ENGR217/266 2022 Q A1 (ENGR5002)

ENGR217/266 2022 Q A1Past paperCurrent spec2:125 marks30 min

ENGR217/266 Summer 2022 Q A1[VALID]

A tank is divided in two compartments as shown in Figure A1-1. The left compartment contains a fluid of unknown density $\rho_f$ (darker area in the Figure A1-1), confined on the left by an L-shaped gate. A beam holding a mass of weight W is hinged to the gate as shown in the Figure A1-1. The rest of the gate is filled with water (light blue region, seawater density $\rho_{sw}$ is $1035\ \mathrm{kg\,m^{-3}}$ ).

Figure A1-1: two-compartment tank with an L-shaped gate, hinged beam (50 cm arm) and weight W; levels 30, 90, 50 and 100 cm.

Formulas you may need

Hydrostatic pressure: $p(z) = p_0 + \rho g z$ ( $z$ = depth) (on the formula sheet)
Force on a plane surface: $F = p_{CG} A = \rho g z_{CG} A$ (on the formula sheet)
Centre of pressure: $l_{CP} = l_{CG} + \dfrac{I_G}{A\, l_{CG}}$ (on the formula sheet)
Rectangle: $A = bh$ , $I_G = \dfrac{b h^3}{12}$ (on the formula sheet)
Moment equilibrium about the hinge: $\sum M_O = 0$ (learn this)

(a)
Using the fluids levels indicated in the figure, determine the density of the unknown fluid $\rho_f$ occupying the leftmost portion of the tank.
rho_f =kg/m^3
[6]
(b)
Using the value of the density $\rho_f$ computed in A1-(a), determine the force exerted by the fluid on the gate (assume that the width of the gate, i.e. its size in the direction normal to the paper is 2 m).
F =N
[6]
(c)
Determine the location of the centre of pressure on the gate.
cp depth below surface =m
cp distance below hinge =m
[6]
(d)
Draw a free body diagram incorporating the weight W and the hydrostatic force exerted by the fluid in the tank into the gate.
[4]
(e)
Determine the minimum value of the weight W that keeps the gate closed (gate does not rotate about the hinge).
W min =N
[3]