GATE 2023 ME – Question 64

Fluid Mechanics · Fluid properties, fluid statics, forces on submerged bodies, stability of floating bodies · 2 marks · Numerical answer

The figure shows two fluids held by a hinged gate. The atmospheric pressure is $P_a = 100$ kPa. The moment per unit width about the base of the hinge is ____________ kNm/m. (Rounded off to one decimal place)
Take the acceleration due to gravity to be $g = 9.8$ m/s$^2$.

a vertical gate hinged at its base. On the left are two fluids: a top layer 1 m deep of density 1000 kg/m$^3$ on a bottom layer 2 m deep of density 2000 kg/m$^3$. The atmosphere at $P_a$ acts on the free surface and on the right-hand side of the gate.

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Correct answer: 56.91 to 57.49

Explanation

The atmospheric pressure acts on both sides of the gate, so only the gauge pressure of the fluids matters. Top layer: the pressure rises from 0 to $1000 \times 9.8 \times 1 = 9800$ Pa, a triangle. Its force is $\frac{1}{2} \times 9800 \times 1 = 4900$ N/m, acting at 2/3 of the depth of the layer from the top, so at $3 - \frac{2}{3} = 2.333$ m above the hinge. Moment: $4900 \times 2.333 = 11433$ N m/m. Bottom layer: the pressure rises from 9800 Pa to $9800 + 2000 \times 9.8 \times 2 = 49000$ Pa. Its rectangular part (9800 Pa over 2 m) gives the force 19600 N/m at 1 m above the hinge, so a moment of 19600 N m/m. The triangular part has the force $\frac{1}{2} \times 39200 \times 2 = 39200$ N/m at $\frac{2}{3}$ of the depth below the interface of the layer, that is $0.667$ m above the hinge, giving $39200 \times 0.667 = 26133$ N m/m. The total is $11433 + 19600 + 26133 = 57167$ N m/m $= 57.2$ kN m/m.