2. we have a liquid column gauge, a U-shaped tube with liquid inside. the two ends of the tube are exposed to different pressures of air. the right side is always exposed to the atmosphere; the left side is exposed to air at a pressure p = p(t) (the “input"). because p changes with time, so does the difference in the liquid level between the two sides of the tube, h = h(t) (the “output"). dynamic model for h = h(t), in deviation form, is: \frac{d^{2} h^{*}}{d t^{2}}+\frac{6 \mu}{R^{2} \rho} \frac{d h^{*}}{d t}+\frac{3}{2} \frac{g}{L} h^{*}=\frac{3}{4 \rho L} p^{*}(t) with R (radius of tube), L (total length of liquid in tube), g (gravitational constant),p(density of liquid), and u (viscosity of liquid) constants. (a) (3 points) derive the transfer function relating the response of the liquid level tochanges in the pressure on the left end of the tube. write the transfer function instandard gain-time constant form.

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