Finite Wing Theory Questions and Answers

Answer: A

Rearranging \( C_{D_i} = C_L^2 / (\pi e AR) \) gives \( e = C_L^2 / (\pi AR C_{D_i}) \). Substituting, \( e = 0.64 / (\pi \times 8 \times 0.030) = 0.64 / 0.754 = 0.849 \). A value close to 0.85 is entirely typical of a real wing; a result above one would indicate an error, since the elliptical case is the best a planar wing can do.

Answer: B

Both statements are true and the second explains the first. Minimising the induced drag integral for a fixed lift and span yields the elliptical loading, and the mathematical signature of that optimum is precisely a constant downwash, so no section is working harder against its own induced flow than any other.

Answer: C

Both are true and the reason accounts for the assertion. Induced drag is a pressure drag produced by the geometry of the flow, and the trailing vortex sheet that produces it needs no viscosity to exist in the model. This is why d'Alembert's paradox, which assumes no wake, does not apply to a lifting wing of finite span.

Answer: D

Both statements are true and the second is the reason for the first. Constant bound strength forces all the shed vorticity into two concentrated tip filaments, and the inverse-distance law then sends the induced velocity to infinity as those filaments are approached. Spreading the shed vorticity along the span removes the singularity.

Answer: A

Both are true and the second explains the first. With no circulation anywhere there is no shed vorticity, hence no downwash and no induced angle, so the wing behaves exactly as its section does. Finite span alters the slope of the lift curve but leaves its intercept on the angle axis untouched.

Answer: B

Both statements are true, but the second does not explain the first. The effective span increases because the winglet displaces and diffuses the tip vortex, weakening the wing's self-induced downwash. The additional wetted area is a separate and unwanted consequence, which is why winglets pay off only where induced drag is a large share of the total.

Answer: C

Both are true and the reason explains the assertion. A sailplane spends its life at high lift coefficient and low speed, where induced drag dominates, so the inverse dependence on aspect ratio is worth a great deal. Aspect ratios above thirty are common, limited chiefly by structural weight and by handling on the ground.

Answer: D

Both statements are true and the second accounts for the first. In cruise the aircraft is fast and the required lift coefficient is small, and squaring a small number makes it smaller still. In the take-off climb, where the lift coefficient is several times larger, the same relationship makes induced drag the dominant term.

Answer: A

The assertion is false and the reason is true. Concentrating the bound vorticity on one line is a fair approximation only when the chord is small compared with the span and the wing is unswept. A low-aspect-ratio delta violates both conditions, so lifting-surface methods such as the vortex lattice are needed instead.

Answer: B

The assertion is false and the reason is true, and indeed the reason is what forbids the assertion. A filament must extend to a boundary or close on itself, which is why the bound vortex on a wing has to continue into trailing vortices and close through the starting vortex left behind at the beginning of the motion.