What does "r" mean in equations for gravitational force and centripetal force?

On one hand, that's easy for me to answer: 

In the gravitational force equation F = GMm/r,² the r represents the distance between the centers of the two objects.  I often use the variable d to emphasize this meaning.

In the centripetal force equation F = mv²/r, the r represents the radius of the circular motion.

These facts are also easy for students to read and recall.  It's not as easy for students to put into practice.  They see the letter r, hear "radius," and plug in any random distance they can pluck out of the problem stem.

And, this <pop> pull-a-radius-value-out-of-their-tuckus method is very often successful in a gravitation problem.  When an object is on the surface of a planet, the r value is in fact the planet's radius.  When a satellite undergoes a circular orbit around a central planet, the orbital radius r is in fact the same as the r distance between the satellite and the planet's center.

So why does it matter if students truly understand the difference between these two meanings of r?  In what possible physical situation in introductory physics does this difference even matter?  Here's one.

Two stars, each of equal mass M, maintain a constant distance x apart and rotate about a point midway between them at a rate of one revolution in every time t₁.

(a) Why don’t the two stars crash into one another due to the gravitational force between them?

(b) Derive an expression for the mass of one star.  Use given variables and fundamental constants only.  You must annotate your calculation – if your response has no words, you will redo it from scratch in consultation.

This is a difficult question for students to conceptualize, especially because while we've done plenty of straightforward orbit problems, students have very often remembered comforting algorithms and not necessarily internalized physical meaning.  And I won't answer questions from students before they turn it in.  (They can discuss and argue with each other as much as they want!)  In class the day this problem is due, I don't start by "going over" how to solve it.  And on that model, I won't simply go into my solution here.  Instead, I'll show you the quiz with which I begin class:

For #1, r in this equation represents the distance between the centers of the two planets.  The problem says explicitly - the stars are always 8.0 x 10¹⁰ m apart.  (Most common misconception:  because in every problem they've previously done a satellite orbits around a central planet, they think that the "point midway between" the stars is the location of a planet of some sort - or that this "midpoint" is what exerts the force on the star.)

For #2, r in this equation represents the radius of the orbit.  The problem says explicitly that the stars "rotate about a point midway between them".  Since the stars are 8.0 x 10¹⁰ m apart, the point midway between them is half that distance from one of the stars, or 4.0 x 10¹⁰ m.  (It's not correct that one star is fixed with the other orbiting around the fixed star.  That's not what the problem says, nor is it how binary stars behave.  And half of 8.0 x 10¹⁰ m doesn't mean divide the exponent by two: half of 80 billion meters is certainly not 4 hundred thousand meters!)

For #3, students know to use this equation when they know the period of an object's circular motion.  It comes from the fact that the orbital speed is constant - for constant speed, speed is distance/time.  The relevant time here is the period, the time for one orbit.  The corresponding distance, then, is the circumference of the circular orbit.  From geometry class, that circumference is 2πr, where r is the radius of the circle.  We want the same distance as in number 2, the 4.0 x 10¹⁰ m radius of the orbit.

For #4, the distractors are practically word-for-word from past student responses.  As I discuss the quiz, I pick an incorrect answer to explain why it's incorrect:

For (A), first we discuss and agree that the gravitational forces on each star are indeed a Newton's Third Law force pair.  Then I go to a student's desk, hand the student a string, and pull.*  Is the force of me on Mr. Chamberlain equal to the force of Mr. Chamberlain on me?  So is that a N3L force pair?  Yes.  Are we orbiting around each other in a circle?  No.  So the logic of choice (A) is not logic at all.

*In the Before Times, I'd clasp hands with a student and pull lightly.

Next I ask students to close their eyes.  All who chose this incorrect answer, raise your hands high.  Now put your hands back down, and open your eyes.  The point is for students to acknowledge their misconceptions. It's okay that they made the mistake - after all, in my class everyone gets an A- until the AP exam, quizzes are given and graded but don't "count".  I don't want students to feel shame for being wrong.  But more importantly, I don't want them to sour-grapes style convince themselves that they knew that and that they didn't really make a mistake.  No.  Own the misconceptions, then don't make them any more!

For (B), first we discuss and agree that the gravitational forces on each star are indeed a Newton's Third Law force pair.  I again go to a student's desk, hand the student a string, and pull. The forces on each of us are equal. Are we orbiting each other?  No?  Then (B) is wrong.

For (D), I ask a student to point to the midpoint between us.  Only objects can exert forces... so what object at the "midpoint" can exert a force?  No object.  So (D) is wrong.

And finally, for the correct choice (C), I draw a picture of the two stars orbiting.  I draw the direction of one star's instantaneous velocity, which is tangent to the orbit.  I ask about the direction of the net force on that star, which is toward the center.  Everyone sees that the velocity is indeed perpendicular to the net force.  And we discuss how that's a restatement of one of our circular motion facts: when an object moves at constant speed in a circle, its acceleration is toward the circle's center.  The velocity will always be tangent to the circle, which by geometry is perpendicular to the direction toward the center.

Link to "AP Live" video playlist from Spring 2020

Folks, from March through May, I presented live, 45-minute shows preparing students for the weird 2020 AP Physics 1 exam.  Although the exam-specific information is relevant ONLY to the 2020 exam - the 2021 exams will be in standard format and include all topics - the content and general exam preparation advice is still useful and entertaining.

I keep fielding questions about whether those shows are still available.  They are!  They're sorta buried on the College Board youtube page, because they don't want this year's students to think that the 2020 exam information applies to future years.  But the videos are still available, because teachers and students do want to access them.  

My Physics C - mechanics independent study students have found these particularly useful, because even though they focus on Physics 1, the concepts in Physics 1 are required prerequisites for understanding the more mathematical Physics C- mechanics problems.  

Here is the link to the playlist!

Josh Beck - who is an awesome physics teacher and AP reader - presented about half of the shows, and I did the other half.  My pet hippopotamus Edna and her friends only show up in my episodes.  Enjoy!

Mail Time: Linearizing a parabolic distance vs. time graph

An APSI participant writes:

I had a question regarding linearization.  We just did the free fall lab and a student graphed their data to get the parabolic curve of d vs t.  When asked what they would do to linearize, they stated to convert d to v and graph v vs. t.  

When they determined v, they did d/t for each point.  Then plotted that value for Vavg vs time.  Is this an acceptable way of linearizing?  

My first instinct is no, because the slope of the v-t graph, while still the acceleration, does not produce the value as it would if you did d vs t^2.  

The slope of their v-t graph was 5.15 m/s/s.  If they would have plotted d vs t^2, they would have a slope of 5.45 m/s/s with an acceleration of 10.9 m/s/s for g.

This is a common approach by early-in-the-year AP Physics students.  They did not do this right... 

By dividing just d/t for each point, they took the average velocity from the beginning - that's not how a velocity-time graph is made, and that quantity is rather meaningless in the experiment you describe.  Basically never divide the values of two data points!  :-)

What WOULD be a reasonable alternative approach to this experiment would be to make a legit velocity-time graph: by taking the slope of a tangent line to each point on the d vs. t curve.  Then plot those *instantaneous* speeds as a function of time.  That's a true velocity-time graph, for which the slope is acceleration.

Of course, that sounds way, way more complicated than just using lab linearization approaches of writing the relevant equation, then plotting data as it appears in the equation, d and t^2.  

Hope that helps!

Notation for Newton's Second Law: F, Fnet, Sigma-F. And Julius Sumner Miller.

Alex writes in:

I was listening to Julius Sumner Miller lecture on YouTube while doing some work last night and noticed that he said F=ma. 

So my question is using the term net force something that teachers just use? Was it something that wasn’t used then but is common vocabulary now?

Glad you reminded me about Julius Sumner Miller (which is still what my brain wants to call my colleague Julius Reynolds, because until him I'd never actually known a Julius other than Orange Julius).  I should watch some of those videos with my AP class when we're online in December!  Make sure you watch the one where he puts random stuff in liquid nitrogen, and forgets some things in the carafe.  JSM is a big, big influence on my teaching style, and I'll bet you can tell.  He didn't have a pet hippopotamus named Edna, though, more's the pity.

That's an interesting history question, Alex... there are people who research the history of physics education.  I wonder... I copied Gardner Friedlander, one of my go-to folks for AP Physics history, in case he has something to add.  (He did - see below!)

Before we start - I am not discussing whether your personal version of Newton's Second Law is the best one.  Of course it is, and all others are inferior.  I will not publish any comments that talk about why one notation is best.  I'm discussing the various notations, their history, their pros and cons.  If you use sigma-F but I don't, that doesn't mean you're better than me or vice-versa.  Really. We all have our reasons for our notation.

I do seem to remember that when I grew up - and I watched JSM in high school physics class all the time, on VHS - the phrase in common use was just "F=ma."  I looked back at the earliest AP solution set I can find, and it says sigma-F = ma.  (There was no equation sheet back then, before the days of graphing calculators.)  The 1983 New York Regents exam says "a=F/m."  I'd love some evidence that my recollection of "F=ma" being the common statement is correct... certainly I think that was the non-physics-class zeitgeist meme that people would say.

When I started teaching in 1996, I always used the notation "Fnet," emphasizing that only the NET force could be set equal to ma.  See, I discovered quickly the common issue that students would pluck a force value from the problem statement, pluck a mass, and smash them together into F=ma to get an acceleration.  I didn't like the sigma-F notation because students tended to add force numbers without reference to direction when they saw that.*

*or they would cluelessly wonder "what's that big ol' squiggly E, I dunno, it's got an F next to it, I guess."

In the AP physics 1 revolution, the curriculum design committee (led by some top rate physics educators) decided to rewrite as a=sigma-F/m, because this emphasizes that acceleration is usually the quantity that is the result of the various forces.  I had some discussions with a big group about whether and why that made sense or didn't make sense, and how that change should be explained.  On the equation sheet now, it reads a=sigma-F/m = Fnet/m.  I don't know, but I suspect, that this dual notation is to accommodate the two dueling camps of physics teachers: those who prefer students to write using vector notation "Fn + (-mg) = ma" because down is the negative direction, getting a negative acceleration if the acceleration is downward; and those who prefer students to not use negative signs but just magnitudes of forces, writing either "Fn-mg=ma" or "mg-Fn=ma" starting with the direction of acceleration.  The former is sigma-F=ma, the latter is Fnet=ma.

Gardner Friedlander says: He quoted several sources from the last 50 years that use all three versions of what should be set equal to ma: F, Fnet, sigma-F.  

His thought - which I agree with, now - is that the difference is probably not so much a historical trend, but rather the intended audience.  Julius Sumner Miller and Paul Hewitt were aiming at a general, non-mathematical audience trying to understand physical concepts, so used just F.  Mathematically based courses preparing students for continuation in the physical sciences used sigma-F.  And those aiming for an in-between audience, like AP Physics B, AP Physics 1, high school honors courses - they tended to go for the in-between notation of Fnet.

And in a final twist... I've started changing my notation to "netF = ma".  I've noticed for years that Fnet is confused with Fn for normal force; and that it's a misconception that "Fnet" is a separate force like the friction force or a tension that should be on a free body diagram.  By using the language of unbalanced and balanced forces (as the Physics Classroom does), the net force is just the unbalanced force; so netF emphasizes something different from Fnet.

Are Kepler's Laws Covered in AP Physics 1?

Kepler's laws are never a starting point for an AP Physics 1 question.  

That said... situations involving gravitation and planetary orbits are part of Physics 1.  AP Physics 1 does include circular orbits, from which one could in principle do the work that derives Kepler's T^2-r^3 law.  AP Physics 1 does include angular momentum conservation, from which one can understand the same consequences as Kepler's equal areas/equal times law.  There's no need to understand elliptical orbits, just circular orbits.  

Physics C includes some questions where the best approach is to cite one of Kepler's laws and logic from there.  Physics 1 does not - the approach to every planet problem starts with Newton's law of gravitation or conservation of angular momentum.

The physics behind a football spiral

I was forwarded this article (wsj subscription required) in which author Jason Gay issues a warning: SPORTSWRITER DOING PHYSICS!  Well, I've done a lot of sportswriting and I've done a lot of physics.  I give Gay enormous credit, because his physics explanation was crystal clear to me.

Gay explains a recent paper in the American Journal of Physics, in which the authors show that gravity by itself is not sufficient to cause a football to spiral in the direction of its motion throughout its flight - conservation of angular momentum suggests that the initial angular momentum can't change without an external torque (which gravity does not supply).  The football's path should bend in a parabola... but the point of the ball should always point in the same direction.  Then air resistance should cause the ball to rotate end-over-end, "like a duck".  Presumably he means a duck in flight who suddenly experiences cardiac arrest, but I won't quibble with sportswriters' analogies.

This paper's authors - Richard Price, William Moss, and TJ Gay - show that contrary to previous models, torque provided by air resistance continually changes direction.  This changes the direction of the angular momentum such that the spiral always points tangent to the football's path.  Cool.  That makes sense.

Now, I'm an experimentalist... my next step would be to hire Patrick Mahomes (the reigning Super Bowl MVP, quarterback for the Kansas City Chiefs) to attempt to throw a spiral in NASA's enormous vacuum building.*  If the paper is right, then even Patrick Mahomes should not be able to throw a proper spiral.  Rather, though the football should spiral, the nose of the ball should continue to point in whatever sorta-upward direction it was spinning when the ball was released.  Without air, there'd be no threat of a dead-duck motion, but the spiral shouldn't gracefully arc throughout the flight as in the NFL Films films.

* Or perhaps on the moon.

Anyone want to write this grant?

Mail Time: I love the AP Physics 1 Workbook. Why isn't there a physics 2 or physics c workbook?

If you teach AP Physics 1, I hope you've discovered the Workbook.  It's great.  But, the question was asked, why only for Physics 1?  Why not provide the same sort of scaffolded, sequential, task-modeling exercises for all physics courses?

There's not a Physics 2 or C workbook simply because of the necessary person-hours - and high-end expert person-hours at that - required to produce the book!  Amy Johnson, who's as expert as you'll meet, spent a truly ridiculous amount of time spearheading that project (and writing much of it herself).  

Then, AP Physics 1 was prioritized because that's the largest and hardest course, the one where it's rather commonplace for a school to tell a biology teacher "oh, you are certified in science, you can teach AP Physics 1, good luck."  I've met so many of these unfortunate folks.  They often become outstanding physics teachers!  But in those first couple of years, they need an anchor.  

The workbook can be that anchor.  While it's in no way good practice, it is nevertheless possible for an overwhelmed and inexperienced teacher to do nothing but assign the workbook, page by page... and their students would have a fighting chance of success in the course.  Then the teacher can build on that foundation in future years.  

And finally, P1 is the most misunderstood AP course.  "1 - Algebra-Based" conveys a sense of simplicity to nonexperts, such that administrators and parents and amateur college counselors routinely push weak students into this course, expecting an easy STEM AP credit - yet P1 is, statistically, the most difficult AP course of all.  Teachers don't always know better, either: so many assign Giancoli calculational problems, teach nothing beyond plug-n-chug, then are surprised and sour-grapes-y when few students pass the exam.  

The workbook is there to demonstrate the kinds of verbal responses that will be necessary on the exam; and to guide students (and teachers) toward building the necessary communication skills.  For strong students, AP C can be picked up from textbooks and Khan Academy-style videos.  Physics 1 isn't so easily mastered, even by top-rate students.

Hope that helps explain... Really, the physics folks involved with the College Board are trying to help everyone.  They ain't perfect, and they can't please everyone, but they're trying!  :-)

position-time and velocity-time graph introductory exercise - simulation for students who aren't in class.

Motion graph simulation by Milo Jacobs

On the very first day of kinematics, I introduce position-time graphs via this come-and-show-me exercise.  Each student gets a different position-time graph, and is asked to use position-time graph facts to justify their answer to two questions:

1. Is the cart moving toward or away from the motion detector?

2. Is the cart speeding up, slowing down, or moving with constant speed?

Then, once I approve their predictions, the student heads to the back of the lab where I have carts and tracks.  They have to choose to use either a regular pasco cart, or one of them motorized bulldozers.  They have to choose how or if to incline the track.  They have to choose where to put the sonic motion detector that makes the position-time graph.  

Finally, they bring me the graph they made.  (I use the detectors that work via bluetooth with the graphical analysis app on a phone; if you're using a labquest or labpro, I just have students take a picture of the graph and show me.)  I check that the graph looks correct AND represents at least one second's worth of motion.  Then I give the student a different graph to go through the process again.

Once the class gets the idea that they are required to begin each response with a fact written verbatim from the fact sheet, they figure out very quickly how to do these exercises.  We can move on to velocity-time facts and exercises very quickly.  

But what if you have students who aren't in your lab?  How can they do this laboratory exercise from home?

I've got a simulation for you. It's certainly not the same as being in lab!  There's a huge difference between clicking buttons on a computer, and actually physically futzing with a track, cart, and motion detector.  Yet, a simulation is better than nothing at all, better than saying "okay, imagine the motion..."

There's no shortage of motion graph simulations around.  But none of them did what I wanted - I want the ability to use EITHER a motorized cart OR a free-wheeling cart.  I want to be able to place the motion detector anywhere I want.  I want to be able to tilt the track in any direction.  And I want to be able to choose whether to give the cart a quick shove, or to just let the cart go from rest.

This summer my family's computer programming department - that is, my son Milo - was stuck at home with no obligations.  I offered him a job programming the exact simulation I was looking for.

Take a look at Milo's simulation here - a screenshot is at the top of the post.  You adjust the track angle, click confirm; click a position for the detector; then click to push, release, or use the motor cart.  The simulation shows the cart's motion, and simultaneously shows the position-time and velocity-time graphs develop - exactly as if you were using a labquest in the classroom!

Then there's a link to download the graphs if you want an easy way to get a .png file to submit electronically.  Here's what they can look like:

I used this simulation this summer in my institutes.  It works - it's not the same as being in lab, but it's as good as I've found for teaching remotely.  Try it, and let me know how it works!
 

Respecting your audience, and Rule 1 of teaching

Rule 1: Never condescend, nor even give the appearance of condescension.

I've had to watch a bunch of videos and zoom sessions already, with more to come, as we prepare for the school year.  These have been / probably will be valuable, in the sense of promoting discussion among faculty, of communicating necessary information to students and faculty.  So far, sessions have been respectful of my time - no one is reading powerpoints at the attendees, for example.  It's been good.

Today's presenter - who turned out to be pretty danged awesome - nevertheless turned me off from the beginning.  

While people were joining the zoom session, the presenter had already shared his screen which included five bullet points:

• Try to be as present as possible
• Remove distractions if possible
• Get a beverage
• Get note-taking items
• Prepare to participate

Wait, Greg, aren't these reasonable cultural expectations for attendees at a zoom presentation?

Of course they are.  That's not the issue.

By telling us that which any professional educator should already know, the presenter communicated to me a sense of distrust.  "I know you naughty teachers will not pay appropriate attention to our meeting..."  

The presenter further communicated a hubris about their own power and influence.  "...but if I give you specific rules about how I expect you to pay attention, then presto, you teachers will follow these rules and pay attention.  Perhaps you just authentically didn't know that you should pay attention in a meeting, so I'm relieving you of your ignorance.  Or, perhaps you will sigh and say 'aww, man, I was planning on playing Candy Crush for the next few hours, but dammit, these are the rules, so I'd better go get my note-taking items."

Greg, you talk about culture building... doesn't culture building start with a foundation of what the interaction rules are?

Culture does begin with a foundation of shared expectations.  For our boarding students, most of whom are new or new-ish to living away from home with other teenagers, we start by discussing specifics about how a dorm community should function.  If I were teaching third grade, I would likely begin the year with an open discussion of how we should behave toward one another.  

However... culture building also considers the audience.  The audience for my physics classes are 14-18 year old students who have been in school for many, many years.  The audience for faculty meetings is a room of professional teachers - if they don't know that during a meeting they should "try to be as present as possible," then, well, they shouldn't have been hired in the first place.

Greg, you also know that in a typical meeting a quarter of the faculty are, in fact, playing candy crush.  Or possibly pac-man.

Quite possible.  There are but two effective remedies for this: (1) Make the meeting so worthwhile that most attendees forget about video games and focus, by choice.  Make the audience want to be attentive.  (2) Find one or two people who are blatantly disengaged, and have a conversation with them.  Do they want to be a part of the faculty, or not?  

These are the same remedies, by the way, that I recommend if you have similar issues with students during your class. 

Unfortunately, today's presenter chose ineffective and maddening option (3): give the audience perky-toned yet condescending "rules" for the meeting.  A commonly-used equivalent is the all-faculty email reminding everyone how meetings are important and we should make an effort to focus.  The candy crush players don't change their behavior in response to option (3).  

Rules do nothing but irritate the professionals.  And that was our first impression of the presenter.  He had a mountain to climb to win me over.

Now, I'll be fair to this presenter - he did win me over.  His 90-minute meeting was fantastic, to the extent that when he stopped I said "oh, wow, that was short, he could've gone on longer." How I feel at the end is a major way that I judge any performance.  So this presenter passed with flying colors.

But he could have made things so much easier on himself.  And, how many other folks in the meeting did he lose just by implicitly questioning their professionalism?

 

What equipment do I use to record my physics shows?

I've fielded the question enough times that it's worth a blog post.  In my AP Live videos in spring 2020, in the upcoming AP Daily videos, in my summer institutes... what equipment did I use for recording?

The main camera is a logitech "Carl Zeiss Tessar" model mounted on a tripod facing my demonstration table.  It helps that my whiteboard is low-glare... the reflection from the windows and lights is not bad.  I "enable high definition" in the video settings in Zoom.

I also have a document camera on the edge of the demonstration table.  To switch between cameras, in zoom I say share screen --> advanced --> content from second camera.  This brings up the document camera.  But then I see a button in the top left of my screen to "switch camera" - click that, and I seamlessly switch from the doc cam to the logitech.

As for audio... both my doc cam and the logitech include a microphone, and I can put audio to the speakers in my classroom.  But the sound quality that way isn't great.  So I use a plantronics headset, as pictured in the top right.  The sound is as good as I've ever heard!  The only disadvantage is that I am wired to a USB port.  I've several times stepped on the wire and accidentally disconnected.

Speaking of USB ports... that's three items that need ports.  I had to get a multi-port hub.  And it's important that my internet connection is hardwired to the desktop computer I'm using - otherwise things can get annoyingly slow on Zoom.  

I'll give a shoutout here to Woodberry Forest's academic technology guru Cronin Warmack.  He's been extraordinarily helpful offering whatever technology I need, and helping to make sure it actually works.  All spring and all summer, I had working computer, internet, printers, copiers, network storage, software applications... that's not a trivial thing.  

At my previous school, the head IT guy was offended that I requested the technology that I had been promised at my interview.  While that school's administrators always had working equipment, it seemed beneath this guy even to respond to a mere teacher's concern, let alone to actually act upon it.  And I am aware that many schools operate exactly this way regarding the most basic tech.

Not here... Cronin (and Aiden and Jason) have kept Woodberry running online.  I appreciate their work.