The General function modeling a hanging cable between two poles is: y=acosh(x/a)

Question

The General function modeling a hanging cable between two poles is:

y=acosh(x/a)+c when x is on [-b,b]

where the parameters a and c depend on the composition of the cable and the height of the poles. The Horizontal distance between the poles is 2b.

1. Use wolframalpha or some other device to get some some plots of the above function for some a few values of a and c. About 4 graphs is enough. I'm not really sure how to input this into wolfram alpha

2. a. Show that thye length of the cable is given by:

L=2a*sinh(b/a)

b. Show that the sag S (the vertical distance between the highest and lowest points on the cable) is given by

S=acosh(b/a)-a

3. Show that if the lenght L and the sag S of the cable are given, then one can find the parameter a of the cable by

a=((L-2S)(L+2S))/8S

Explanation / Answer

2.) length of the cable= length of curve = L = (int_{-b}^{b} sqrt{1+ (dy/dx)^2} dx) here we have dy/dx= a*sinh(x/a)*(1/a) = sinh(x/a) also we have cosh^2 x - sinh^2 x=1 so using above formula L = integral(-b,b) sqrt(1+ sinh^2 (x/a)) and this on solving gives us L = integral(-b,b) sqrt(cosh^2 (x/a) ) and hence L = integral(-b,b) (cosh(x/a) which on solving gives you sinh(x/a)*a between the limits (-b,b) = asinh(b/a)-(-asinh(b/a)) = 2a*sinh(b/a)

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