Re: [Phys-l] Significant figures -- again

["Quist, Oren" <Oren.Quist@SDSTATE.EDU>]

John,

I like your relativity problem, as the expansion leads directly to energy terms.

And, you don't like my problem because there is no mention of sound.

But, it is right out of Resnick & Halliday, chapter 20 on Sound Waves (problem #6 at the end of the chapter).  So, sound is a given.  I just did not mention this in my earlier post.  Obvious clarifications are necessary for other textbooks as you point out.

This topic was encountered in the first semester (or quarter) of a student's physics study.
     I used it as an assigned problem.
     I used it to demonstrate the folly of strictly following significant figures rules.
     I used it as an excellent example of the power of expanding something.
              Most students knew how to expand things, but few had seen any practical use of this.
     I used it to introduce the idea of iteration (converges in just two or three calculations).

But, keep up your comments and criticisms, they are appreciated.

Oren Q.

________________________________________
From: phys-l-bounces@carnot.physics.buffalo.edu [phys-l-bounces@carnot.physics.buffalo.edu] on behalf of John Denker [jsd@av8n.com]
Sent: Tuesday, April 10, 2012 7:28 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] Significant figures -- again

On 03/29/2012 04:29 PM, Quist, Oren wrote:
> One of my favorite problems, before I retired, was the "drop a stone
> down a well and listen for the splash."  Calculate the well depth
> from the time delay.   Following the normal sig figs rule eliminates
> the answer.  Students did not know what to do next!
>
> "Expand" the square root (from the quadratic formula).  This shows
> precisely what is happening.  The "big" terms cancel out - exactly -
> leaving the answer in the (quickly converging) series.  Sig figs
> takes on a whole new meaning in this problem after the expanding
> algebra is complete.

I'm collecting examples in that category.  Examples include
 *) The aforementioned stone-in-well.
 *) Relativistic KE, when the velocity is smallish.
 *) The pH of a dilute solution of strong acid, as discussed
  below.

If anybody has any other favorite examples, please let me know.

Here's the nicest example I can think of, namely finding the pH
of a moderately dilute solution of a strong acid.  The details
can be found at:
  http://www.av8n.com/physics/uncertainty.htm#sec-ph-quadratic

I like this example because:
 -- It is obviously a real-world situation.  There is nothing
  contrived about it.  Anybody who has taken high-school chemistry
  knows what pH is.
 -- The "pH versus concentration" curves are interesting, with
  some non-obvious but understandable behavior, as discussed at:
    http://www.av8n.com/physics/uncertainty.htm#fig-ph-scan
 -- The term that is in danger of being wiped out by roundoff
  is not some minor correction term;  it is the entire answer.
 -- The algebra to derive the key equation is reasonably simple.
 -- et cetera.

In contrast, I'm somewhat less happy with the stone-in-well example,
because it's not so clearly a practical real-world problem.  Specifically:
 -- The term that is in danger of getting wiped out due to roundoff
  is a small correction term, and
 -- It is hard to come up with a realistic reason to include the
  speed-of-sound correction but not the air-drag correction.

In the /simplified/ numerical pH calculation, sig-figs doctrine would
require rounding off to a single digit.
 *) One digit is not enough.
 *) Carrying three "extra" digits is not enough.
 *) Carrying ten "extra" digits is not enough.
 *) You need at least 12 digits to get the right answer with
  acceptable accuracy.

In the non-simplified (cubic) calculation, even IEEE floating point
(roughly 16 decimal digits) is not enough.

Situations like this come up all the time, in many different situations,
ranging from astronomy to zoology.

To put it bluntly, if you see a number such as

       ⎛   1.497925297894696 ... ⎞
  X =  ⎜ ± 0.01                  ⎟
       ⎝                         ⎠

you should not assume it is safe to round off.  It may be that the
number already has too /few/ digits.

On 03/12/2012 09:04 PM, John Mallinckrodt wrote:
> > And yet I'd still be flabbergasted.

Even still?
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