A rotating cylinder of radius $R = 0.5\text{ m}$ is placed in a uniform air stream of density $\rho = 1.20\text{ kg/m}^3$ and velocity $U_\infty = 30\text{ m/s}$. If the cylinder has a bound circulation $\Gamma = 25\text{ m}^2/\text{s}$ generated by the Magnus effect, the aerodynamic lift force per unit span $L'$ (in $\text{N/m}$) acting on the cylinder is:

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### 1. Strategy & Theoretical Principles By the Kutta-Joukowski theorem for 2D inviscid, incompressible flow: $$L' = \rho_\infty U_\infty \Gamma$$ where: - $L'$ is the aerodynamic lift force per unit span - $\rho_\infty$ is the freestream fluid density - $U_\infty$ is the freestream velocity - $\Gamma$ is the circulation around the closed contour enclosing the body. ### 2. Analytical Formulation & Calculation Given: - $\rho_\infty = 1.20\text{ kg/m}^3$ - $U_\infty = 30\text{ m/s}$ - $\Gamma = 25\text{ m}^2/\text{s}$ $$L' = 1.20 \times 30 \times 25 = 36 \times 25 = 900.0\text{ N/m}$$ ### 3. Distractor Forensics - Including cylinder diameter or radius in the Kutta-Joukowski formula ($L' = \rho U \Gamma R$) is a common error; the theorem depends solely on $\rho, U, \Gamma$. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Kutta-Joukowski applies to arbitrary 2D body shapes regardless of radius or camber. - **Standard Reference:** Anderson, J.D., *Fundamentals of Aerodynamics*, Chapter 3.

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