Q$1.(25$ marks$)$

The dynamic behavior of the liquid level in each leg of a manometer tube, responding to a change in pressure, is given by:

$\frac{d^2{h}^{\text{'}}}{d{t}^2}+\frac{6}{R^2}*\frac{d{h}^{\text{'}}}{dt}+\frac{3}{2}*\frac{g}{L}{h}^{\text{'}}=\frac{3}{4L}{p}^{\text{'}}(t)$

The output is the level of fluid measured with respect to the initial steady$-$state value, the output is the pressure change in the manometer. $R,L,$ and $$ are constants.

a$.$ What order is the TF of this system? Why? $(2$ marks$)$

b$.$ Find all the parameters of that TF $($gain$,$ time constant and damping coefficient$).(15$ marks$)$

c$.$ For what values of the damping coefficient does the height oscillate? Why? $($I marks$)$

d$.$ If density of the fluid increases, would the height oscillate more or less? Why? $(3\mathrm{.}5$ marks$)$

e$.$ If the viscosity of the fluid increases, would the height oscillate more or less? Why? $(3\mathrm{.}5$ marks$)$

For the last $2$ parts, please indicate in a rough plot, as to how the response to a step input of this underdamped system will change when the density increases $($all other parameters constant$)$ and viscosity increases $($all other parameters constant$),$ in separate cases.