Consider a bridge deck weighing S pounds per linear foot and supported by a cable, which is assumed to be of negligi-ble weight compared to the bridge deck. The cable section OP from the lowest point (the origin) to a general point P(x, y) is shown in Figure 13.
The forces acting on this section of cable are
H = horizontal tension pulling at 0
T = tangential tension pulling at P
W = (r = weight of x feet of bridge deck
For equilibrium, the horizontal and vertical components of T must balance H and W, respectively. Thus,

That is,

Solve this differential equation to show that the cable hangs in the shape of a parabola. (Compare this result with that for the unloaded hanging cable of Problem 53 of Section 6.9.)

Tsin " = tan = dv= y(0) = 0 , T sin T cos H O .x