Re: [Phys-L] determine k

[John Denker <jsd@av8n.com>]

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

Let's imagine that this started out as a nonlinear problem, 
with uniform spread on the distribution of measured period
values.

By squaring the period, we linearize the problem, which is
good, but now there is a non-uniform distribution on the
squared readings.  We have successfully traded a large-ish
problem for a much smaller problem.

  a) In the introductory course, we could pretend we didn't
   notice the non-uniformity ... but in any other setting, 
   it would be quite unprofessional to do an unweighted fit.

  b) One downside is that the weights increase the cognitive
   workload.

  c) Another downside is that typical muggle-oriented curve
   fitting tools, such as the linest() spreadsheet function,
   do not take weighting into account.

  d) However, we're in luck, because it is ridiculously
   easy to trick the linest() function into performing a
   weighted fit.  At the ith point, just multiply the
   observed data Y(i) and each of the basis functions F(i)
   by a factor of 1/σ(i).  For details, see
     https://www.av8n.com/physics/linear-least-squares.htm#sec-weighting

In my spreadsheet,
  https://www.av8n.com/physics/measure-k-oscillator.xls
the first sheet does an unweighted fit, just to keep 
things simple.  The second sheet does a weighted fit.