$(6).($Response of a manometer$)$: The trasnient response of a U$-$tube manometer to changes in pressure

is described by the initial value problem

$\frac{d^{2}h}{d{t}^2}+\frac{24}{d^2}\frac{dh}{dt}+\frac{3g}{2L}h=\frac{3}{4}\frac{p}{L}$;$,t>0,h(0)=0$;$h^{\text{'}}(0)=0$

where $h$ is the deviation of the manometer liquid level from the equilibrium position, $t$ is time, $g$ is the

gravitational acceleration, $d$ is the tube diameter, $$ and $$ are the density and viscosity of the manometer fluid,

$L$ is the total length of the fluid column and $p$ is the change in pressure $($a$)$ From the above model, determine

the critical tube diameter above which the response is oscillatory $($b$)$ Cast the equation in dimensionless

form and show a schematic plot of the transient response of the manometer for a step change in the pressure

when the tube diameter is twice the critical value determined in $($a$).($c$)$ If the viscous damping term can

be neglected, determine the period of oscillation $($or the natural frequency$)$ for the case where the total

length of the liquid column is $2$ meters $($d$)$ Examine the eigenvalues of the homogeneous equation with the

damping term and calculate the critical diameter of the tube below which the system does not oscillate

when the manometer fluid is water at $20C$ with density ~~$1\mathrm{.}0\frac{g}{c}{m}^3$ and viscosity ~~$1$ centipoise $(=0\mathrm{.}01$

g$.($:$c{m}^{-1}{s}^{-1}$