PASCO.

GCJ

Mail Time: Approaching energy concepts in junior-level general physics

Erik Born, a colleague here at Woodberry, teaches our junior-level general physics course.  We aim this course essentially at the New York Regents exam, with a few tweaks of topics (e.g. we teach an astronomy unit, and we don't teach electric or magnetic fields).  Erik wrote to me the other night:

I was wondering if you can give me a short overview of how you approach energy concepts. I am used to starting with work, and moving from that into energy, but I'm following the order of your problem sets, which has started out with introducing KE and PE first. I was wondering if you could give me some advice about conversions between KE and PE without talking about work or total mechanical energy. Maybe you have a specific blog post on it?

Well, I do now, I suppose :-)

At the AP / Honors level, Erik knows that I introduce the definition of work, and then my own version of the work-energy theorem:  W_NC = (KE_B – KE_A) + (PE_B – PE_A). You can see my reasoning for why and how I use this formulation at this post.  

But at the general level, I approach energy concepts very differently.  I ask the class to state and write out all energy conversions IN WORDS.  

For example, a ball drops to the floor -- how fast is it going?  They must start by writing, in longhand, (gravitational potential energy) converts to (kinetic energy).

Or, a waterskier takes off from water level at 14 m/s, and ends up going 13 m/s at his max height.  They must write: (kinetic energy) converts to (gravitational potential energy) and (kinetic energy).

If a block is moving fast and stops on a rough surface, they must write (kinetic energy) converts to (thermal energy).  If you want them to call this (kinetic energy) converts to (work done by friction) that's fine, too.

We haul up a block by a rope:  (Work done by the rope) is converted to (potential energy).

In each case, once they write the words, then they translate the words into an equation:  KE = 1/2 mv² , PE = mgh, spring PE = 1/2 kx², and work = Fd.  Then they solve for whatever needs solving.

Erik continues:  How do you deal with positive and negative work?

In regular physics, I *don't*.  If we're writing the energy conversions in words, like I described, then we don't have to worry about positive and negative work.  The sign of the work comes out in the wash of the equation written from the description.  Examples:

Block is moving, comes to rest on a rough surface.  (kinetic energy) converts to (work done by friction). *   1/2 mv²  = (Ff)d.  Yes, I know that kinetic friction does negative work, but that's not necessary in this formulation.  The equation and statement in words provide both conceptual understanding AND a path to solution for any relevant value, with no negative signs involved.

* I let them know that a source of "thermal energy" is "work done by friction."  The distinction isn't important at this level; however, recognizing that the relevant equation for thermal energy here is (force of friction) x distance *is* important.

Block is pulled up by a rope.  (Work done by the rope) converts to (gravitational potential energy).  (tension)d = mgh.  Once again, it's not important that work done by the rope in this case is positive:  The sign comes out in the wash.

As a comparison for a teacher (not for the students), many textbooks teach a similar approach to calorimetry:  don't use positive and negative signs, write out the heat gained or lost in words.  For example, write (heat lost by hot iron) = (heat to raise liquid water to 100 degrees) plus (heat necessary to convert water to steam).   Then mcDT_iron = mcDT_water + ml_steam.

Hanging Mass on an Incline lab

Mr. Johnson shows off his hanging mass setup

In my general physics labs, we follow a familiar formula each week.:

(1) Students collect data, constructing a linear graph as they go.

(2) They take the slope of a best-fit line using far-separated points on the line that are not data points, including units on the slope.

(3) They use a relevant equation to relate the physical meaning of this slope to a measurable physical quantity.

That's it.  I keep things as simple as possible so as to work on just these three skills.  The trick is, of course, finding straightforward yet interesting experiments which lend themselves to this approach.  Oh, and these experiments must stick within the topic areas in general- (approximately Regents-) level physics.

The procedure is essentially the same each lab day.  I demonstrate a method of data collection.  I write on the board the graph that students are to make.  Students collect and graph their own data in groups of two.  Once I've approved a group's graph, then I hand them a two-page homework assignment for them to work on either the rest of the lab period, or that night if they need more time -- no one leaves early.

Ideally, I'm designing an experiment such that I know what the value of each group's slope should be, but the students do not.  This is a bastardized version of the "recurrent lab" as described by Mikhail Agrest.  Students earn credit either for predicting something using their slope, or for matching their slope to an independent measurement.

So, Greg, give us an example of such a laboratory activity.  Sure -- see the picture.  I have students set up an inclined track, on which they place a Pasco cart.  The cart is connected over a pulley to a hanging mass.    The hanging mass is adjusted until the system hangs in equilibrium.  The angle of the incline is what they are eventually going to predict; I surreptitiously come around during the period to measure each incline with my iPad clinometer app.

We graph the mass of the hanging stuff on the vertical, with the mass of the Pasco cart on the horizontal.  Some mass is added to the Pasco cart, the hanging mass is adjusted to equilibrium, and another data point goes on the graph.  Rinse and repeat for an easily obtained straight-line graph in about 30-60 minutes (including setup and cleanup).

The homework assignment based on this lab activity is available here.  In sum, students take the slope, and then are guided to identify the slope as the sine of the incline's angle.  Most groups easily match my measurement.  

This is one of my better lab exercises, because (a) it fits the formula we've been using all year with no deviation, (b) it allows for accurate prediction of a measurable but initially unknown quantity, and (c) it reinforces the content in the problem solving portion of the course.  Try it... post comments or questions.

GCJ