E$)$ As will be discussed later, the volume rate of flow, $Q,$ through a pipe can be determined by means of a flow nozzle located in the pipe, as illustrated below. The nozzle creates a pressure drop, $p_A-p_B,$ along the pipe that is related to the flow through the equation $Q=k\sqrt[\phantom{2}]{p_A-p_B},$ where K is a constant depending on the pipe and nozzle size. The pressure drop is frequently measured with a differential U$-$tube manometer of the type illustrated.

Determine an equation for $p_A-p_B$ in terms of the densities ${}_1$ and ${}_2,$ and the various heights indicated.

Determine $p_A-p_B,$ if ${}_1=1000k\frac{g}{m^3},{}_2=1590k\frac{g}{m^3},h_1=1$ m $,$ and $h_2=0\mathrm{.}5m.(30$ points$)$