Wednesday, June 20, 2012

Holy Flaming Burritos, Batman!

Another Redditor made me this....

It's been over a week since I did my Reddit AMA, and I want to say thank you to every one who asked questions. As evidenced by the giant spike in page views, it had a bit of an effect on my blog viewership.

So thank you to everyone who asked questions (even I didn't get a chance to answer yours.)

There were, however, a few questions that I said I'd get back to people on, and I wanted to address those.

(1) One of the questions I got and couldn't answer at the time was the following:
A physicist! I've been waiting for one. I've been wondering this for a while, but can't come up with a solid answer. 
If I was in space and I attached an LED light to one corner of a cube, is it possible for me to push/toss/throw/rotate the cube in such a way along a linear path that the LED light's pattern would never repeat itself (aka, there would never be a period)?

Sadly, I still don't have a good answer to this one. The best I can do for now is say that another Redditor provided me with this link to his work on the chaotic motion of rigid bodies. It seems pretty cool, but I'm too bogged down in summer research (and moving to a new state!), so I can't devote a whole lot of time to this one.

(2) Yum_Krill (an appropriate username if I ever heard one) asked,
How large would a creature have to be to make a sound like The Bloop? The largest creature on earth, the blue whale, would not be able to.
I had to look up the Bloop.1  Apparently, it's some weird animal-like deep ocean sound that is both loud and low-frequency:

According to at least one source, the Bloop reaches about 246 decibels. For context, normal human conversation occurs around 60-70 decibels, the loudest rock music reaches about 150 decibels, and blue whales can reach about 188 decibels.

Decibels are a logarithmic scale, which means that a human conversation at 60 decibels would have a power ratio of 106 and a blue whale singing at 190 decibels would have a power ratio of 1019. The Bloop would have a power ratio of about 1025, or roughly a million times bigger than a blue whale.

Whatever the Bloop is, it presumable has some object that mechanically oscillates to produce sound. The amplitude of this oscillation grows as the square root of the power. Since the Bloop's power is 10times bigger than the blue whale's, its amplitude of oscillation would be about 1000 times larger than a blue whale's. If the rest of the Bloop remains proportional in size, it should be about 30 kilometers long and weigh close to 180 billion metric tons.

(3) In response to a message about a flying Pegasus, I replied
We need to consider two things here: wing area and wing flapping rate. I did a similar problem for Mothra's wingspan. Horses weigh about 500 kg, which gives a downward gravitational force of about 5000 N. If you assume her wings flap 2 meters down and do so once every second, she'd need winds that were about 1000 m2 in area. A 2 meter wide wing would need to be about 5 football fields long.
After posting the problem, I got a message that read as follows:
You say Horses couldn't possibly fly unless they had unrealistic, impossibly long wings (up to 5 football fields long), then how come Pterosaurs weighted up to 400 pounds could fly at blazing fast speeds with reasonably sized wings?
I understand the difference in weight, but what was the main difference here to allow this other heavy animal to fly easily, is it wing-flapping speed?
I was unaware of overweight pterosaurs when I was doing the problem, but I still had reason to be concerned with my result because of the earlier problem I did on Mothra. However, in an effort to keep up with the questions, I was working pretty fast and (stupidly) had my normal skeptical filter on silent. That said, I think I know what wrong here.

A horse weighs about 500 kilograms, which means it feels a downward gravitational force of about F = 5000 Newtons. You need at least this much force to have liftoff.

To generate this force, Pegasus pushes air down. The mass of the air pushed depends of the density of the air (ρ =1.2 kg/m3) and the volume V of air swept out with each downward thrust:

m = ρ V = 2 ρ L w h

where L is the length of the wing, w is the width of the wing, and h is the vertical height the wing is thrusted downward. The factor of 2 comes in because there are two wings.

The factors affecting thrusting force are the frequency of flapping f (in Hertz), the distance h the wings are thrust down (in meters), and the mass m of the air pushed (in kilograms). Using dimensional analysis, we can estimate the magnitude of the upward force as

F = 2 ρ L w h2 2.

This can be solved for the length,

L = F / (2 ρ w h2 2).

If we use, as I did previously, the values F = 5000 N, ρ =1.2 kg/m3, w = 2 m, h = 2 m, and f = 1 Hz, you find that the wings must must be 260 meters, or roughly two and half football fields long. I had forgotten the two, but that's not the only problem.

There's two other potential problems I foresee with this. First, dimensional analysis is great for figuring out how certain physical quantities scale with other physical quantities, but if you're looking for actual numbers you can be off by quite a bit. Putting that aside, there's still a second problem. Whenever you do an estimation, you should go back and check to see if your original assumptions make sense. Now if your wings are five footballs long, it doesn't make sense that they're thrusting only two meters downward. In the video below, they appear to be thrusting about 4 meters, which reduce the result to roughly a 65 meter wingspan.

Moreover, if I up Pegasus's wing flapping rate to 3 flaps per second, we further reduce the wingspan to a much more believable 7 meters. I apologize if I mislead any Pegasus fans out there.

I'm not sure what this says about pterosaurs. I know very little about them except that they varied in size from 10 inches (~25 centimeters) to over 40 feet (~12 meters). Just from physical considerations, I would have guessed the largest of these were more like ostriches (i.e. flightless), but maybe there's some evidence to the contrary.

There's an important take home point here. Whenever you reach a conclusion, whether it be for an estimation or a general conclusion about life, you always need to go back and ask what assumptions you made to get there. You've got to make sure the assumptions you made still make sense by the time you get to the end!


Thanks once again, Reddit. If there are more questions you want answered, message me. I can't guarantee a response (I'm still reading some from last week), but I can guarantee that if I see your question and it intrigues me, I'll answer it on the blog and give you credit.

[1] Fun Fact: Previous Diary of Numbers guest Chris Moore's book, Fluke, or, I Know Why the Winged Whale Sings, features the Bloop.

Aaron Santos is a physicist and author of the books How Many Licks? Or How to Estimate Damn Near Anything and Ballparking: Practical Math for Impractical Sports Questions.   Follow him on Twitter at @aarontsantos.

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