Re: [Phys-L] Timing Statistic + Moiré patterns

[Paul Nord <Paul.Nord@valpo.edu>]

John,

Thanks for the phase plot.  That makes it very easy to explain to my
colleagues how particular values of angular velocity may show a large sigma
and others may show small sigmas and be consistently high or low.

Back to the original question...

This is a lab for college freshmen taking either the college physics or the
university physics course.  This statistical analysis is pretty far beyond
them at this point.  In fact, the lesson is about conservation of angular
momentum.  Our lab is designed to make students think critically about
error analysis and how well any measured value describes a physical
system.  In this lab we hope they discover that the error in the
measurement of the radius of the disk contributes more to the error in the
calculation of the moment of inertia than does the error in the measurement
of the mass.  The angular velocity measurements and their errors are used
to calculate the angular momentum before and after a change to the moment
of inertia.  After the full error propagation they calculate a Z-value and
we ask them to state, in good scientific terms, whether angular momentum
was conserved.  Generally these Z-values are between 0.5 and 1.5.  Though a
few students have poor technique and touch the system while it's spinning.
(The Z-value becomes smaller when they screw up at least one of the 5
trials... and that's a whole other statistical discussion.)

If I wanted to give the students a reasonable description of the
capabilities of the Vernier transducer, is it reasonable to tell them: The
precision of the measurement looks like +/- x but you shouldn't trust
that.  We've studied the device more carefully and you should assume that
the reported angular velocity measurement is really +/- y?

Is there a recommendation for a modification to the internal magic of
Vernier's software?  Most of their clients are high school teachers who
would like a device which reports a reliable angular velocity in a small
number of time steps.  Yes, for you and I it would be interesting to get
precision time values for each digital step in the internal encoder.  But
most people won't want to see that level of detail.  Is there something in
between?  Perhaps precise times at a few uniformly-spaced positions in the
rotation?

Paul


On Tue, Nov 13, 2018 at 10:55 AM John Denker via Phys-l <
phys-l@mail.phys-l.org> wrote:

> On 11/12/18 4:08 PM, Paul Nord wrote:
>
> > The residuals in the phase values show a really cool Moire pattern.
>
> Right.
>
> That answers the question posed at the start of this thread.
> Are the residuals random noise?  Nope.  There's almost nothing
> random about them.
>
> Students may find this easier to understand if you show them
> a Moiré pattern produced by the model on the basis of no data
> whatsoever.  It arises simply because the rotation period is
> incommensurate with the sampling interval.
>
> I added a page of spreadsheet to do this:
>   https://www.av8n.com/physics/spindown.gnumeric
>   https://www.av8n.com/physics/spindown.xls
>
> And for convenience I separated out the graphs:
>   https://www.av8n.com/physics/img48/spindown-moire.png  [overview]
>   https://www.av8n.com/physics/img48/spindown-moire-z1.png [zoomed in ->
> prettier]
>
> Particularly interesting is:
>   https://www.av8n.com/physics/img48/spindown-moire-vs-v.png
>
> which plots the residuals as a function of angular *velocity*
> so you can see that when the rotation rate (in degrees) is
> at or near an integer multiple of the sampling rate, the pattern
> is single-valued.  When it is at or near an odd half-integer
> multiple of the sampling rate, the pattern is two-valued.
> And so on.  This leaves no doubt that we have identified
> correctly the fundamental physics (aka math) issues involved.
>
>   (These patterns are related to the "aliasing" phenomenon
>   that you see in Fourier transforms when the sampling
>   interval is incommensurate with the underlying physics,
>   but that's probably more than the students want to know
>   at this point.)
>
> We need to come up with a better name for these "residuals".
> They were residuals with respect to an ultra-simple model,
> but our fancier model can predict them more-or-less exactly,
> so they're not residuals with respect to the fancy model.
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