Re: Error Analysis

[finkenthl@gak.gat.com (Daniel F. Finkenthal)]

> What error analysis should Academic and AP Physics high school students
> be doing?  Thanks for any imput.

I think error analysis is not only very important, but also very insightful
into the world of laboratory science and the art of measurment in general.

I conduct a lab in which the students attempt to measure the free-fall
acceleration by timing the falling of a ball from a height of two meters.
First, however, I have the students measure their reaction time using a
falling ruler through the thumb and forefinger bit. This reaction time
represents a systematic error in any measure of time they intend to make. I
actually have them do five trials, take the average, and then calculate the
deviation of the mean:

dt = sqrt( ( <t^2> - <t>^2 ) /(N-1) )

The total uncertainty in a measured time, then, becomes

         dt_tot = dt_sys + dt_ran

where dt_sys = average reaction time, dt_ran = the deviation of the mean
calculated above.

Now,  they  measure the free fall acceleration five times by timing the
fall time and calculating using g=2h/t^2. The values, of course, are way
off published values of 9.8 m/s^2. But are they within the range of
experimental error? For this they have to think about error propagation. I
have them use just the strait sum, rather than the quadratic sum:

      Relative error in accelertion = dg/g + dt/t + dh/h + 2(dt/t)

where dg/g is the RANDOM error they get by measuring g five times:

        dg = sqrt( ( <g^2> - <g>^2 ) /(N-1) )

dh is the estimated relative error in the height, (about 1cm/200 cm) and we
use 2(dt/t) because the acceleration depends on t^2

By examining each of the contributions to the total uncertainty it becomes
clear that the measured time introduces the most uncertainty, since the
reaction time is very large compared to the small times being measured. We
get uncertainties of around 70%!

This motivates the students to find a better technique of measuring g, and
we do so using a pendulum in which the timing error is dispersed over a
large number of periods, and then using an electronic timing device in
which reaction time is eliminated.

This all sounds complicated but it really isnt once you go through it. It
really underscores the art involved in measurement, and what it actually
means to make a good measurement (reduce the uncertainty) Its not always so
obvious to students coming into the sciences that things arent always neat
and clean (and dry), and brings understanding about the pathologies of
error propogation...

I can give you a write up of my lab if you want...

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|      Dr. Daniel Finkenthal                   (619) 455-4664        |
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