as the size and volume of flying birds increase, surface area of wings cannot increase proportinal to general body surface area. is this a true statement? explain your answer using words and math. bird size seems severely constrained wheras aircraft size is not...why? do the constraints on bird size apply to bats? if so, why aren't there swan-sized bats?
To answer the first part of your question, let's consider the relationship between body size and surface area in flying birds. Surface area is proportional to the square of the linear dimensions, whereas volume (and body size) is proportional to the cube of the linear dimensions.
If we assume that the wingspan of a bird scales proportionally with body size, then the wing area would increase with the square of the linear dimensions. However, as the bird's body size increases, the volume of the bird increases much faster because it is based on the cube of the linear dimensions.
Let's represent the linear dimensions of a bird as "L" and the wing area as "A". In a simple proportional relationship, we would expect:
A ∝ L^2 (Equation 1)
On the other hand, the volume of the bird is represented by "V" and scales as:
V ∝ L^3 (Equation 2)
Now, let's compare the ratio of wing area to body volume:
A / V = (L^2) / (L^3) = 1 / L
This suggests that as the bird's size increases (represented by "L"), the ratio of wing area to body volume decreases. In other words, the wing area cannot keep up with the increase in body size and volume proportionally, supporting the statement that the surface area of wings cannot increase proportional to the general body surface area.
As for the second part of your question, aircraft and birds have different evolutionary and engineering constraints. Aircraft are man-made and designed with consideration for aerodynamics, structural integrity, and engine thrust. They don't face the same biological limitations as birds.
Now, regarding bats, they have similar constraints to birds. While their wings are constructed differently, the same principle applies. Bats also have limitations in wing size as their body size increases. However, bats have adapted different strategies compared to birds. They tend to have higher wing loading (body mass supported by wing area) than birds, which allows them to generate more lift with smaller wings. Additionally, the flight mechanism and requirements may differ between birds and bats due to their distinct evolutionary histories and ecological niches.
In summary, the constraints on bird size are related to the scaling relationship between wing area and body volume, whereas the constraints on aircraft and bats involve different factors like engineering design, aerodynamics, and evolutionary adaptations.