Sunday, July 28, 2013
The catenary and the parabola.
If the two ends of a chain are held fast at the same height, what is the shape of the hanging section? This question has been around for a long time. Galileo speculated it was a parabola, but some experimentation showed him that wasn't accurate.
The name of the shape is the catenary. The simplest equation for the shape is the hyperbolic cosine function, y = ½(e^x + e^-x). Unlike sine and cosine that oscillate us and down, hyperbolic cosine has just the one lowest point, like a parabola, and hyperbolic sine looks something like the cubic. The thing that makes them sort of like the trig functions is that each is the derivative and anti-derivative of the other. (I write "sort of" because the derivative of sine is cosine, but the derivative of cosine is negative sine.)
As pretty as the tangent line representations can be, looking at the actual functions is clearer in this case.
Red curve is the catenary from x = -3 to 3.
The blue shape is y = x² + 1.
The 1 added so they coincide more closely. A catenary grows slightly more slowly than the parabola at the vertex, but it is an exponential function, so when it starts to grow faster, it will overwhelm a quadratic function very quickly.
This drawing is the catenary in red and the parabola in blue yet again, but this time from -5 to 5. The shape of the hanging chain depends on the ratio of the length of the chain to the distance apart of the points from which the shape hangs. To get the exact comparison of the catenary and parabola with those values given is a harder thing to compute and describe, as it involves the arc length formula for the parabola and the catenary, which turn into messy integrals.
Tomorrow we will look at the top of the bell shaped curve before the second derivative goes to zero and compare it to the top of a circle.