Calculating the curve of a hanging rope.

Question

I happen to know that the curve of a hanging rope isn't a parabola.

But that's irrelevant, is there a way of calculating the curve of a rope by using its weight, flexibility, stretch, and length.

Have no fear, this isn't homework, I'm just curious.

Answer 1

While it is true that the curve defined by a hanging rope is a catenary expressed mathematically as:

y = a cosh(x/a)

The difference between a catenary and a parabola is that; in the catenary the weight is dstributed uniformy along the curve while in a parabola the unit weight is distributed uniformly along the span.

Thus when the sag of the rope is too small compared to the span, the catenary is approximated by a parabola. In which case the relationships of the weight, stretch, tension and legth are as follows:

Let:

w = unit weight per unit length of the rope

s = span or distance between the hitch points

l = length of the rope from on hitch point to the other

h = the sag of the rope

T = tension on the rope

wl/2(s/4) = Th

T = wl/2(s/4)/h

But s is approxmately equal to l hence

T = wl^2/(8h)

If we now replace l with x and h with y, then:

y = wx^2/(8T)

:

Answer 2

In physics and geometry, the catenary is the theoretical shape of a hanging flexible chain or cable when supported at its ends and acted upon by a uniform gravitational force (its own weight) and in equilibrium. The chain is steepest near the points of suspension because this part of the chain has the most weight pulling down on it. Toward the bottom, the slope of the chain decreases because the chain is supporting less weight.

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