weekend math project

[reible]

Weekend project


Please read only if you like math and want some information about how far a cut is off when the angle is wrong.


As an example if you were to set your miter gauge to 90 degrees but the setting is off by 1 degree, you cut a 10 inch width of wood and put a square to it and the error shows up. How far off is the cut at the exit point of the blade?


Let's do the math. If we picture this as a triangle with the end nearest the miter gauge being an angle of 1 degree (call in angle “A”). If for the moment we think of the triangle being a right angle triangle we know that the length of cut should have been 10” or the width of the wood (this side we will call “b”. With a known angle and a known side we get to solve this using a=b*tan A. “a” is the length of he side that gives us the amount of error.


b = 10”
A = 1 degree


a = 10 * Tan A
a = 10 * .0175
a = .1746 (remember in fractions this is near 3/16” or .1875”)


Yea that is a major error when you are out 10”. So what happens if you are off by a half degree? The problem is the same except the Tan A is = .0087 and the error comes up as .087” and yea you might have guess it, that is near 3/32” or .0938”. So all the math majors already know what will happen when we have an error of only .1 degree. Tan of .1 is .0017 and the error is .017” or close to 1/64” or .0156”. You are now with in a “hairline” of perfect.


Now say we are cutting a piece that is only 3-1/2” wide. In this case the 1 degree error is .0611”, and recall that 1/16” = .0625”.... yea still bad. Just for completeness sake ½ degree error is .0305”, and .1 deg has an error of .0061”. [A sheet of paper is in the .004 range depending on the paper.]



The same sort of thing happens when you look at the saw table and the blade. If you have some 3/4” stock to cut up and the error is 1 degree. it is .0131” off, ½ degree error is .0065”, .1 degree error is .0013”.


Now I'm not going to draw and conclusions from this but I will give you one more board to cut, this time it is a 4'x8' sheet of plywood that you want to cut in half the 4' way. You have the miter set with only a .1 or 1/10 degree error... That should be close enough right??? OK for the last time in this post the math, and the answer is .0838” or a bit less then 3/32” (.0938”). How bad would it have been if you were off my 1 degree??, yea you can cheat, it is 10 times worse or .838!!!!!!


Ed

[charlese]

That math is interesting, but---- an offset of 1 degree on a miter gauge is pretty big, unless set on purpose.

I find an easy way to see if your meter gauge is set at zero is to do the following with scrap wood; Set the miter gage at 0 (90 degrees to the saw) or pretty close. Cross cut your piece. Flip one of the cut pieces front to back and put the two pieces together again at the new saw cut. Any offset of the miter gauge will be shown a gap in the mating edges. Assuming I flipped the right hand piece; if the gap is on the far side, I need to rotate the miter gauge clockwise by one half the angle of the gap. Don't need degrees here. Just lay one of the sawn pieces against the saw blade (remembering I want to rotate clockwise) putting the long part of the newly sawn edge to the right, and adjust the gauge to be flush with the sawn edge.

Try the whole thing again to see if your offset has been corrected. If not
make a second adjustment.

Now back to the math;

My offset crosscut produced a similar triangle as the 1 degree description above. This illustration points out that you can correct any offset by the long leg of the right angle.

Or if you wish, you can say that the offset is corrected by the angle subtended by the altitude of the isosceles triangle formed by putting the two pieces back together again.

[reible]

Well I see this is going some place other then where I thought it would.... well, I guess that's OK, go where you want with it.

I was kind of thinking more on the line of exceptable errors... is 1/10 degree off OK with everyone? Anyone settle for 1/2 degree off? Is 1/10 degree still way too much error?? Do you do projects where you have to be closer then 1/10 degree?? Now we are talking wood working here so there is a point of diminshed return for your efforts, right?

Ed

[dusty]

I have been accused of changing subjects within a thread and I don't want to do that (at least not intentionally). But here goes.

I don't know if eds' input was prompted by discussions about the Wixey angle indicator or not. The Wixey proclaims to be accurate to within 1/10 of a degree.

I attended a Traveling Academy where the instructor proclaimed that he attempted to do work accurate to within 1/32". He further proclaimed that 1/16" was not accurate enough for him in most of his personal projects.

This thread just sort of plays into both of those discussions. Whether deliberately or accidently I don't know. But having heard or read both statements myself, I have to ask.

If I need accuracy to 1/32", is the Wixey (accurate to .1 degree) of any value to me?

or asked another way

If I align the Shopsmith to within 1/10th of a degree, will my cuts be accurate to within 1/32 of an inch.

Am I making fine jewelry boxes? Am I doing door/window casings for the barn? Am I creating architectural wonders for my living room? Do the answers to those questions matter in this discussion?

Incidently, I know that if you are laying a 50 feet long rock wall and you set a line that is off by less than a 1/4 degree at the start you end up with an unsightly error to deal with at the end. Fractions of a degree do matter!

How much it matters is a function of each particular craftmans' pride.

[reible]

I have been accused of changing subjects within a thread and I don't want to do that (at least not intentionally). But here goes.

I don't know if eds' input was prompted by discussions about the Wixey angle indicator or not. The Wixey proclaims to be accurate to within 1/10 of a degree.

I attended a Traveling Academy where the instructor proclaimed that he attempted to do work accurate to within 1/32". He further proclaimed that 1/16" was not accurate enough for him in most of his personal projects.

This thread just sort of plays into both of those discussions. Whether deliberately or accidently I don't know. But having heard or read both statements myself, I have to ask.

If I need accuracy to 1/32", is the Wixey (accurate to .1 degree) of any value to me?

or asked another way

If I align the Shopsmith to within 1/10th of a degree, will my cuts be accurate to within 1/32 of an inch.

Am I making fine jewelry boxes? Am I doing door/window casings for the barn? Am I creating architectural wonders for my living room? Do the answers to those questions matter in this discussion?

Incidently, I know that if you are laying a 50 feet long rock wall and you set a line that is off by less than a 1/4 degree at the start you end up with an unsightly error to deal with at the end. Fractions of a degree do matter!

How much it matters is a function of each particular craftmans' pride.

OK I have to admit I was thinking about the Wixey when I decided to have a look at what a 1/10 deg. really looks like. Of course my mind never stays on one subject for all that long anymore.... I think it has been about a year since I first read about the Wixey... in another discussion group but not until now did I do the math.

I also wanted to see how much the distance or length of cut had to do with things. Things like if you were to use a miter gauge to cut 48", how far off is the cut then??? Like wise when ripping 3/4" stock what is that error.... Things like a 1/32" are relative when dealing with angles, the longer the cut the larger the error becomes.

I also ask myself about why a company makes a tool with say 1/10 deg readings... or the miter gauge that shopsmith sells that steps in 1/2 deg steps, what is the reasoning???

Ed

[dusty]

ed - you wrote:

I also ask myself about why a company makes a tool with say 1/10 deg readings... or the miter gauge that shopsmith sells that steps in 1/2 deg steps, what is the reasoning???

Do you mean why 1/10 degree rather than something different or just why 1/10 degree? Why stop at 1/10 degree accuracy, why not 1/100 degree or whatever? I think it all boils down to money. How much does it cost to engineer and how much demand will there be in the chosen market.

If you give me a digital miter gauge that is accurate to 1/100 of a degree I will gratefully (and eagerly) accept. If you want me to pay the going price for that degree of accuracy - I don't need it. The incra fence sort of falls in this category. I am impressed but I don't do that type of woodwork. If I did, I would probably have an Incra fence - or if you have one you want to give me.

[charlese]

I am amused by the claims of quite a few manufacturers about machine and device accuracy to .001" or 10th of degrees. What really matters is how the woodworker produces projects, or project parts. Better yet - What does the finished project look like.

To me it is hilarious to measure angles of a miter gauge in 10ths of a degree. I suppose I can understand half of a degree. It is 30 minutes! On the other hand 1/10 of a degree would be 60 minutes divided by 10, or 6 minutes. Who would measure an angle of 6 minutes or multiples thereof? If measuring angles by mills (as the artillery) then 1/10 of a mill would make more sense. But I don't think even Wixey can achieve that accuracy. (1/10 of a mill would be 1/64000 of a circle. while 1/10 of a degree would only be 1/3600 of a circle. If we would all switch to mills rather than degrees just think! One mill is already a smaller angle than 1/10 of a degree. (An angle of 22.5 degrees would be exactly the same as 400 mills)

These minute angles are all quite useless anyway. The previous paragraph should illustrate this fact. A degree is only the measurement of an angle. As the legs of an angle are lengthened the distance between the two widens. DUHH! If firing an artillery shell 20 miles a small error in azimuth or elevation can make a big difference. No wonder they use mills (6400/circle). If one is cutting a board degrees make relatively little difference. That's why I posted above (the 2nd post). For example, if the table saw blade is sloped 1/2 degree and the jointer is also sloped a small amount these slopes will make absolutely no difference if those boards were to be joined. The angles would offset each other to yield a perfectly flat joint.

Yes, it should be obvious that square cutting saws and jointers are the goal, and squared boards are usually the goal. However,what really matters is how does the project fit together and what does the finished project look like. The measurements are trivial! To say that 1/64" is my personal tolerance is also almost useless. A cabinet door may be off as much as 1/8" corner to corner. That door, may work out just fine if it is an overlapping door and the corner measurements are large. On the other hand an insert drawer must be very close to square and so does the cabinet drawer opening. 1/8" off doesn't make it here. Neither does 1/32" or 1/64th inch. To look right the insert drawer has to fit perfectly.

Measurements, whether they be 1/1000" or 1/10 degree are meaningless by themselves, except they are both pretty small. When you consider that huge rockets that lift the space shuttles diameter was ruled by the width of the width of two royal war horses -Here's the story in a couple of sentences - (the width of the booster rockets was limited by the wheelbase of railroad cars as the rockets had to be transported by rail from Utah to Florida). What determined the width of rail road's tracks? Roman chariots, then wagons, then railroads were all built to fit the tracks of the previous. All these things were built to the size to fit two horses rear ends. The measurements just got carried on from the old Roman days, through wagons to rail roads. (Curious? It works out to be 4ft. 8 inches) No rulers, throughout the ages - just story sticks. I seriously doubt if these wagon wrights used anything that measured in tenths of degrees or thousands of inches. If you want this true story in detail see the January 2000 issue of the Buggy Builders Bulletin or get a copy of Jim Tolpin's "Woodworking Wit and Wisdom from Popular Woodworking books.

[tchwrtr55]

It strikes me there could be an engineer, or two, here somewhere. I deal with engineers on a daily basis. I run into this a few years back somewhere. If there are no engineers, thats okay, hope you enjoy this anyway. If there are engineers, I hope you are not offended.

A Definition of an Engineer
An engineer is a person who passes as an exacting expert on the basis of being able to turn out with prolific fortitude of infinite strings of incomprehensible formulae, calculated with micromatic precision from vague assumptions which were based on debatable figures taken from inconclusive experiments carried out with instruments of problematical accuracy by persons of doubtful reliability for the avowed purpose of annoying and confounding a hopelessly chimerical group of practical and mechanically minded and experienced personnel who are referred to most frequently as “The Shop,” that group of people who the engineer feels “cannot even speak this language.”

[charlese]

Your definition of an engineer is perfect! Thanks! I'll have to see if my son the engineer has seen that one!

[reible]

"... Measurements, whether they be 1/1000" or 1/10 degree are meaningless by themselves, except they are both pretty small. When you consider that huge rockets that lift the space shuttles diameter was ruled by the width of the width of two royal war horses - Roman chariots, then wagons, then railroads. All these things were built to the size to fit two horses rear ends. The measurements just got carried on from the old Roman days, through wagons to rail roads. (Curious? It works out to be 4ft. 8 inches) No rulers, throughout the ages - just story sticks. I seriously doubt if these wagon wrights used anything that measured in tenths of degrees or thousands of inches. If you want this true story in detail see the January 2000 issue of the Buggy Builders Bulletin or get a copy of Jim Tolpin's "Woodworking Wit and Wisdom from Popular Woodworking books."

Of course you could read this:
http://www.snopes.com/history/american/gauge.htm

Perspective is interesting isn't it.

Ed