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Mechanical Properties of Fluids

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Question : 14 of 31
Marks: +1, -0
In a test experiment on a model aeroplane in a wind tunnel, theflow speeds on the upper and lower surfaces of the wing are 70m/s70\,\text{m/s} and63m/s63\,\text{m/s} respectively. What is the lift on the wing if its area is 2.5m2?2.5\,\text{m}^2? Take thedensity of air to be 1.3kg/m31.3\,\text{kg/m}^3.
Solution:  
Let v1 and v2v_1 \text{ and } v_2 be the speeds on the upper and lower surfaces of the wings of the aeroplane respectively, P1 and P2P_1 \text{ and } P_2 be the pressures on the upper and lower surfaces of the wings respectively.
Here, v1=70ms1;v_{1}=70\,\text{m}\,\text{s}^{-1};
v2=63ms1;v_{2}=63\,\text{m}\,\text{s}^{-1};
ρ=1.3kgm3\rho=1.3\,\text{kg}\,\text{m}^{-3}
The level of the upper and lower surfaces of the wings from the ground may be taken same.
h1=h2\therefore h_1 = h_2
area of wing, A=2.5m2A = 2.5\,\text{m}^2
Thus from Bernoulli’s Theorem,
P1+ρgh1+12ρv12=P2+ρgh2+12ρv22P_{1}+\rho g h_{1}+\frac{1}{2}\rho v_{1}^{2}=P_{2}+\rho g h_{2}+\frac{1}{2}\rho v_{2}^{2}
or P2P1=12ρ(v12v22)(i)P_{2}-P_{1}=\frac{1}{2}\rho(v_{1}^{2}-v_{2}^{2})\ldots(i)
This pressure difference provides the lift to the aeroplane.
If F be the lift on the wing, then
F=(P2P1)×A=12ρ(v12v22)×AF=(P_{2}-P_{1})\times A=\frac{1}{2}\rho(v_{1}^{2}-v_{2}^{2})\times A[by using (i)]
=12×1.3×(702632)×2.5=\frac{1}{2}\times 1.3\times(70^{2}-63^{2})\times 2.5
=12×1.3×931×2.5=1512.9N=\frac{1}{2}\times 1.3\times 931\times 2.5=1512.9\,\text{N}
=1.5×103N=1.5\times 10^{3}\,\text{N}
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