Question $3-$ Fluid Mechanics

$[$Total $25$ Marks$]$

a$)$ A closed tank contains compressed air and oil as shown in Figure Q$3$a$.$ A U$-$tube mercury manometer is connected to the tank. Derive an equation for the pressure reading $($gauge$)$ at the top of the tank in terms of the column heights $h_1,h_2$ and $h_3,$ and the specific weights of ${}_{Hg},{}_{\text{o}\text{i}\text{l}},$ and ${}_{\text{a}\text{i}\text{r}}.$

$[5$ marks$]$

b$)$ A $3$ m wide, $8$ m high rectangular gate is located near the bottom of an open tank that contains water $($see figure Q$3$b$).$ The gate is hinged at the bottom and held close by a horizontal force $F=2000kN$ applied at the centre of the gate. The centre of the gate is located at an unknown depth $h.$ the specific weight of water $=9\mathrm{.}8k\frac{N}{m^3}.$

i$.$ Determine the force exerted by the liquid on the gate as a function of $h.$

$[5$ marks$]$

ii$.$ Determine the location of the centre of pressure on the gate as a function of $h.[5$ marks$]$

iii. Sketch a free body diagram of the forces applied on the gate.

$[3$ marks$]$

iv$.,$ Determine the water level $h$ above which the gate will start opening if $F$ remains unchanged.

$[7$ marks$]$