So, if a 2 inch wide board with square edges is cut to the curve of a circle whose radius is 20" what will be the depth of material removed from the edges of the board. Thanks
That curve could go in any of an infinite number of places, so there is no determinate answer. (Assuming I’ve understood “depth of material” correctly.)
-FrL-
If I understand the question properly…
Sqrt (20[sup]2[/sup] + 1[sup]2[/sup]) - 20
= Sqrt (401) - 20
= 0.024984 inches, approximately, (call it 0.025") milled off each corner of the board, assuming the center of the edge of the board will be touching the circle at a single point, so the edge of the uncut board forms a tangent.
If you want me to show my work… well, that’s a bit tricky.
Expanding on this, if the board is 2" x 40" (the diameter of the circle), shaving a mere fortieth of an inch off the four corners will allow the board to fit inside the circle. Of course, the board is three-dimensional and the circle is two-dimensional, so if there’s more to this question, it’ll need a better description.
It’s not a significant difference from Bryan’s, but I begin with the fact that since we’re assuming a pie-shaped wedge with a 20" radius, we should be going with 20"- sqrt (20[sup]2[/sup] - 1[sup]2[/sup]) , or 20" - sqrt (399) = 0.02501564456.
About a fortieth of an inch milled off the corners.
Here’s another approach:
I’m going to assume that you have access to a scientific calculator *(the trigonometric functions are your friends here).
You begin with an isosceles triangle, the base of which is 2", and the sides are 20". If you calculate the height of the triangle, and subtract that from 20" you have the depth of your arc.
1 / 20 is the cosine of the base angle; so if you enter 0.05 and do an inverse cosine function, you have the actual value of the base angle 87[sup]o[/sup] 8’ 2.458".
Now, if you calculate the sine of that angle, you can multiply that back by 20 (or divide by 0.05, it’s the same thing) to get the height of your triangle (~19.975"). Subtract that from 20" and you have 0.025", or about 1/40th of an inch, just like Bryan Ekers said.
I think that makes about 5[sup]o[/sup] 43’ 55.1" of arc at the milled end.
*Most Windows operating systems I’ve encountered have a calculator in the accessories menu.
Ok, you guys have lost me. Let me try again with a visual. I have a guitar with a 2" wide, flat neck that I want to reshape to match my favorite. I double checked said favorite and it turn out to be a 12" radius, not a 20". So, how much material do I need to take off the ends to have a curve the radius of which would be 12" Thanks again; your earlier answers are probably right but I can’t follow the steps so I want to make sure we’re on the same page.
Can’t you just use a pencil to draw the appropriate curve onto the neck, then cut along the line?
I am still not clear what you’re asking, I guess.
-FrL-
Where are you measuring this 12" from?
I’m going to guess that regardless of how you cut something, you’re just not going to get something 2" wide to fit a 12" radius circle.
I’m sorry for the confusion, it seems so obvious to me, but I’m looking at the item. Guitar fingerboards are often shaped with a curve to facilitate fingering. The arc of this curve is expressed as the radius of an imaginary circle, the larger the radius, the shallower the curve. I want to shape a flat fingerboard to match the curve on my favorite guitar, since the fingerboard is a thin strip of hardwood glued to the neck, I worry about taking off too much material. Thus, if a flat 2" board with square edges is shaped to the curve of a circle with a radius of 12", how much material will have to come off the edges, thanks
Oh, OK. I was picturing you shaping the end of the neck. Of course the calculation would be the same, just the imaginary center of the circle shifts.
By my calculations, the edges will wind up shaved down by an amount equal to:
12 - sqrt(12[sup]2[/sup] - 1[sup]2[/sup])
This is approximately 0.042"
So you have a 2" square inside a 12" circle, but with the outer edge of the square tangent to the circle? The the question is what is the distance from the outer corners of the square that the circle intersects it at.
Don’t know the answer, just clarifying the problem. Maybe there’s a guitar manufacture/remodeling site out there that has the answer for a standard guitar.
I know nothing about guitars, but you’re talking about taking a 2" wide piece of wood and curving it in a manner similar to a parenthesis i.e. ( ?
If so, why not place the wood with its ends on pieces of cinderblock and put some bricks in the middle, weighting it down so it curves. Dump water on the wood now and then and sooner or later it’ll warp into the shape you want.
I believe he wants to crown the surface of the neck of the guitar to get his fingers around it more easily that he could a flat playing area. The board would stay flat but the long edges be planed/sanded to get them slightly curved.
Hey, outlierrn, shaving down a fretboard to change the radius is not for the casual dabbler. :eek: If you want to change the radius you probably need to go to a luthier and have them remove the fretboard, curve the neck, and install a new fretboard and frets. Without seeing the guitar I am guessing the fretboard is probably too thin to shave it down to that radius from flat. And shaving the wood down won’t matter if the frets stay flat–you will have to remove the frets and recut the slots and replace them, hammering them into the new radius (although I’m guessing you’re planning to remove them anyway to plane down the fretboard).
This is not a DIY job unless you have lots of experience building necks or you have a really cheap guitar so it won’t break your heart if you ruin it.
Short lesson on guitar neck radius
The neck is about 2" wide (that sounds like a narrow neck to me but I digress), and on this guitar the surface of the neck (the part where your fingers merrily dance) is flat. Most guitars have a fingerboard that is slightly curved so as to be a section of the surface of a cylinder with its axis parallel to the length of the neck. The radius is the distance from the surface to the axis, and is the spec used to define the curvature. The larger the radius, the less the curvature. (You are talking about warping of the neck to make it a curved piece of wood; this is what we call “relief”.)
Without checking through the maths, I just popped in to say that the dimension requested is properly called the sagitta. (from the latin word for bow) Knowing that might help with googling.
Here is a diagram in a Word document to show the problem. The neck is shown in cross section. The neck radius is in red. Diagram is not to scale. We need to solve for x.
Actually you could just draw this to scale and measure off the answer. That would be quicker than trying to remember high school trig. 
What’s the 2" measure? Don’t we need to know both length and thickness of this board in order to tell you “how much” (by which I guess you mean “what volume?”) wood has to come off?
-FrL-
I think it’s now reasonably clear that what’s sought is a dimension, not a volume. I believe CookingWithGas’s diagram shows the problem.