posted
I have a math problem, geometry to be exact.
I am trying to figure out a geographical structure made primarily of obsidian. The structure would be a rough, inverted cone. Think of a giant obsidian spear rammed into the ground. I figure it is about 1000 feet tall. Tip of the cone would be deep underground. On the flat top is a town that also spills over the sides. Anybody know how to figure out how big the base would need to be? Or how much of an angle I'd need? This feel like this is one of those problems with the flagpole and shadow thing.
posted
My geometry is pretty rusty, but I'll give it a whirl to dust it off if nothing else. Your base wouldn't have a set size based on the height alone. Essentially a cone is just a right triangle rotated around in a circle using one edge as an axis. In this case the axis would be your height edge, 1000ft. The right angle is the one between your height and the base, that one is 90 degrees.
You can figure out the base or the angle of the slopes if you know what either of them is. In other words if you know you want your base to be 3000ft long your right triangle has one side that is 1000ft and one side that is 1500ft (3000ft base cut in half). So your remaining side, the slope of the cone, is the square root of 1000 squared plus 1500 squared or 1803ft (rounded). There is a formula to figure out the angles too, but it escapes me at the moment. Someone else may know it, or you can probably Google something like "how do I find the angles of my right triangle given the sides" and I'm sure you'll find it.
posted
Your question is not a geometry question, but instead, an engineering question. I know very little about engineering, but I can tell you that any inverted cone will be less stable than a cube. Also, the smaller your base, the less stable the structure. I would imagine that the stability of the base would depend on a percentage of the top and also based on the height. Any engineers in the house?
Just reread coralm's explanation of cone - the right angle comment threw me off for a minute. Edited to remove comment about cylinders, cones and angles.
[This message has been edited by philocinemas (edited October 07, 2010).]
posted
Thanks. I know the cone-ish structure is not as stable as a cube, but this isn't a construct. I am thinking more like a rock formation.
Posts: 823 | Registered: May 2009
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posted
In case you haven't found a site to help you figure out the angles yet here is one I found in my travels today that explains things in an easy to understand way.
If you look at the picture at the top of the page you can see what I attempted to explain. Spin that right triangle around and it's a cone, so you can use the same formulas they go through on that site to figure out any dimension you need to know.
posted
All right, babooher, I did a little experiment for you. I took a roll of duct tape and placed it on top of two cups that both came to about 10 inches with the roll of tape - multiply by 100. From what I can tell, the stability of the structure was primarily determinant of the size of the base in relation to the top of the simulated cone.
One of the bases was 50% of the diameter of the tape, and the other was 40% of the tape's diameter. The 50% base was more stable.
The diameter of the roll of tape was approximately 50% of the height of the cup (the one with the 50% base).
So, here is my suggestion: For an inverted cone 1000 ft high, you could safely have a top diameter of 500 ft and a base of 250 ft. As you increase the top diameter you would need to equally increase the base diameter to maintain stability. Also you need to be conscious of the diameter of the base in relation to height - the base should probably be a minimum of 25% of the total height of the object. The greater this percentage, the more stable the cone, but if you increase the diameter of the top of the cone, you would also need to equally increase the base.
[This message has been edited by philocinemas (edited October 08, 2010).]
posted
Yeah, my wife walked into the room and looked at me as if I was trying to build a replica of Devil's Tower out of mashed potatoes.
Posts: 2003 | Registered: Jul 2008
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posted
This is a neat idea. The math here looks good, but I see some problems with the physical engineering aspects. Obsidian is not a crystal mineral, but a volcanic glass. This makes it look nice, but gives it unusual strength characteristics. Its tensile strength (strength under tension, being pulled) is extremely high. Its compressive strength (strength being crushed under weight, for example) is quite low. It's very brittle. It is quite easy to knock pieces off of it. I'd be concerned that an inverted cone of obsidian couldn't even hold up its own weight. So, depends on how 'real' a reality you're looking for.
Of course, you could always work it into the story that pieces keep threatening to shear off.
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