how are parabolas and catenarys different and alike??

Answer 1

Parabolas and catenaries are mathematically distinct types of curves. However they do have some things in common.

One important thing that parabolas and catenaries have in common is that both of them can be generated by hanging a flexible cable or chain between two fixed points. The difference between them arises from different ways of distributing weight along the length of the chain.

If weight is distributed evenly along the length of the chain, the result is a catenary. This occurs when the only weight attached to the chain is the weight of the chain itself.

If weight is distributed evenly along a horizontal line, in the manner of a suspension bridge, then the result is a parabola. See the link below for photographs of this type of bridge. The Severn bridge is a particularly good example.

The equation of a catenary is:
y = a cosh(x/a)
or equivalently:
y = a ( e^(x/a) + e^(-x/a) ) / 2
where a is a constant

The equation of a simple parabola is:
y = a x^2
A more general parabola is given by:
y = a x^2 + b x + c
where a, b, and c are constants

The word "catenary" comes from the Latin word "catena", which means "chain".

Answer 2

A catenary is formed when a uniform, flexible chain hangs by its own weight under the influence of gravity. If you glued another chain under it, the weight distribution along the length of the chain would be essentially the same, and so the two chains together would still form a catenary shape. The various physical forces on each part of the chain combine to yield the equation y= cosh(x), which I'll not go into, except to say the relevant thing, which is, that the horizontal component at all points of the chain is constant. Now in a suspension bridge, the main cable is attached to many smaller cables that are attached to the brige deck, so the main chain is not only supporting itself, but also supporting the weight of the bridge, which, relevantly, is a uniform load in the horizontal direction. The weight of the chain is relatively insignificant to the weight of the bridge, so when all the forces are combined, the equation that pops out is that of a parabola. Its horizontal component is not constant along the length of the chain, so it is different from a catenary. Of course, all things aren't perfect, so the shape of a suspension bridge is probably somewhere between a catenary and a parabola.

Answer 3

They have aprox.similar shapes but the equations are totally diferent.
y=x^2 is the equ.of a certain parabola and y=(e^x+e^-x)/2 is the equ. of certain catenary

Answer 4

hmm... alike:

both are concave curves, have only one min or max, both have no limit as "x" increases without bound in either the positive or negative direction.

different:
parabolas have a focus point. Otherwise, I think their differences are mainly in the equations themselves.