Re: [Phys-L] determine k

[John Denker <jsd@av8n.com>]

On 02/09/2015 02:19 PM, Bill Nettles wrote:

> Do you want to impart the idea that the k in the static spring is
> different from the k in the oscillating spring,   or do you want to
> say that the k belongs to the spring and something else (like an
> unknown moving mass contribution) is making them appear to be
> different?
>
> If you prefer the latter approach, then when you discuss the
> oscillating behavior (period = 2pi\sqrt{m/k} for example) ask them
> what the "m" actually represents. They should distinguish this from
> the masses used in the static case if you present the static as
> |\Delta F |= k |\Delta X| for a linear spring where you begin \Delta
> X=0 with a stretched spring.  (Be aware that springs which have no
> gap between the coils  may actually be stretched from their real zero
> force lengths. That's why I start with a slightly stretched spring.)
>
> When they figure out that part of the spring is oscillating, then you
> can set m = m_bob + m_unknown, take several data points, plot
> period^2 versus m_bob. The ratio of the T^2 intercept to the slope
> yields the contribution of the spring's mass, m_unknown.

That's a better way to do the analysis.
I modified my spreadsheet to do it that way.
  https://www.av8n.com/physics/measure-k-oscillator.xls

> I don't think this is too hard for HS students and it gives a good
> opportunity for modeling an unknown and then extracting it.  It's
> also an example of "linearizing" a system which is a good first order
> tool.

Shifting now to the college level, if you do the data 
analysis properly, this is tremendously instructive
example.

Doing it right means calculating the covariance matrix.
I do this at the bottom of my spreadsheet.  It is not 
particularly hard to do, if you know the tricks, as
outlined here:
  https://www.av8n.com/physics/nonlinear-least-squares.htm
Here we are doing linear least squares, but the ideas are
the same, so don't obsess over the name of that document.

In this case, as usual, it really pays to do the singular
value decomposition (SVD) of the covariance matrix.  It
turns out that the two fitted parameters (extra mass and
k) are highly correlated.  The σ^2 for the sum of them is
30 times larger than the σ^2 for the difference, roughly
speaking.  (The mixing angle is not quite 45 degrees,
but you get the idea.)

> From this we learn that you can get a much more accurate
value for k if you determine the effective mass /without/
fitting and then do a one-parameter fit to extract k.

  In contrast, if you do a two-parameter fit to extract
  both quantities, the uncertainty in one pollutes the
  other, to a remarkably large extent.  The model has a
  horrifying "mechanical disadvantage".

Note that even if the raw data is uncorrelated and IID,
it is extremely common to find that the fitted parameters
are correlated in nasty ways.

Bottom line:  For two or three parameters, doing the SVD
is easier than you might have guessed ... and is tremendously
informative.