An ideal, incompressible fluid of density \(\rho_f\) flows steadily through a horizontal pipe. The pipe has a narrow section with cross-sectional area \(A_1\) where the fluid speed is \(v_1\) and the absolute pressure is \(P_1\). The pipe widens to a section with cross-sectional area \(A_2\) where the fluid speed is \(v_2\) and the absolute pressure is \(P_2\). A U-tube manometer containing a liquid of density \(\rho_m\) (where \(\rho_m > \rho_f\)) is connected between the bottom of the narrow section and the bottom of the wide section, as shown in Figure 1. The fluid in the pipe and the liquid in the manometer do not mix.