## Monday, July 30, 2012

### M and I: Young's Modulus

After looking at the relatively involved apparatus used by the authors and the \$1100 Pasco apparatus, I was pretty worried about measuring the Young's modulus for materials. After a very fun meet-up dinner with M and I adopters (seriously - when's the last time that you, Halliday, Resnick, and 20 excited adopters had dinner?) last night, I got three different methods to measure it from folks there:
•  First, from Mark Hammond, the idea of just hanging a wire from an I beam in the ceiling, putting a pointer on it, and measuring.  I hadn't gotten around to trying this, but I was thinking that it'd be a good last-ditch substitute.
• Second, from Aaron Titus (do you have a blog Aaron?), the idea of doing that but with a motion detector underneath the mass hanger.  If the wire stretches enough, this will measure the distance more accurately than the first method (which is apparently possible - I thought that the stretch wouldn't be much relative to the motion detector's resolution, but I guess that I'm underestimating one of the two).
• Third, also from Aaron, attach one end of the wire to a clamp, run it horizontally (I ran it a little more than 6 meters to maximize the stretch amount) over a Pasco rotary motion sensor.  Have a short length of wire hanging over the pulley and wrapped around a mass hanger.  It only took about 500 g of mass to get a nice range of stretch.  Even a 50 gram difference caused an easily measurable stretch for the sensor (around .05 radians/50 grams).
For the third method:
• No knots: wrap the wire around both connection points - you don't want kinks and knots slipping.
• I measured the tension force with a force probe as well as calculating it from mg.  They were pretty equivalent in practice (as well as in theory, of course), so I'll just use mg.
• I used thin enameled magnet wire (.17 mm diameter - don't know the gauge); I'd like to get aluminum wire as well, so that I can predict resonant frequencies for the singing rods.
• Edit: put a catch box underneath.  As I was writing this, the wire gave way with 500 g hanging from it.
• Here's the data - 118.9 GPa (117 on Wikipedia). I'll surely take that. I'll use this to predict v for a transverse wave, too, and see if I can get agreement there, but this is seriously encouraging.

## Wednesday, July 25, 2012

### M and I: Speed of Sound from First Principles

I'm excited about using the Matter and Interactions curriculum to connect the micro-world to the macro-world. Two of the big candidates are speed of sound and specific heat. Right now, I'm looking at the speed of sound.

Using the interatomic bond length and stiffness to determine an expression for the speed of sound (longitudinal) in a material, the authors arrive at:
Here, k is the interatomic bond stiffness and d is the interatomic bond length (modeling the solid as a cubic lattice of atoms connected by Hooke springs), and m is the mass of one atom.

What I really want is to be able to put the physics students (who cover sound and know the standing wave stuff and can experimentally determine the frequency of a standing wave and calculate the speed of sound) together with the AP Physics students (who can calculate the speed of sound from experimental measurements, if I can get a Young's Modulus apparatus) to see if the approaches agree.

While I don't teach it, this made me think of the speed of sound in a wire expression:
Here, T is the tension and mu is the mass per unit length.  This has the obvious dependence of speed on tension explicitly and made me think about the M and I expression - tension's not (obviously) there.  Is this an approximation or will the approach only work under no tension (as the value was calculated)?

Actually, the two (pretty much) reduce to each other. Starting at the speed in a wire expression:
Comparing the tension in the wire to the tension in an individual "strand" of atoms:
Substituting, simplifying, rearranging:

The approximation was that one fourth equalled one sixth, from the volume/area substitution. The problem is that this really only works if we call the tension force between two atoms kd, which isn't really what it should be.  I'm not sure if we're dealing with a limitation of the M and I model, some unstated assumptions in the wire model, or both.  ...or a longitudinal/transverse mismatch. Can anybody help?

I'm encouraged about the possible agreement, though it's not really necessary for the expression to agree for me to test the prediction made by the M and I model.  I tried to test it out a bit using a 17 cm long 2 cm thick copper cylinder.  I dropped it on its end on the floor and recorded the high-pitched ringing while it was in the air.  The frequency spectrum:
The fundamental frequency for that bar (using all of the available simplifications) should be around 15 kHz, but it's quite a bit lower (less than 3 kHz).  The spacing's not even here, so I'm thinking that I have an issue with the cylinder having some odd vibrational modes, but I can't find a reference on what to expect.  In any event, I'd like something with well-behaved harmonics that my physics students can analyze successfully. A copper wire seems great, but the waves are the wrong mode, and definitely wouldn't agree.

Another idea that I had: a long aluminum rod (one of those Arbor Scientific dealies).  It's 74.5 cm long, and I recorded this spectrum from the demo video on the Arbor site (I can't find the rosin):
These peaks are nice and harmonic, with a fundamental right around 3346 Hz. This means a speed of sound of 4986 m/s, which should agree nicely with the value that we'll get from using the bond strength of aluminum (OK - I need to get some aluminum wire.  Hopefully that's a thing.).  It looks like this is the way to go.

Other suggestions, anyone?  I'm going to try thumping a marshmallow, too, since the modulus is easy to determine, but I'd have to do some voodoo with the calculations, since it's not an elemental substance.

## Monday, July 23, 2012

### M and I: chapter 1 - Relativistic vs. Classical Momentum

In my last post, I promised to share thoughts and materials.  Here's the first!

Chapter 1 is a lot of review for my second-year students. The new bit is relativistic momentum, and the revelation that our definition of momentum from last year was really just a special-case limit.

One of my themes for the year is 'pulling back the veils' on all of those more complicated bits that we approximated and hand-waved away last year. I want this to be a major selling point of the course. Unfortunately, as presented, I think that the momentum definition isn't accessible or real to the kids.  We can't really discover it, and it's just an abstraction at first.

I wrote a VPython program to give this some context. You can run it twice for the students - in each case, a 60 g block is pushed from rest by a constant 1 million Newton force. The momentum and velocity of the block are graphed as time goes on.

Here's where another pillar of the course is going to come in to play: fundamental laws of the universe. Really, we didn't deal with too many last year, at least not on a really visceral and important level. Here, though, the universal speed limit becomes important. Look at the v graph pass right by it without blinking, when the program uses the classical definition of p (and finds v from there):
That's a problem. The problem's with our model, though: momentum just doesn't really equal mass times velocity.  Uncomment the gamma definition in the program to use the real relationship now:

Here's a great way to distinguish which is the fundamental principle here: the momentum principle still holds just fine, even in the relativistic case, but our definition of v (as determined from the momentum) was broken. With the relativistic relationship, the velocity approaches the universal speed limit.  Two days in and kids not only know a fundamental law of the universe, but they know how it really does fit in with what they knew about momentum and what the missing piece in their model was.

Script:

 from __future__ import division, print_function
from visual import *
from visual.graph import *
scene.background = color.white
scene.height = 50
scene.width = 50
scene.x = scene.y =0
## Josh Gates 2012, started with kernel of Ruth Chabay program
print ("""
Click anywhere in display window to start.
Click to stop.
Momentum and velocity are shown as .6 kg block is pushed with a constant 1,000,000 N force.
Uncomment gamma line to remove classical p def'n/assumption""")

## CONSTANTS
delta_t = 0.01  ## for 100 steps
mblock = 0.06
c=3e8
Fnet = vector(1e6,0,0)

## OBJECTS & INITIAL VALUES
pblock = mblock*vector(0,0,0)
gamma = 1

# start time at 0
t = 0
scene.center = (0,.1,0)  # move camera up
scene.range = 0.15

## GRAPH STUFF
gp = gdisplay(background=color.white, foreground=color.black, y=0, x=250, height=300,
title='Momentum vs. time: block with constant F', xtitle='Time (s)', ytitle='Position (m)',
ymax=3e7,xmax=30)
blockp = gcurve(color=color.magenta)
blockp.plot(pos=(t,pblock.x)) ## initial pos.x of block

gv = gdisplay(background=color.white, foreground=color.black, y=300, x=250, height=300,
title='Velocity vs. time: block with constant F', xtitle='Time (s)', ytitle='Velocity (m/s)',
ymax = 3e8, ymin=0, xmax=30)
blockv = gcurve(color=color.blue)
blockv.plot(pos=(t,pblock.x/(gamma*mblock))) #initial velocity
asymptote=gcurve(color=color.red)

scene.mouse.getclick()
stopper='go'

a=.1
while a<1000:
asymptote.plot(pos=(30/a,3e8))
a=a+1

## CALCULATIONS
while stopper=='go':
rate(500)

pblock = pblock + Fnet*delta_t

# comment to make classical approximation
#gamma = (1-(pblock.x)**2/(mblock**2 * c**2+(pblock.x)**2))**-.5

# update time
t = t + delta_t
# plot pos.x, velocity.x of block
blockp.plot(pos=(t,pblock.x))
blockv.plot(pos=(t,pblock.x/(gamma*mblock)))
##  print t, block.y
if scene.mouse.clicked:
stopper='stop'



I'm using Matter and Interactions for the first time with my AP class (AP Mechanics C) this year. Because of the mismatch between M and I's goals and the AP curriculum's goals (both of which are good in their own ways, but put together will really be a complete understanding of mechanics, I think), I'm trying to work ahead to figure out where to supplement with  more of the traditional problems that students will need to be able to solve for the AP exam and which should be a part of every student's arsenal.

I'm also going through the whole curriculum to better orient myself to the vastly different narrative than the usual mechanics treatment. This, combined with the emphasis on computational representations, is the reason that I'm so excited about this curriculum, even though I'll have to do extra prep work to make it work for the context of AP C.

Another wrinkle is that M and I is designed as an intro course, and my students have already had a year of algebra-based (mostly) mechanics. They have good FCI scores and can solve some algebra-based motion, force, momentum, and energy problems well.  Some of the information here is review, some is review in new terminology, some is a new look at something they've seen before, some is a new technique for dealing with a concept that they know for situations that they couldn't apply it to before, and some is totally new.  These all require different approaches, and it's an uphill battle.

Bruce Sherwood has already been an infinite factor more accessible than Walker, Giancoli, Halliday, etc.: I've been in a G+ chat with him and he personally sent me a DVD with the new instructor materials (the Pearson rep jealously guarded these until I could send her a receipt for the purchased books).

I'll try to record and share as many of these strategies and tools as I can.

## Monday, July 9, 2012

### Student perspectives on the internet as a resource

I teach an online AP Physics B course during most of the year.  It's a challenging course, both because of the (overly) broad content of that course, as decided by the College Board, and because students generally need to be independent learners.

There are readings from Walker's intro text and some (usually decent) video lectures (for most topics), provided by the Thinkwell folks.  I supplement that with some links to relevant Hippocampus tutorials.

As much as I'm into reform pedagogy, this is basically the opposite - direct instruction with (optional) questioning of me via email.  We do have some live discussions which are group conceptual problem-solving.  I've also designed the labs to be better than the typical cookbook, but they're not modeling, in terms of discovering paradigms, by any means.  They're all applications of material that they've read about/watched.  It's not what I'd like to teach or take as a physics course, but it's similar to (honestly) all of the courses that I've ever taken, and the same is true for the students.  It's not that this is worse than a 'normal' physics course, it's just that it's lacking in all of the advantages of what I've been doing in my live classroom for so long now that it seems antiquated by comparison.  The medium (exclusively online and asynchronous) really hems you in.  In combination with the AP B syllabus, there's no other choice.

This course is, though, exclusively targeted at very high-achieving students.  It's tied to a larger recognized program that they must gain admission to first.  At the end of the day, I get a lot of very bright (some very young) students, and they're the folks that lecture always worked best for, so it goes relatively well.

Some students were chatting in the online message board today.  One was asking the others for ideas about extra resources, because he/she was finding the end-of-chapter problems fairly difficult.  There were some offers of resources, I reminded them that they can ask questions (some just don't, regardless of what you say, as usual), and a student posted something about typing the questions into Yahoo! Answers.

I responded with a warning:

"This is also skirting illegal assistance. While talking to each other we can expect that we all know that general talk about the principles involved is OK and giving a solution is not OK, no such agreement exists on Yahoo Answers, etc.

Posting the text of questions - homework and especially test - to such a forum is _not_ allowed because of the types of answers that you're likely to get back (complete solutions)."

The "finding the answer" bit won out over understanding the material here; while I can (try to) prohibit it, I'm frustrated that I don't have a great mode in this online course to do better education about mindset.  I send a lot of stuff out (I do include some tips on mindset and taking an online course effectively), but there's also a great deal of logistics to deal with, but I'm lucky if half of them read any given thing that I send out (a story as old as teaching).

Another student (I think before my post had been sent out), though, replied beautifully:

"For this reason, xxxx, I found it helpful to use online resources much more generally. Although I do think the homework problems are much more difficult and involved, I've usually been able to work through most of the problems on my own with just the notes and the book and do pretty well on the assignments generally - it just takes AWHILE. But sometimes I'll work through a problem three times and still not understand what I'm doing wrong. In that case, I'm usually overlooking a concept, so visiting online resources like hippocampus or looking for educational tutorials to a problem of the same general type (the way our video lectures explain sample problems) on youtube or physics websites I stumble across on the internet will help clarify the principles for me. That way, it helps point me in the right direction without explicitly giving me a solution - I'm still applying the principles and working out the homework problem on my own. Good luck!"

This student appreciated the value of the struggle and was not interested in finding _the answer_ as an end unto itself. It's the kind of student response that you put in a drawer to look at when you're having a bad day.

The idea about differentiating between the point at which productive struggle becomes unproductive struggle was also a good one, and I think this is something that we all could do a better job educating students about when we're looking at mindset (I've shared my evolving set of mindset talks/lessons/activities here).

Late update:

After I agreed with him/her vehemently, student posted this:

"Absolutely! I feel I absorb an enormous amount from the homework. It's difficult a good way if that makes sense, and the "struggling" usually leads me to the highest level of understanding. Thanks smile"

## Monday, July 2, 2012

### Chains of Reasoning: Standing Waves and Tension

I'm clearing out a few "meant to" posts from the year.  Here's a chain of reasoning problem I had kids do about a slinky hanging from the ceiling. I asked this as the 'advanced' question on their big waves assessment, and we came back to it in groups the next day. Students were asked about what would happen to the frequency, wavespeed, and wavelength of the waves from the top of the slinky to the bottom, and then to draw a standing wave diagram to reflect that.  I prefaced it with a question that elicited from almost everybody that the tension in the slinky was greatest at the top (almost 100% success even though they never studied forces - you just have to ask it specifically for them to realize it).

Since most had trouble with the question, I wanted them to work through it, rather than just forget it and move on.

The whiteboards are below. I was very particular with them writing down all of their reasoning - mostly "how do you know that's true?" and "what did you observe or assume to get to that?".  If I stayed on them, they did well and their answers were all correct! It's a year-long process to get them to internalize that process. It's not that they don't have the ability, but being a true self-critic is much more difficult than giving up.

Hmmm... I can't get this one rotated - Blogger issue.  The original's fine!