The mathematical (and other) thoughts of a (now retired) math teacher,
Thursday, 24 March 2011
Will It or Won't It
In my last blog I tried to solve a problem about a rigid steel rail a mile plus one foot long that was forced into a space one mile long. I came up with an answer of about a 45 foot bulge that agreed with other mathematical answers, but then I got a comment from Jonathon who suggested, "I think you need to take the compression of the steel into account. A mile of steel rail would be pretty heavy and at that small angle (if it could be held perfectly in a line above the ground) it would probably compress itself until almost flat... "
My first thought was that he was probably right. I even commented that I thought so... and then... I begin to wonder... Would the steel flatten? I imagine a fine steel wire rod, or maybe even a hollow tube.. let's say 1 inch in diameter and the given length....Steel has pretty remarkable strength... Would it bend or would it compress along its length and sag to the ground?
If I take a wooden yardstick and bend it to fit a 35.5 inch distance (I know, not the same scale, but the same idea) it will bulge up quite well, and I think of wood as being equally strong under lateral compression... but I know that when we scale up the length, the masses will scale by the cube of that ratio.. so ???
ANYONE, ANYONE, .....Bueller?
So if anyone out there knows how to decide with appropriate technical engineering skill how much such a rod would sag... chime in here... And what would you do if you wanted it to arch supported only at the extermes and reach a height of 45 feet... Can that be done with one piece of steel. I guess I could scale the problem down, but at lengths I could manage, the height to the top of the bend would be pretty small. If I scaled down the one mile to 52.8 feet then the extra length would be .1 feet or about an inch and a quarter, and the height of the bend would be .45 feet or just under 5.5 inches. However I can't afford a 53 foot lenght of metal pipe, and another scale of ten reduces the problem to about 1/2 an inch of bow...