The motion of a liquid in an open tank is that of a combined vortex consisting of a forced vortex for \(0 \leq r \leq 2 \mathrm{ft}\) and a free vortex for \(r>2 \mathrm{ft}\). The velocity profile and the corresponding shape of the free surface are shown in Fig. P6.39. The free surface at the center of the tank is a depth \(h\) below the free surface at \(r=\infty\). Determine the value of \(h\). Note that \(h=h_{\text {forced }}+h_{\text {free }}\), where \(h_{\text {forced }}\) and \(h_{\text {free }}\) are the corresponding depths for the forced vortex and the free vortex, respectively.

Figure P6.39

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10 vo. ft/s 0 2 12 h r, ft. 2 r, ft