The vertical and horizontal forces (reactions) at joints \(A\) and \(B\) of a typical cable of the suspension bridge shown in Fig. 8.27 are given by \(F_{x}=2.8 \times 10^{6} \mathrm{~N}\) and \(F_{y}=1.1 \times 10^{6} \mathrm{~N}\). The cables are made of steel with a weight density of \(76.5 \mathrm{kN} / \mathrm{m}^{3}\). If the effective diameter of the cable is \(25 \mathrm{~cm}\), find the first two natural frequencies of vibration of the cable in the vertical direction. State the assumptions involved in the solution.

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Fy A' Tower Cables T B i h = 50 m h Floor 1 = span 500 m FIGURE 8.27 A suspension bridge. B' Tower D'