Suppose that two dragonflies of different mass but identical

Suppose that two dragonflies of different mass but identical body plans each fly in circles with the same wing tilt angle of 200, and that each is flying at its maximum power output (limited by its muscle mass). If the smaller dragonfly has mass m1 and flies in a circle with radius R1, and the bigger dragonfly has mass m2, what will the radius of orbit for the bigger dragonfly? (Hint: derive an allometric relationship for the radius as a function of mass.)

Solution

The muscle force–velocity relationship and its derived n-shaped power–velocity curve offer important insights into muscular limits of performance. Given the power is maximal at 1/3 V_max, geometric scaling of muscle force coupled with fluid drag force implies that this optimal muscle-shortening velocity for power cannot be maintained across the natural body-size range. Instead, muscle velocity may decrease with increasing body size, conferring a similar n-shaped power curve with body size. Here we examine swimming speed and muscle function in the aquatic frog Xenopus laevis. Swimming speed shows an n-shaped scaling relationship, peaking at 47.35g. Further, in vitro muscle function of the ankle extensor plantaris longus also shows an optimal body mass for muscle power output (47.27g), reflecting that of swimming speed. These findings suggest that in drag-based aquatic systems, muscle–environment interactions vary with body size, limiting both the muscle’s potential to produce power and the swimming speed.