Fundies Math

[Rick Inman]

A very simple example of dimensional analysis that only involves length would be a conversion from 1.2 mile to xxxx centimeters.

You just setup the equation as
1.2 miles * 5280 ft/mile * 12 inches/ft * 2.54cm/inch = ______ cm.

When you go through and match up units in the dividends and the divisors, the miles, feet, and inches should all cancel out, leaving only the cm. If it doesn't work out that way, then you haven't set up the equation right.

Dimensional analysis is just the fancy way of saying "when you cancel out units, you ought to have the right ones left over".

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Dimensional analysis can often provide some nice insights.

For example, it is easy to show that the real units of SAC aren't cfm or lpm, but instead are truly cu ft per minute per ata. It just happens that at 1ata the last part drops out. If you include the /ata at the end you can easily see for example that
1 cu ft/min/1ata = 2 cu ft/min/2ata.

Oh.... my..... GOD!!!! I actually understand your explaination.


I'm becoming a nerd....:umnik: :read:

 

[amascuba]

I'm a nerd....:umnik: :read:

There, that's better.

 

[lamont]

Oh.... my..... GOD!!!! I actually understand your explaination.

now what are the units on G?

F = G m1 m2 / r^2

[ newtons] = G [ kg ] [ kg ] / [m]^2

 

[dsteding]

For example, it is easy to show that the real units of SAC aren't cfm or lpm, but instead are truly cu ft per minute per ata. It just happens that at 1ata the last part drops out. If you include the /ata at the end you can easily see for example that
1 cu ft/min/1ata = 2 cu ft/min/2ata.

This is why I think of things in ATA/minutes. From there, gas consumption calculations get easy:

60 fsw for 50 minutes equals 150 ata/minutes. If my SAC is .75cf/min/ata (mine is about 0.6-0.7 but I still use 0.75 to be conservative), I know that three quarters of my ATA minutes is the gas I'll need (in this case, about 112 cubic feet).

A couple other handy items are PSI per 5 minutes at 100 and 60 feet, having those numbers in the back of your head for the tanks you are using (250 psi and 200 psi respectively for my double 100s with a SAC at 0.75) allows you to do gas planning on the fly if need be.

 

[Phil K.]

60 fsw for 50 minutes equals 150 ata/minutes. If my SAC is .75cf/min/ata (mine is about 0.6-0.7 but I still use 0.75 to be conservative), I know that three quarters of my ATA minutes is the gas I'll need (in this case, about 112 cubic feet).

Totally agree with these formulas. However, your example ignores a basic math skill regarding "significant figures." 60 fsw (a measurement with 2 significant figures) is approximately 2.8 ATA (a measurement with 2 significant figures). 2.8 ATA for 50 minutes (a measurement with 2 significant figures) = 140 ata/minutes (a measurement with 2 significant figures). The gas consumed is determined by mutliplying 140 by .75 (a measurement with 2 significant figures) = 105 cf

Your example rounds up 2.8 ATA to 3.0, which replaces a measurement with 2 significant figures with a measurement that has 1 signficant figure and lowers the precision of the result. 112 is more than 6% greater than 105 and is based on an error in the measured depth of 10% (60 fsw instead of 66 fsw). What if our dive was to 3.1 ATAs or 70 fsw. Shall we round up to 4 ATAs and base our calcs on a dive to 100 fsw? The discrepancy here is too great to just ignore as imbedded conservatism. Gas planning should be more precise than that.

There's no need to invent some protocol like rounding up to the nearest 25 hundredths. If were focused on precision, mathematics already gives us one. Keep the number of significant figures the same. If you like .75 as your SAC then use ATAs to two significant figures.

For those who think computing ATAs is mentally challenging, try converting to meters first. (# fsw/10)*3. Then divide the result by 10 and add 1. Voila ... ATAs.

So 60 fsw = (60/10)*3 = 18 meters. 18 meters = (18/10) +1 = 2.8 ATAs.

 

[Rick Inman]

Oh.... my..... GOD!!!! I actually understand your explaination.

now what are the units on G?

F = G m1 m2 / r^2

[ newtons] = G [ kg ] [ kg ] / [m]^2

...oh, well...:sad:

 

[Charlie99]

This is why I think of things in ATA/minutes.

Ooops, you forgot to do your dimensional analysis. :banana:

It's really ata-minutes (both in the numerator).

I kind of prefer ata-minutes because you can figure out the ata-minutes of a profile, ascent, or stops and then quickly and easily adjust it for different SAC assumptions.

For quick and dirty gas usage estimates all I need to remember is that I'll be somewhere between 15psi/min/ata (loafing, drift dive, or just slowly puttering around a reef) or 20psi/min/ata (1kt transit) when using an AL80. If diving with someone who doesn't know his SAC, I'll use 30psi/min/ata, which corresponds to about 0.75cfm.

 

[Charlie99]

Totally agree with these formulas. However, your example ignores a basic math skill regarding "significant figures." ........

Your example rounds up 2.8 ATA to 3.0, which replaces a measurement with 2 significant figures with a measurement that has 1 signficant figure and lowers the precision of the result. 112 is more than 6% greater than 105 and is based on an error in the measured depth of 10% (60 fsw instead of 66 fsw).

SAC's easily move back and forth +/-10%, or even 20%. IMO this sort of rounding is reasonable considering that it is a rough plan than can be changed as the dive goes along.

For those who think computing ATAs is mentally challenging, try converting to meters first. (# fsw/10)*3. Then divide the result by 10 and add 1. Voila ... ATAs.

So 60 fsw = (60/10)*3 = 18 meters. 18 meters = (18/10) +1 = 2.8 ATAs.

I sometimes describe my conversion from feet to meters as "go in the WRONG direction by multiplying by 3; then drop a zero". Same as your equation, just phrased differently. And obviously, to get from feet to ata, multiply by 3, and then drop 2 zeros or divide by 100.

Going from meters to feet (at least in recreational depths) I find it easiest to just work from 3m=10' and figure out how many 3 meters there are. 18 meters is obviously 6 "3 meters", so therefor is 60'.

Even 10's of meters are even integer ata's, which we have all memorized. 20m = 2 ata = 66'.

 

[dsteding]

Totally agree with these formulas. However, your example ignores a basic math skill regarding "significant figures."

Err, with a Ph.D. in geochemistry I better understand significant figures. Of course, they are constrained by the precision of the measurement (here that is arguably your SAC, and given the slop in measuring that I'd propose that at best you can know your SAC to one significant figure anyways). I prefer working with even numbers for quick calculations . . . but I recall that you've been through this with Rjack at length before, so let's agree to disagree.

Charlie, you got me on dimensional analysis, I wrote that, thought about it and said f-it, people will get the idea.

 

[SparticleBrane]

I don't know about that.
I said "ATA per minutes?! ...what the heck? "