# Doing calculus with a bandsaw



## mklotz (May 14, 2008)

I have a program on my webpage (DIPSTICK) which can be used to calculate the amount of fluid in tanks of various shapes from a dipstick reading. One of our British cousins posed the problem of a cylindrical tank tilted slightly from horizontal (to allow sediment to collect in the lowest area) and wished to have a dipstick program for such a tank.

To write this program, it was necessary to generate a formula for the volume of an ungula.
If you put a small amount of water in a cylindrical water glass and then tilt the glass so the water no longer completely cover the bottom of the glass, the shape assumed by the water is termed an ungula. Ungula is the Latin term for a cow's hoof (you may recognize Ungulata as the class term for cattle) and the shape looks something like a hoof, hence the strange name.

I did the calculus to derive the formula and came up with an expression* that's a bit on the complex side. The trick now is to verify if this formula is indeed correct.

I took a cylindrical piece of aluminum and bandsawed an ungula from it. You can see it on the left in the photo below. I also cut a cylinder from the same rod.







Using a jeweler's scale (handy shop item), I could determine the weight of the cylinder and, by measuring it, calculate it's volume. Given the weight and volume, it's simple to find the density of the aluminum (density = weight/volume).

I can then weigh the ungula and use the calculated density to calculate its volume (volume = weight/density). It came out to 0.0916 in^3.

Now, I could measure the ungula, plug these measurements into my formula, and see what the calculated volume was. The formula yielded a volume of 0.0910 in^3.

With an error of less than 0.1%, I called it good and pronounced my formula verified. The program was written and dispatched to the requestor, who, being an ungracious berk, never confirmed receipt or use of it.

--------------------------
* if
b = width of ungula base at widest point
h = height of ungula
r = ungula radius
V = ungula volume

then:

a = sqrt(2*b*r - b^2)
phi = pi/2 + arctan[(b-r)/a]
V = h*r^2*[3*sin(phi) - 3*phi*cos(phi) - sin^3(phi)]/[3*(1-cos(phi)]


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## DickDastardly40 (May 14, 2008)

What an ungrateful git! Maybe he thought it was a formula you just happened to have lying around (like you do!)


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## Divided He ad (May 14, 2008)

Marv,

 As you are probably aware this is mostly over my head ??? But that said, the bit that certainly is not is the bit we should have all been brought up with ... The Ability to thank someone!! 

We are not all like that I assure you, Most of us were raised correctly. ( I add please and thank you onto the end of strangers requests etc in shops and pubs... Gets me some odd looks but it does annoy me when people are ignorant!)


Ralph.


P.S. I really like the new word of the day... 'ungula' will now be inserted into various conversations for no other reason than to sound smarter than I was before I read this post!! :big: :big: Thank you. ;D


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## mklotz (May 14, 2008)

Yeah, Ralph, armed with "ungula" and "sagitta" the birds at the local pub will be all over you.  Tell them that your job is searching out sesquipedalia (which also comes from Latin and translates literally to a word that is a foot and a half long).

Of course, being British, I'm sure that you, like the major general in Penzance, "know precisely what is meant by commissariat". 

Please note, guys, that I'm not picking on the Brits. Lord knows I've had plenty of Americans do the same thing even when I've asked for feedback to confirm that the program works properly.


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## Lew Hartswick (May 14, 2008)

And I always thought Bill Buckley used a lot of big words. 
  ...lew...


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## BobWarfield (May 14, 2008)

I thought the ungula was that odd looking thing that hangs down in the back of your throat!

(just kidding)

An interesting derivation. I'm more of the empirical school, so I was glad to see the formula was verified!

Best,

BW


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## Divided He ad (May 15, 2008)

Now your just showing off Marv   :big: 

I know a little and it gets me by! now I know a little more and it will help me a little further! ;D

Am I correct in figuring that the use of 'sagitta' in this instance would be the trigonometric version not the astrological? 



Ralph. 


P.S. don't you just love to google everything you don't have a clue about!! :big:


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## tel (May 15, 2008)

Being a mathematical cripple, I'm completely Ungulated by all that.


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## andrewh (May 15, 2008)

Of no use, but possibly of interest, is the fact that ungula is Latin for a nail (as in fingernail)

Hence the horse family are the Monugulates since they run around on one fingernail.

Every day, if we're not careful we learn a new thing
andrew


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## mklotz (May 15, 2008)

Divided He ad  said:
			
		

> Am I correct in figuring that the use of 'sagitta' in this instance would be the trigonometric version not the astrological?



Yes, Ralph, the trigonometric usage. If you draw the radius that bisects a chord, the portion of that line between the chord and the circle is known as the sagitta. And now, for extra credit points, what is the name for the other portion of that line - the part that extends from the circle center to the chord?


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## gilessim (May 15, 2008)

I just noticed this post, interestingly enough, since we're getting into Latin ,Sagitta is a Latin word for arrow or arrow head as in "Sagittarius"(the archer) ,also used by masons as the name for a keystone in an arch and by calligraphers as the term to describe the downstroke of the letter "Y", it translates into modern Italian as "saetta", also meaning arrowhead but with the added meanings of; lightning strike, Gnomon ( the bit that casts the shadow on a sundial)and "fast as lightning" etc.

I'm more of an etymologist than a mathematician so I can't answer Marv's question, without looking it up, but that's just my 10p's worth if anyones interested!

Giles


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## derekm (May 15, 2008)

mklotz  said:
			
		

> Yes, Ralph, the trigonometric usage. If you draw the radius that bisects a chord, the portion of that line between the chord and the circle is known as the sagitta. And now, for extra credit points, what is the name for the other portion of that line - the part that extends from the circle center to the chord?



apothem


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## mklotz (May 15, 2008)

Very good, Derek. It's satisfying to see that there were a few people who didn't sleep through geometry class.

Your prize, a years supply of round tuits, will be in the mail soon.


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## tel (May 15, 2008)

Aw Gee! I'm in more need of Round Tuits than 'im.


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## Divided He ad (May 15, 2008)

Hey That's not fair!!!  

That was my Question. I was out all day working my --- off earning a crust!! :

Still, used ungula and ungulata more than once ;D 


Ralph.


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## CrewCab (May 15, 2008)

I wouldn't worry Ralph  : , if your anything like me you've probably got enough "round tuits" already ;D ............... in fact I'm sure I've got enough so send Tel a few months supply .............. at some stage  

Dave


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## tel (May 16, 2008)

.... like when you get a round tuit?


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## georgeseal (May 16, 2008)

:big:

Well guys this was a very educational post with a lot of humor and no back stabbing. That's one reason I like this site


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## zeeprogrammer (Jul 1, 2009)

Trying to learn a bit about bandsaws and came across this thread.
Sheesh. Should I worry?
I think I'll start a thread.


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## Cedge (Jul 1, 2009)

Hmmm... 
I thought the apothem was the name for the only natural born marsupial found in in the US....(grin)

Steve


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## kvom (Jul 2, 2009)

I like the bandsaw method of verifying a mathematical formula.  ;D

Assuming that the bottom is completely covered, then I believe a dipstick measurement in the center of the cylinder would give the same value for any angle of tilt.


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## mklotz (Jul 2, 2009)

kvom  said:
			
		

> Assuming that the bottom is completely covered, then I believe a dipstick measurement in the center of the cylinder would give the same value for any angle of tilt.



That might be true if the axis of the cylinder were vertical but in this case it was tilted slightly from horizontal. Moreover, the port where the dipstick enters is at the high (i.e., tilted up) end of the tank. While admittedly practical, it makes for some nightmarish mathematics.


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## radfordc (Jul 2, 2009)

mklotz  said:
			
		

> That might be true if the axis of the cylinder were vertical but in this case it was tilted slightly from horizontal. Moreover, the port where the dipstick enters is at the high (i.e., tilted up) end of the tank. While admittedly practical, it makes for some nightmarish mathematics.



Marv, are you saying that kvom's statement is not true? I did a quick drawing of an open top cylinder half full of liquid with the dip stick in the exact center of the cylinder. No matter what angle you tilt the cylinder the level of the liquid seems to stay at the same point on the dip stick...assuming the angle of tilt isn't so extreme as to spill the liquid.

Charlie


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## mklotz (Jul 2, 2009)

Perhaps I'm misunderstanding what you folks are suggesting but, just to be sure, take a look at the attached sketch and see if we're all visualizing the same thing.

Edit: Note that, in the sketch, I've exaggerated the tilt of the tank. Offhand, I don't remember the exact angle but it was only a few degrees.


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## Foozer (Jul 2, 2009)

mklotz  said:
			
		

> Perhaps I'm misunderstanding what you folks are suggesting but, just to be sure, take a look at the attached sketch and see if we're all visualizing the same thing.



As long as the subject is on

Not that any one would have a tank setup as your dwg depicts, but for S&G wouldn't the surface tension of different liquids affect the actual dipstick reading. Its picking salt out of the sugar shaker for sure, just curious if their is a (cant explain it properly) table of surface tension height difference against a known standard that could be applied to the basic formula that would increase its accuracy over a variety of liquids measured.

Silly experiment with water and dipstick, the water creeps up the stick a bit, a drop of soap in the water breaks the surface tension and the creep also altered. Same amount of water but two different readings.

I'll go back to my room now

Robert


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## mklotz (Jul 2, 2009)

Foozer,

This was a big tank on a farm. I can't imagine anybody who is worried about minor inaccuracies due to surface tension measuring liquid levels with a dipstick. 

I did the problem for the guy because it was an interesting math problem.

The real point of the thread was to introduce the concept of using metalworking to verify a derived formula and the idea of finding the volume of some odd shape by weighing it - a more accurate technique than the old practice of measuring the amount of water it displaces when immersed.


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## radfordc (Jul 2, 2009)

mklotz  said:
			
		

> Perhaps I'm misunderstanding what you folks are suggesting but, just to be sure, take a look at the attached sketch and see if we're all visualizing the same thing.
> 
> Edit: Note that, in the sketch, I've exaggerated the tilt of the tank. Offhand, I don't remember the exact angle but it was only a few degrees.



I was visualizing something completely different.

Charlie


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## kvom (Jul 2, 2009)

My concept was the same as radford's.

While I'm OT, I can recall doing a contract project 20+ years ago involving the large petroleum storage tanks where tanker trucks fill up. Each of these tanks is "surveyed" to give a volume for each 1/16" of depth. The bottom of the tanks are usually conical so that any water in the tank will settle, and can be drained off from a valve in the lowest point.

Typically the contents of the tanks would be adjusted each time a truck filled up or fuel was delivered via pipeline or barge. Then each night the operators would record the reading on a float valve indicator. This was accurate enough for daily use to detect any discrepancies, However, once a month (usually midnight on the 1st) the operators would have to climb to the top of the tower carrying a tape measure and a plumb bob. He would coat the tape with a couple of inches of a clay-like substance (using the gauge value), and carefully lower the plumb bob through an access hatch at the top. When he felt it hit bottom, he would pull up the tape. The petroleum would have made a very accurate mark in the clay, and with this depth measuremant and the tank survey data, the volume in the tank could be determined.

I had the opportunity to climb up a tank and watch the measurement being done. It was summer in Atlanta, so no problem. But they also have to do it in mid-winter where it's cold and those steps are icy.


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## zeeprogrammer (Jul 2, 2009)

zeeprogrammer  said:
			
		

> Trying to learn a bit about bandsaws and came across this thread.
> Sheesh. Should I worry?



Yes. :big:


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## mklotz (Jul 2, 2009)

Hmmm, responding to your own posts. Isn't that a sign of insanity?

Although I must admit that anybody who can meaningfully combine 'dipstick', 'calculus', 'ungula' and 'bandsaw' in one thread probably needs some form of counseling as well.

Two sure ways to break blades on one of these saws.

Cutting thin stock across the thin dimension... If you want to cut a piece off some 3/16 x 1" stock you cut across the 1" dimension, not the 3/16.

Part not held rigidly (there's that rigid thing again). The saw blade exerts a surprising amount of force parallel to the blade. If the part is pulled crooked (to the cut) in the vise, the blade will bind and break. As on the mill, it's a good idea after clamping the part to grab it and, using all the force you can muster, try to break it free.


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