GATE 2022 AE – Question 51
A wing has $\Gamma(y)=\Gamma_0(1-2|y|/b)$ for $|y|\le b/2$. Span b=10 m, peak circulation 20 m²/s, density 1.2 kg/m³ and speed 80 m/s. Wing lift (nearest integer N) is ________.
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Correct answer: 9600
Explanation
**Kutta-Joukowski theorem.** The lift per unit span at a station is $L^\prime(y)=\rho\,U_\infty\,\Gamma(y)$, so the total lift is
$$L=\rho\,U_\infty\int_{-b/2}^{b/2}\Gamma(y)\,dy .$$
**Integral of the circulation.** $\Gamma(y)=\Gamma_0\left(1-\dfrac{2|y|}{b}\right)$ is a **triangle** with height $\Gamma_0=20$ m²/s and base $b=10$ m:
$$\int\Gamma\,dy=\tfrac12\,b\,\Gamma_0=\tfrac12(10)(20)=100\ \text{m}^3/\text{s}.$$
**Lift:**
$$L=1.2\times80\times100=\mathbf{9600\ N}.$$