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Suspension bridge curve

A similar analysis can be done to find the curve followed by the cable supporting a suspension bridge with a horizontal roadway. If the weight of the roadway per unit length is w and the weight of the cable and the wire supporting the bridge is negligible in comparison, then the weight on the cable (see the figure in Catenary#Model of chains and arches) from c to r is wx where x is the horizontal distance between c and r. Proceeding as before gives the differential equation

This is solved by simple integration to get

and so the cable follows a parabola. If the weight of the cable and supporting wires is not negligible then the analysis is more complex.