I need to elevate a slab 6° on one side and 6° on the adjacent (90°) side. Then I need to drill a hole straight down. However, for setup reasons, I want to elevate only the corner. I need to know what angle to elevate the corner.
Essentially, I need to know the angle formed by the intersection of two planes. Both with 6° elevation and with the planes 90° to each other.
Does anyone have the solution and possibly the derivation of the formula?
I'm not sure if I understand what you're trying to do, but here goes:
Assuming the 'original' methodology is: elevate one side (call it AB)of the slab, so that a line from it to the other side (call it CD) is 6 degrees above horizontal. then elevate side BC so that a line from _it_ to the opposite side (AD) is 6 degrees above horizontal.
And what you want is the angle above horizontal of the line BD. B is the high corner, and D is still at the original horizontal.
Taking things piecemeal, corner "A" is above the original horizontal plane by sin(6 degrees)*length of side AD. corner "B" is above corner A" by sin(6 degrees)*length of side AB. thus corner "B" is above the original horizontal plane by (sin(6 degrees)*length of side AD) + (sin(6 degrees)*length of side AB) And the distance from D to B is given by sqrt(AD*AD + AB*AB) So the angle is
(sin(6 deg)*len of side AD) + (sin(6 deg)*len of side AB) theta = arcsin( --------------------------------------------------------- ) sqrt(AD*AD + AB*AB)
if the slab is a _square_, things get a _lot_ simpler:
theta = arcsin( sqrt(2)*sin(6 deg) )
which is approximately: 8.5009361422462 degrees
Note: when dealing with small angles like this, one _can_ simply disregard the trig functions, and be accurate to within a fraction of a percent sqrt(2) * 6 degrees == 8.48+ degrees, which is within 0.25% of the precision answer, above.
I agree with Horatio's answer, if you elevate one edge (AB) and then elevate (BC) you essentially raise the height of the top corner twice and you get
8.5 degrees elevation along the diagonal. However, I interpreted the question slightly differently. If you raise AB by 6 degrees, then the high corner is at some height X. Now put the slab flat again, elevate BC by 6 degrees, and the height of the top corner is again at the same height X. In order for the high point to be at the same position for both cases, the slab must be a square. Not surprisingly, with my interpretation of the problem, the elevation along the diagonal is 1/2 of Horatio's answer. (4.24 degrees.)
Assuming you've got a square slab with corners 1,2,3 & 4 with corner 1 remaining on the original plane and you tilt corner 2 up 6 degrees then holding corners 1 & 2 at this position and elevate corner 3, diagonally accrossed from corner 1, up 6 degrees the resulting angle between corners 1 & 3, relative to the original plane is 8.5 degrees or close enough to 8.5 degrees
Anyone want to confirm my results so I don't feel too bad if Preston uses 8.5 degrees and I'm wrong. Had to remember how to convert degrees to radians and radians back to degrees.
At the risk of flogging a dead horse a few days late, here's an easier way to get the 4.24 degree answer:
Consider a square labeled like a baseball diamond. We want to raise the 2nd base corner while leaving home plate alone. From the original question, I assume that we want to tilt the square such that the 1st, 2nd and 3rd base corners are all raised to the same height as each other. If the distance from home plate to 1st base is L, then the distance from home plate to 2nd base is L*sqrt(2). Both 1st and 3rd base will be elevated by L*sin(6 degrees), while 2nd base will be elevated by L*sqrt(2)*sin(theta). We want to know what value of theta will make the heights the same (L*sin(6 degrees) = L*sqrt(2)*sin(theta)). This occurs for theta = arcsin[ sin(6 degrees) / sqrt(2) ] = 4.2388 degrees.
Re-reading my response I don't think I described the process correctly and what I was trying to say would most likely be mis-interpreted due to its poor explanation.
I was primarily trying to show an easier way to do the calculation that Horatio presented in the piece meal approach. There is also the problem of more than one interpretation of the problem. So here is a brief description of the easier calculation for either interpretation:
1) Horati's interpretation: the "top corner" is effectively raised twice as high as the corners that are defined by the 6 degree incline. If "L" is the length of the side of the square, this interpretation means that L*sqrt(2)*sin(theta) = 2*L*sin(6 degrees), where theta is the angle if inclination for the top corner and the length of the diagonal of the square is sqrt(2)*L. This gives theta = arcsin[ 2 * sin(6 degrees) / sqrt(2)] =
8.5009 degrees.
2) Michael's interpretation (which I described poorly) was that we wanted to raise the top corner the same amount as the other corners when they are raised to an inclination of 6 degrees. In this case, L*sqrt(2)*sin(theta) = L*sin(6 degrees), which occurs for theta = arcsin[ sin(6 degrees) / sqrt(2)] = 4.2388 degrees (as I described in the previous post).
I am afraid that my description in the original post implies that 1st, 2nd and 3rd base are raised to the same height at the same time; this would just raise the square, not tilt it.
I hope you have already figured this out, but here it goes. You need a center line for your slab. Mark your hole location and draw a line through the center of that hole perpendicular the slab center line. Since your angles are both six degrees your sight line will be a line that extends from the hole center to the slab center line at 45 degrees to the hole center line perpendicular. Here is the tricky part. You dont elevate a corner. You elevate the far side of the sight line. To get 6 degrees on both planes tilt your drill pres table to
8.5 degrees. Keep the sight line perpendicular to the table tilt and you will be fine. I got this information out of the back of Drew Langsner's book The Chair Makers Workshop. It has tables for all the possible rake and splay angles and their sight lines. It also has a better explanation of the process than I have given here.
HomeOwnersHub website is not affiliated with any of the manufacturers or service providers discussed here.
All logos and trade names are the property of their respective owners.