Re: [Phys-L] determine k

[John Denker <jsd@av8n.com>]

On 02/10/2015 11:05 AM, Paul Lulai wrote:
> This brings things back full circle with uncertainties.  

OK.

> In the past, the 3 crank method has been mentioned. 

OK, good.

> On some days it is ok and other days it is unacceptable.

Huh?  Where's that from?  In tricky situations 3 is not 
enough, but the idea behind crank-three-times is not 
wrong, and the idea will never have to be unlearned.

>  Stating measurements or raw data have no uncertainty may be the most accurate. 
>  But, I don't see how these conditions (below) can be met at the high school
> level or college freshmen level and still teach physics.

> The lab problem is fairly straight forward.  
> Determine k & compare.  
> I can't really spend more than 3 days in lab for determining k.
> Roughly 1 day of prelab, 1 day of lab (data collection), 1 day post
> lab. That would be a lot for this lab. Kids that haven't had
> statistics (and that is not a problem that will change within this
> timeline). 

OK.

> I like Joe's method. 

a) I don't have a problem with that as is.

b) It is even better as part of a /slightly/ larger effort;
 see below.

> It is something that is understandable by first exposure students. It
> includes some method of addressing variances in data and the fact
> that two different k values could be the same k value and possibly
> the same spring (2 groups doing this at different times with the same
> spring), or not.

Right.  In the simplest case, each particular student 
measures the same mass 5 times.  This makes it easy to 
compute mean and standard deviation.  So far so good.
Easy peasy.  The only downside is that it provides no 
handle on the systematic error.  This is however a 
fixable problem;  see below.

Conversely, if each student measures 5 different masses,
it is easy to spot the systematic error, but hard to
calculate the mean and standard deviation.  Presumably 
they learned in HS chemistry how to draw a straight 
line through a bunch of data points, but that gives 
them only the best-fit parameters, without much of 
an estimate of the associated uncertainties.

> Is there a way to do a spring constant lab in 2 or 3 50-minute
> periods while stating the data are exact (but can account for two
> different k values possibly being the same k)?

The data is the data.  The data may be drawn from some
/distribution/ that has some width.  As always, the width
is a property of the distribution, not of any particular
data point that may have been drawn from the distribution.

Also the data may be subject to systematic error, which
cannot be detected by statistical methods.

> Is there a way to do a spring constant lab in 2 or 3 50-minute
> periods while stating the data measurements have uncertainty?

Again: the data is the data.  The data may be drawn from some
/distribution/ that has some width.  As always, the width
is a property of the distribution, not of any particular
data point.  Putting little error bars on the raw data
points is just wrong.

Also the data may be subject to systematic error, which
cannot be detected by statistical methods.

===========

Possibly constructive suggestion:  As I have said before,
I do not expect students in the introductory course to
carry out a sophisticated data reduction.  However, the
teacher can do it for them, so they can see what it looks
like.

How about this:  Each student measures the oscillator
five times, using the same mass ... but different students
have different masses.  Each student does a simple
calculation of mean and standard deviation.  Then the
teacher collects all 5N data points and plugs them
into a spreadsheet and does a real industrial-strength
least-squares fit, to produce best-fit k and best-fit
extra-mass correction.

Setting up the spreadsheet is a pile of work, but once
it is set up, plugging in the data is easy ... and the
output is a nice easy-to-interpret graph with a slope
and an intercept.

At the college level I would point out that there is
a nasty degree of correlation between the fitted k
and the fitted extra-mass parameter, which means the
idea of "error bars" is dead on arrival.  However at
the HS level that is asking too much.  I don't expect
them to know what a matrix is, much less a Hessian
matrix of second partial derivatives!