Beat it, Mothra

The WWE (formerly the WWF) has a huge following.  This modern day morality play took in $475.16 million in revenues last year, but there’s one way to improve on this already booming business: incorporate anthropomorphic Japanese monsters battling it out in a cityscape arena.  You could even have flying monsters like the great Mothra.  In real life, how fast would Mothra have to beat her wings to stay afloat?

According to Wikipedia, Mothra weighs about 22,000 tons in adult form with a wingspan of 250 m.  From this wingspan, we can estimate the total area of Mothra’s wings to be about 63,000 m².  Assuming her wings move 100 m down with each thrust, she will push a total volume of 6.3×10⁶ m³ of air downward.  Using 1.2 kg/m³ as the density of air, we can compute the total mass of air propelled down with each thrust,

air mass =  (density) · (volume)

=  (1.2 kg/m³) · (6.3×10⁶ m³)

= 7.6×10⁶ kg.

Pushing the air down will result in an equal an opposite force up on Mothra.  This force must at least balance the force of gravity if she is to fly.  The force of gravity can be computed as follows,

gravitational force = (mass) · (gravitational acceleration)

= (22,000 tons) · (9.8 m/s²)

= 2.0×10⁸ N.

Using dimensional analysis, we can construct a formula for the upward reaction force needed to keep Mothra afloat and set this equal to the gravitational force,

gravitational force = (air mass) · (range) · (frequency)².

We can then solve for the frequency,

frequency = {(gravitational force) / [(air mass) · (range)]}^1/2

= {(2.0×10⁸ N) / [(7.6×10⁶ kg) · (100 m)]}^1/2

= 0.5 Hz

This number is a lot smaller than I expected.  For example, a hummingbird’s wings oscillate at 50-200 Hz.  Since Mothra is so much larger than birds and insects, I expected she’d need to beat her wings really fast just to hover.  Her mass—hence the gravitational force pushing down on her—will grow as her length cubed.  The reaction force I calculated also grows as her length cubed since it’s proportional to the volume of air she pushes down.  That may be true.  After all, insects’ wings do appear to beat more frequently than birds’ wings, but I’m still a little bit skeptical.  If it were correct, then large flying creatures would only need to beat their wings at the same frequency as small flying creatures to stay afloat, and that would mean that large birds could fly just like small birds, albeit with greater energy requirements since it would take more energy to move their massive wings.   At any rate, I would like to hear readers’ opinions to see if my answer is crazy.