Assume that the wing area of an airplane is proportional to...

Assume that the wing area of an airplane is proportional to the square of the wing span and the volume, and thus the weight, is proportional to the cube of the wing span (this is the square-cube law). Find the wing loading of the aircraft as a function of wing span, $b$. Using these relationships explain why very large aircraft (like the Boeing 747 or the Airbus 380 ) have to fly with very large wing loadings. If you wanted to re-design an existing aircraft that weighed $82,000 \mathrm{lbs}$ with a wing span of $65 \mathrm{ft}$, what wing span would you need to add an additional $20 \%$ to the weight?

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Key Concepts

Square-Cube Law

The square-cube law explains that as an object's linear dimensions are scaled, its surface area increases with the square of the scaling factor while its volume (and thus, for a given material, its weight) increases with the cube. In the context of aircraft, this means that as the wing span (a linear dimension) increases, the weight increases much faster than the wing area, leading to higher wing loading.

Wing Loading

Wing loading is the measure of an aircraft's weight per unit of wing area. It is a crucial performance metric because it influences takeoff, landing, and overall maneuverability. When scaling up an aircraft, if weight (which scales with the cube of the wing span) increases faster than the wing area (which scales with the square of the wing span), the resulting wing loading increases linearly with the span.

Scaling Relationships in Aircraft Design

In aircraft design, scaling relationships illustrate that geometric scaling (wing area ? wing span² and weight ? wing span³) forces larger aircraft to contend with increased wing loading. This is why very large aircraft often have to operate with high wing loadings and why any increase in weight necessitates a proportionate increase in wing span to keep performance characteristics within acceptable limits.

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