Posted by admin | Posted in archery | Posted on 25-06-2010
Tags: 5 degree taper reamer, 7 degree taper reamer, 8 degree taper, degree taper, degree taper per foot calculator
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Tapered shafts and all the other wonders of trigonometry?
Machine shop calculations // A tapered shaft has a diameter of 5 centimeters at the small end and is 15 centimeters long. The taper is 3 degrees. Find the diameter d of the large end of the shaft.
FIGURE
there is a picture of a cylinder that is 15 cm long, on one side, the end is 5 cm in diameter. there is a line coming from the end and it shows an angle of 3 degrees. at the other end, there is a "d" variable and it shows a little more than 5 all the way around.
i'm sorry if this is vague.. it's fuzzy in my mind too..
any and all help is greatly appreciated.
thanks!
Consider a side on view of this tapered shaft. It will look like a cone with the top sliced off:
.....___
..../......
../..........
/_______
Diagram 1
You can really split this into a rectangle and two triangles:
........___
..../|..|......|....|
../..|..|......|....|..
/__|..|___|....|__
Diagram 2
What you want to do now is to find the length of the triangle's base (the part on the 'floor' in the diagram). You have been given the hypotenuse (15cm), and you' been given the angle at the top (3 degrees). In addition, you can see that this is a right angle.
So, solving for the base (the 'opposite' side to the known angle):
sin 3° = base / 15
base = 15 sin 3°
base = 0.785 cm
Now, looking at Diagram 2 again, you'll see that there are two triangles. Thus, the TOTAL length of the base (in Diagram 1) is the length of the rectangle + 2 times the length of the triangle's base:
Length = rectangle base + 2 x triangle base
Length = 5 + 2 x 0.785
Length = 6.57 cm
Incidentally, this length is the diameter of the larger side, thus your problem is solved.
Oh yes, I can't tell from your description whether the the 15cm applies to the slanted side or the perpendicular distance between the two ends. I've solved it assuming it's the slanted side. If it's the latter case, use tan 3° instead of sin 3°.
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