(Catenary) In our Suspension Bridge Cable example we neglected the weight of the cable its...

(Catenary) In our Suspension Bridge Cable example we neglected the weight of the cable itself relative to the weight of the roadbed. At the other extreme, suppose that the weight of the roadbed (or other loading) is negligible compared to the weight of the cable. Indeed, consider a uniform flexible cable, or catenary, hanging under the action of its own weight only, as sketched in the figure. Then Fig. 8 still holds, but with ΛW= μΛs, where y, is the weight per unit arc length of the cable.

As a partial check on these results, notice that they should reduce to the parabolic cable solution in the limiting case where the sag-to-span ratioH/Ltends to zero, for then the load per unitxlength, due to the weight of the cable, approaches a constant, as it is in Example 2, where the load is due entirely to the uniform roadbed. The problem that we pose for you is to cany out that check. HINT: Think ofLas fixed andH tending to zero. ForHto approach zero, in (3.3), we needCL/2to approach zero - that is,C →D. Thus, we can expand the coshCx −1 in (3.2) in a Maclaurin series inCand retain the leading term. Show that that step givesy(x)≈Cx2/2,and the boundary conditiony(L/2) = Henables us to determineC.The result should be identical to (15).

3.3

3.2