Perhaps the likeliest use of this equation is for very long anchor chains for ships at sea. If the chain is long enough (think Marianas Trench @6.8 miles), then most of the strength in the upper chain is going to support the lower chain, so the chain needs to taper exponentially. http://en.wikipedia.org/wiki/Mariana_Trench Anchor chains with uniform links describe a catenary. What shape do exponentially tapering chains describe? http://www.petersmith.net.nz/boat-anchors/catenary.php At 12:33 PM 1/1/2015, Henry Baker wrote:
The recent discussions of Gaudi prompted me to think about stalactites.
I wanted to understand how long they could get.
Suppose a stalactite is made out of structural steel.
Suppose that the stalactite is a solid body of revolution, and that the only forces are the vertical tensile forces that oppose gravity, and that the weight of the stalactite is uniformly distributed over the circular face of the base of the stalactite. We also assume that the steel doesn't stretch at all (bad assumption, but it probably doesn't change the weights very much, even though the lengths will grow somewhat).
Structural steel has a tensile strength of approx. 250MPa = 250x10^6 kg m/sec^2.
Structural steel has a density of approx. 7800 kg/m^3, and g = 9.81 m/sec^2.
The optimum shape (with these constraints) for a stalactite appears to be an exponential function R(h)=exp(-h/B)/A, for suitable constants A,B>0.
Let's consider a stalactite whose base area is 1 m^2. Then A=sqrt(pi).
A steel base with area 1 m^2 can support 250*10^6 Newtons. We need to compute the volume of steel which weighs 250*10^6 Newtons.
A 1x1x1 m cube of steel weighs 7800x9.81 = 76518 Newtons. So our base can support hanging 3267 m^3 of steel.
So 3267 = integrate(pi*(exp(-h/B)/sqrt(pi))^2,h,0,inf) = integrate(exp(-2h/B),0,inf) = B/2, so B=6534.
But our stalactite is infinitely long, which can't be right. Suppose that it ends with a single iron atom of radius 126 picometers = 126*10^-12 m. Then h = 145201 m = 89 miles.
And this is ordinary structural steel; specialty wire steels are 10x stronger for the same density.