MadSci Network: Engineering |
938120564.Eg An interesting question, and one that has been posed by others for a similar reason. To answer the first part of the question, yes, there are physical limits to the height of such a structure. Basically, it boils down to the ratio of the strength of the structural material to the density. If you started with just a balloon with a wire hanging down to the earth's surface, the weight of the entire wire will be carried by the very top strand. The limit the wire can carry is described by its breaking strength, usually given in terms of physical load divided by its cross sectional area. For a round wire, this load, P will be: P/ (pi*r*r) = breaking stress where pi = 3.14159 and r = radius of wire The weight of a length of L wire below it can be calculated from the density, the radius, and the length. P = pi*r*r*(density)*L setting the above two loads, P equal and solving for the length gives L = breaking stress/density For a high strength steel like spring wire, the breaking strength can exceed 200,000 pounds per square inch and the density is = .29 lb/cubic inch. So the wire can be about 200,000/.29 inches long, or about 10 miles. If we built a truss like structure like the Eifel Tower, we can minimize the wind loads and other factors that will tend to topple it, but you can't get around the weight effect. I can't explain the problem in 3D so easily, but imagine building a truss structure in 2 dimensions made of triangles like the one below. Each triangle is made up of 3 legs or segments. Adjacent triangles share a segment. Such a design can put all of the load uniformly into the triangles below. (I don't show the horizontal segments.) /\ 1 triangle of 3 segments (two sides and the bottom) /\/\ 3 triangles of a total of 6 segments /\/\/\ 5 triangles of a total of 9 segments /\/\/\/\ 7 triangles of a total of 12 segments If you add up the weight of the segments above each layer, you will find that the 2nd layer has to hold the weight of 3 segments, the 3rd layer has to hold up (3+6)=9 segments, the 4th layer has to hold up (3+6+9)=18 segments, etc. If each segment of the truss is carrying an equal share of the weight above it, then the table below shows the weight carried by each segment as you go up down the structure: Layer segments total segments above weight above that layer per segment in layer 2 6 3 3/6 = 0.5 3 9 9 9/9 = 1.0 4 12 18 18/12 = 1.5 5 15 30 30/15 = 2.0 As you can see, the trend is increasing the load per segment due to the weight above. In reality, the structure is 3D, so the number of trusses below can be greater or less than that shown, but no matter what, the weight becomes a factor eventually. If we could fabricate a metal bar with helium trapped inside, perhaps the bouyancy of the helium would counteract the weight of the bar. Otherwise, even lightweight materials like composite plastics will eventually succomb to the weight limit. Even a mountain is presumably limited by the compressive strength of the soil or rock from which it is made. The second part of the question asks whether such a structure would benefit a space launch. Although rockets tend to use less fuel if launched at a higher altitude, this is usually done by carrying the rocket on an airplane and dropping it before ignition. Presumably, your tower would have a similar benefit. The velocity effect of the earth's radius would be minimal. We launch rockets near the equator to take advantage of the speed gained by the earth's rotation, as opposed to the north pole, where the velocity is pure rotation. I think I recall that the earth is roughly 8000 miles in diameter, so the velocity on the ground is: radius * rotational velocity of earth = 4000 miles * (2 pi radians)/(24 hours) = 1047 mph Even if you build a 200 mile high tower, the radius will be only 4200 miles, or about 5% more. The velocity will increase only 5% as well. I seem to recall scientists posing the possibility of putting satelites in stationary orbit and then hanging thin wires from them down to the earth's surface. Small payloads could presumably be run up the wire to orbit. Not sure how serious they were about the topic, but the practical limits are severe, as you can see. Moderator's Note: The excellent novel "The Fountains of Paradise" by Arthur C. Clarke deals with such a "space elevator." If you could build one, it would lower the cost of transfer to orbit to pennies per pound. The catch is that you would need superstrong materials, stronger than anything we have now...
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