In order to maintain uniform flight, smaller birds must beat their wings faster than larger birds. It is suggested that the relationship between the wingbeat frequency, \(\omega\), beats per second, and the bird's wingspan, \(\ell\), is given by a power law relationship, \(\omega \sim \ell^{n}\).

(a) Use dimensional analysis with the assumption that the wingbeat frequency is a function of the wingspan, the specific weight of the bird, \(\gamma\), the acceleration of gravity, \(g\), and the density of the air, \(ho_{a}\), to determine the value of the exponent \(n\).

(b) Some typical data for various birds are given in the following table. Do these data support your result obtained in part (a)? Provide appropriate analysis to show how you arrived at your conclusion.

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Wingbeat frequency, Bird Wingspan, m beats/s Purple martin 0.28 5.3 Robin 0.36 4.3 Mourning dove 0.46 3.2 Crow 1.00 2.2 Canada goose 1.50 2.6 Great blue heron 1.80 2.0