Comparing Springs

We went through simple harmonic motion derivations two days ago in AP. Writing the second-order diff eq. was easy for them, but it was the first one that they've solved, so we had a nice long guess-and-check for possible solutions. Ultimately, it was pretty clear to them how to take that trig function and get what they needed from it, but the process of using that diff. eq. solution as a model to solve similar ones (especially the hanging spring) was more difficult for them. I don't have any awesome inspirations on teaching that, but looking at a few other situations (two springs, etc.) helped quite a bit, and I assigned the "fall through the Earth" problem last night, which is a great application of SHM.

The point here, though, is to talk about some of the realizations about SHM that they had, and how Python helped that happen. They don't do oscillations in their first course, so this is the first that they've seen it in a physics context. The idea of amplitude independence is always tricky for students, so I just let them make a prediction about what would happen to the period of a spring when I changed the mass, or the spring constant, the amplitude, or g.

It's easy enough to see those results by looking at the derived period function, but it was always difficult to demonstrate some of those things cleanly - springs wobble, students imagine slightly different periods, they can't tell exactly what the amplitude or equilibrium point are (and damping doesn't help), I can't turn off gravity, etc.

I modified our second programming assignment of the year to have two springs instead of one, and demonstrated all of those cases very easily. Because they know how the program works (they basically wrote it), there's more faith in it (well, not faith, since it's based on reason, but they put more stock in it), and it's much easier to see.  The springs will oscillate perfectly forever and I can listen more to the discussion than worry about restarting the springs all the time.

Here's the script - use with attribution and an email:


from visual import *

scene.height=600
scene.width=600
scene.scale=(.6,.6,.6)
scene.center=(0,-.7,0)

# scales
Fscale = .1
vscale =.4

# objects
roof1=box(width=3,height=.05,depth=3,color=color.white,pos=(-.5,0,0))
roof2=box(width=3,height=.05,depth=3,color=color.white,pos=(.5,0,0))

ball1=sphere(radius=.1, color=color.green, pos=(-.5,-.5,0), velocity=vector(0,0,0))
ball2=sphere(radius=.1, color=color.orange, pos=(.5,-.7,0), velocity=vector(0,0,0))
#ballvvec=arrow(pos=ball.pos, color=color.blue, axis=vscale*ball.velocity,shaftwidth=.05)

spring1 = helix(pos = roof1.pos, coils=15, axis=(ball1.pos-roof1.pos),radius=.1)
spring2 = helix(pos = roof2.pos, coils=15, axis=(ball2.pos-roof2.pos),radius=.1)

#forcevec=arrow(pos=ball.pos, color=color.red,shaftwidth=.05)


# physical constants
k = 20
restlength = 1
mass1=.6
mass2=.6

deltat=.001
t=0

while 5 <= 10:
    rate(300)

    # forces
    gravity1=vector(0,0*mass1,0)
    springforce1 = -k * (mag(spring1.axis)-restlength) * spring1.axis / mag(spring1.axis)
    netforce1 = gravity1 + springforce1

    gravity2=vector(0,0*mass2,0)
    springforce2 = -k * (mag(spring2.axis)-restlength) * spring2.axis / mag(spring2.axis)
    netforce2 = gravity2 + springforce2

    # v and r update
    ball1.velocity = ball1.velocity + netforce1 * deltat / mass1
    ball1.pos = ball1.pos + ball1.velocity * deltat

    ball2.velocity = ball2.velocity + netforce2 * deltat / mass2
    ball2.pos = ball2.pos + ball2.velocity * deltat

    # vector and graphical updates
    #ballvvec.pos = ball.pos
    #ballvvec.axis = vscale*ball.velocity
   
    #forcevec.pos = ball.pos
    #forcevec.axis = Fscale*netforce

    spring1.axis=(ball1.pos-roof1.pos)
    spring2.axis=(ball2.pos-roof2.pos)

    # time update
    t = t + deltat
   

 

Speed of Sound

OK, so we're finally there in AP - the speed of sound calculation. Matter and Interactions makes a hand-waving argument based on dimensional analysis to get the longitudinal speed of sound from the interatomic bond length and spring constant and the atomic mass. That's fine, but I wanted to get to the real derivation, which involves a very interesting couple of linear approximations and some possibly bogus calculus, but it's doable.  The transverse one is basically the same story.

It's more guided and less inquiry, but it's a tough bit of stuff. I'm OK with there being less discovery here, because it's going to be a gigantically powerful result - predicting the frequency of a sound using the molar mass, density and Young's modulus? Awesome.

We're applying both of the expressions today in lab.  I'm spitballing the work flow here:

• Wave intro, animations, etc. Define wavelength, frequency, v relationship by analogy to traffic
• Longitudinal wave derivation:
• With known molar mass, modulus, and density, they determine k, d, the atomic mass, and v (using the dimensional analysis result)
• The real derivation - super-guided by me
• Demo singing rod, have them calculate its frequency
• Sing, record, FFT
• Party
• Transverse wave derivation:
• Animations - difference between longitudinal and transverse
• Derivation
• Give them sample of wire - they determine the tension needed to have the sound be an octave below a resonance box (128 Hz); help them with the standing wave's length
• Use force wire strung between two poles (one with a force probe on it) to measure tension. Other end of wire is still on the spool, which is on the rod. Kids twist spool so that the sound is in tune; reveal tension (or vice versa?)
• Party

The setup with the spool (yellow string added so that you can see the path of the wire):

Here's my drawing of the whole process:

Illuminations for Models

In the packets that I make for each unit (generally broken up by physical model), I include a little illumination-esque drawing, using the model's abbreviation as a backdrop to illustrate situations where the model applies. It's fun, and hopefully a kid sees something at some point on one that makes something click.

I'm drawing them (since BFPM) with SMART Notebook and my tablet PC (ThinkPad X220). I've finally found a use for Notebook!

Feel free to use them (with attribution and an email :) if you like them.

These are the ones that I have for the first term:

Physics and Honors Physics:
CVPM (Constant Velocity Particle Model (or Motion, as you prefer))

CAPM (Constant Acceleration Particle Model)

Honors Physics:
BFPM (Balanced Force Particle Model)

UFPM (Unbalanced Force Particle Model)

Physics:

OPM (Oscillating Particle Model)

Yellow to Green

My grading scheme this year for AP Physics (C: Mechanics) has, so far, worked out the best that I've had for that course. My physics/honors physics scheme involves (basically) one or two standards per model, so the standards are fairly coarse. The problems are easy (enough) to generate, and reassessment at more frequent intervals is good for reinforcement, practice, and experience for those first-time learners. Applying that to the AP course in the past has been problematic, given the headaches of senior scheduling and senior motivation. It's also another group of reassessments for me to prepare, schedule, and grade. I posted about this before, but I've made a few tweaks and it's started to actually get used.

Here's my scheme this year:

• The standards are more grainy (see the list here), compared to honors physics (here)
• There are (generally) two assessments per unit in class
• There's one reassessment available per unit, covering the whole unit
• I made a chart showing the acceptable evidence for each standard. Typically, it'll be two strong showings in assessments/reassessments for that standard or one strong showing and several out-of-class successes (problems) or a capstone or several out-of-class successes and a capstone
• Everyone's required to do at least one capstone per term, even if you rocked all of your assessments
• Students submit this slip when they have some evidence to show for a standard.  There may be some revision needed before it's accepted.

The extra outside effort needed to find and do problems and/or capstones motivates them to do good work on assessments, without me needing to wade through another sea of reassessments.  The capstones are my favorite part, and they're coming up with some great ones (more on those as they mature).  Overall, I'm liking the balance between giving them flexibility with their busy senior lives, keeping them accountable, and respecting the fact that, even though there aren't so many grades 'in the book' right now, they're still working.

Today's a big day, because the first student went from yellow (NY: "not yet") to green (P: "Proficient").  Many more to come of those, as well.

Balanced and Unbalanced Forces

We're starting forces tomorrow in Honors Physics, after having done 1D CVPM and CAPM motion (both algebraically and graphically).  In general, I'm pleased with how quickly and deeply they've picked up these concepts and how well they're already starting to tie them together.

I'm doing a new version of the lab that I call "Balanced/Unbalanced." Last year, I did a hoverpuck thing similar to Kelly's, but I wanted to try something a little different this year. I'm trying to surprise them a bit out of the gate and get them to reference data to (probably) disprove their own predictions - we'll see if they see what's there or what they want to see:

I'm setting up Atwood machines from my ceiling mounting rack, with about 130 g of mass on one side and a force probe on the other side:

The students will observe the motion of the force probe and its reading in several situations, including:

• at rest (130 g of mass balances the probe nicely)
• moving upward (still 130 g)
• moving downward (still 130 g)
• moving downward (70 g on the other side)
• moving upward (190 g on the other side)
• moving upward (70 g on the other side)

In each case, they'll note the reading on the force probe (some surprises there, for sure), draw a well-scaled force diagram, and determine what type of motion model describes the motion. The desired UFPM/CAPM (or at least not CPVM) and BFPM/CVPM connection should come quickly, but I'm hoping that the second, third, and sixth cases are very stark messages about the difference between net force direction and motion direction. 

Part of my thinking here is that students can doubt their reasoning a lot. Even if they go through what we'd understand as a very convincing linear argument, that's not always enough to convince them in their guts. Not being able to see force magnitudes (exactly) using the hoverpuck seemed to play into that a bit for my kids last year - there was this mysterious realm of stuff going on that we weren't measuring and, even though we had a good story about what was going on, who knows if there wasn't other stuff happening that they didn't know about - what if I was lying? What if their information was bad? What if their reasoning was wrong? All of that was a possibility, if they haven't really bought into, you know, how science works (and the fact that it works). This computer, though, it wouldn't lie to them (well, that seems to be how they think).

We'll see.