Do you believe in physics? Heavy and light carts hitting the bottom of an incline at the same time

I covered my colleague's honors physics class this morning while he was at a track meet.  (Yes, we have Saturday classes.)  The students were correcting their incorrect answers to a recent test.

One question was the classic about two carts of unequal mass at the top of a smooth incline -- which has more speed at the bottom?  Several people initially couldn't understand why their instinct wasn't right, whether that instinct was that the heavier cart or the lighter cart should be faster.

I gave my standard advice -- forget your instinct, start from first principles, write out the full conservation of energy equations for each cart.  Sure enough, everyone quickly solved to see that mass doesn't appear in the equation for speed at the bottom, so that both carts should move with the same speed.

But I could read faces.  They did NOT believe in physics.  I asked* if anyone would be willing to bet, say, $100 that a heavy cart would beat a light cart to the bottom of an incline, or vice versa.  They initially said "yes," then looked at my face. "Well, no, 'cause you would win, wouldn't you."  

*facetiously, of course, and always in the subjunctive mood... too bad, though, because I could make SO much money gambling on physics.

Of course I'd win.  But I don't want to win the argument with a thought experiment, or via my status as the teacher.  I want to win the argument with an actual demonstration of the right answer.  

So I quickly grabbed two PASCO carts, sticking some extra mass on one of them -- you can see the setup in the picture.  I elevated one side of my PASCO track, which is pretty danged close to being frictionless.  I let the carts go... they reached the bottom at the same time.  Physics works.

"Now, do you believe?"  I asked.  To his credit, one student said no:  "Switch the carts, please.  Put the heavy cart in front of the light one."  Great point.  Not that the switch changed the answer, but he did a good job asking a key question.

Anyway, everyone was glad that they didn't bet with me, even figuratively.  They can try to argue with math, but students have a really hard time arguing with an experiment that's performed right in front of them.

Mail Time: Where's that pressure sensor for the static column experiment?

Tennesseeeeeeeean Wendy Stallings, who attended one of my workshops last summer, writes in:

I'm having trouble finding a pressure sensor to immerse in liquid.  Unless I'm mistaken, I can't find the one you used in the static fluids demo last summer even though I've scoured the Vernier catalogue.  Can you point me in the right direction?

Certainly.  Wendy is referring to a quantitative demonstration I do:  I fill up a tall graduated cylinder with water.  We use the equation P = Po + ρgh to calculate the pressure at the bottom of the cylinder, knowing the pressure sensor reading of atmospheric pressure.  Then, I measure the pressure at the bottom; voila, the pressure is just as we predicted.  Physics works.

The sensor is the standard issue Vernier gas pressure sensor, as pictured, and as available on the Vernier website for about ninety bucks.  (Ask around your school, first -- biology teachers use these, and you may be able to borrow one.)  Thing is, I don't actually submerge the sensor.  

Look at the picture, at the right-hand edge of the sensor.  That's an attachment for a tube, one somewhere in the neighborhood of 5-10 mm diameter.  A bit of this tubing, with various connectors, comes with the sensor, at least according to the Vernier website.

I connect the 18" tube to the sensor, and submerge the tube.  Sure, the tube gets wet, but the sensor doesn't.  The sensor reads the pressure at the end of the tube, giving me the experimental verification for the P = Po + ρgh prediction.  I don't know whether or not Vernier recommends such an experiment; I know I've done this one for years with no adverse effect on my sensor.

And yes, this is the same pressure probe I use for the heat engine and gas law experiments.  I love equipment with multiple uses.

GCJ

Posted: The units of acceleration are...

On our school email server, I've created a conference for my classes.  Students are expected to read messages posted to this folder on a regular basis.  They're also encouraged to post questions for me or for each other, though they don't do that so much.  I post messages to the class, including electronic copies of the problem sets, announcements of upcoming assignments, and my thoughts as I grade student work.

The trick, of course, is to get the students to actually read messages, and to read them all the way through.

Occasionally I've put a statement at the bottom of a long message saying something like "Extra credit will be awarded to each student who hands me a folded up piece of paper in class with the word 'cow' on it. You may not discuss this extra credit opportunity with anyone."  That's been effective... I certainly have gotten better attention since I started occasionally including these "easter eggs." 

Tonight I graded a test correction in which way too many students said "Objects in free-fall gain or lose 10 m/s of speed every second.  At the top of its flight, the ball is still in free fall; so its* acceleration must still be 10 m/s."

* Sanitized -- a ninth grader would, of course, write "it's."

Aarrgh.  Instead of pulling my hair out about the units of acceleration, I just posted a different sort of easter egg:

Consider your test correction to problem 7.  The problem states, when a ball thrown upward at 30 m/s reaches the peak of its flight, what is the ball's acceleration?

Many of you said that the acceleration is 10 m/s.  That's not right -- the units of acceleration are m/s per second.  The acceleration of an object in free fall, even at the top of its flight, is 10 m/s per second.

I will give you an extra point on tomorrow's quiz if you write at the bottom "The units of acceleration are m/s per second.  The units of speed are m/s."

This is even better than asking students to write some random code word to prove that they read my email; they have to write the correct units for acceleration, which can not be done enough times.

That dang cart-and-pulley experiment

As we introduce Newton's second law, the obvious experiment is to measure the net force on an object, and the object's acceleration; a plot of net force vs. acceleration has a slope equal to the object's mass.  Easier said than done.

The trick is providing a truly constant force, one that can be easily varied and easily measured.  Most introductory classes default to the cart on a track attached to a hanging mass, as shown in the picture: easily varied, easily measured.  The motion detector can measure the acceleration of the cart as it's speeding up (or slowing down).  

Problem is, though, the hanging weight is NOT the net force on the cart.  If you just add and subtract mass from the mass hanger, the a plot of hanging weight vs. acceleration will not be linear.  Two experimental solutions come to mind:

(1) Measure the tension in the string, which is the net force on the cart.  You can put attach a Vernier force probe to the cart and to the string; the reading on the probe is the tension in the string.  As the cart is accelerating, you'll see the probe reading drop from a value equal to that of the hanging weight to something slightly smaller.  Use that on the vertical axis, and the acceleration from the motion detector on the horizontal; the slope will be the mass of the cart.

I do this demonstration in honors/AP physics, in which we predict the tension in the string and the acceleration of the cart.  However, this is a complicated way to do the experiment.  The easy way is...

(2)  The hanging weight is the net force of the entire set of objects including the cart and the hanging stuff.  So, the trick is to add about 100 g of masses in small denominations to the top of the cart.  Then, when changing the hanging weight, just redistribute mass from the top of the cart to the mass hanger.  Now the mass of the cart-and-hanger system is remaining constant.  So the plot of hanging weight vs. acceleration will have a slope equal to the total mass of the cart plus the 100 g.

Recognize that the distinction between treating the cart-hanger system as a unit, and treating the cart and hanger separately, is very subtle and difficult for even top level introductory students.  I don't even want to broach the subject with my freshmen, who are doing this experiment next week.  But that's okay... I don't have to tell them WHY they're keeping all the mass either on the cart or on the hanger.  If anyone asks, I'll explain that we need to keep the total mass of the moving stuff constant.*

* The super-smart-and-curious freshmen who would have demanded and understood a deeper explanation have long since been siphoned into an honors section, where they will learn all of the relevant subtleties.  

With the setup in the picture, my freshmen will be able to graph the hanging weight and the acceleration of the cart.  Interestingly, our new Vernier Labquest 2 is giving me trustworthy acceleration vs. time graphs!  This clever new device will spit out an average acceleration with reasonable ease.  I'll set up the carts ahead of time -- half the class will get the 250 g PASCO cart with 150 g of added mass... the other half will get a silver 500 g PASCO cart with 100 g of added mass.  We'll all use the same set of prepared axes, so we'll all see at a glance that the setup with larger mass produces a steeper line on the Fnet vs. a graph.

Motion: Information to memorize for Conceptual Physics

No good single resource for conceptual physics exists.  Sure, the Hewitt text ain't horrible, but try putting yourself in the mind of a 14-year-old who is utterly new to physics, and to academic life as well.  All Hewitt's clever little asides do nothing but distract; his vocabulary (even in "The High School Physics Program") is more suited to seasoned law clerks than to general-track freshmen.

Nevertheless, I have to teach freshmen rigorous conceptual physics.  As we've moved into kinematics, I've ditched the textbook entirely in favor of a typed sheet of "information to memorize."  Take a look at the sheet in this link:

Motion:  Information to memorize for Conceptual Physics

While this two-page sheet covered about three weeks of class, I handed it out in steps.  During the first week, I gave them only the eight facts listed under "motion diagrams" and "position-time graphs."  Those eight facts were plenty to do serious experimental work with spark timers and motion detectors, and to answer any conceptual question about position-time graphs, even up to the low AP level.

Then in the second week, I redistributed the sheet, adding the rest of the first page including "velocity-time graphs" and "questions to answer when describing motion."  

Finally, I passed out the back page, which stated four facts relating to acceleration, and the three relevant equations for distance traveled.*

* One for constant speed, two for speeding up from or slowing down to a stop.  

This piecemeal approach mitigated one of my major issues with textbooks: too much information.  Rather than asking students to wade through pages of densely-paragraphed hard copy to find the important facts -- if they even know which facts are important -- I just wrote the important information by itself.  

I'm sure some idealists, education professors, and textbook publishers will argue that my approach substitutes dry facts for rich context.  Yes, yes it does.  It is incumbent upon me to provide interesting and useful context through class activities, experiments, demonstrations.  That's what I'm paid for.  If merely reading a perfectly-written textbook could possibly be sufficient for most students to understand physics, then all of us physics teachers should find another job; we'd be useless.  

Since my school expects me and my students to show up each day for class, I'm going to simplify the background reading to its bare-bones form.  Then out-of-class time can be used to practice problem solving; in-class time is used to ensure that the out-of-class work was done carefully, and to provide in living color the active physical context that not even the best-written textbook can hint at.