[old] dwarfs measure time and space

2016.01.18 19:18
[old] dwarfs measure time and space

From time to time, I will post things here on the blog that I wrote a long time ago. Today, a text about dwarfs, about 2001:
Once I asked on pl.sci.physics:

Sometimes I wonder how dwarfs perceive reality. This thought was triggered by the observation that some insects can walk on puddles, supported by surface tension, while I cannot. Thus, for that insect, water appears to be a completely different substance than the one I know. But delving into what such an insect might think is not easy, so I use the concept of a dwarf, who has a body structure similar to mine but is proportionally smaller (let's assume this is possible). That is, the dwarf is 18 mm tall - 100 times smaller than me, with a mass 100^3 times smaller, etc. - but its brain and sense organs work on the same principle as mine. It's just a 'me' shrunk 100 times.
So how does such a dwarf perceive reality? First of all, everything seems 100 times bigger to him than it does to me. For me, a 5 cm tall flower is small; for him, it is twice his height - as if it were 5 dwarf meters tall. Simple and understandable, right?
Weight. For the dwarf, everything is heavier than it is for me. How much heavier? One might say that since the linear dimensions are 100 times greater for him, the volumes are 1,000,000 times greater, and so are the weights. Therefore, what weighs 1 gram for me weighs 1 ton for the dwarf. But is that really the case? We judge weight by lifting the object in our hand. Obviously, if I were twice as strong, everything would seem (apparently?) twice as light. And how much weaker is the dwarf compared to a human? I think it's probably not 100^3 but 100^2 times weaker. Because muscle strength is proportional to their cross-sectional area, not volume. I think. I would need to ask on pl.sci.biology. Yes, it makes sense... But I won't explain it because I'll get tangled up... The point is, I understand. Besides, I think I read about this in "Szkiełko i Oko." But in that case - since volumes are a million times greater and weights only ten thousand times greater, all specific weights for the dwarf are a hundred times smaller! No wonder ants can easily carry large - compared to their size - needles, sticks, etc. So for the dwarf, water has a density of 10 grams per cubic decimeter. A liter of water weighs 10 grams... A bucket of water, which might hold 25 liters, weighs 250 grams - a quarter of a kilo. So that's probably why dwarfs are so hardworking. In our world, a car weighs about a ton. But the dwarf's car weighs 10 kilos for him.
Strange, but that's how it really turns out. Fairy tales need to be revised... Besides, the dwarf itself weighs about half a kilo.
And how does the dwarf perceive time? Does time pass slower, faster, or the same for him as it does for me? Oh, that's a rather complicated issue... Because basically, one might think that since he has a smaller brain, the nerve impulse will travel from one end of the brain to the other faster, so his brain will work faster than mine, and the outside world will seem slower to him... And a hundred times slower... Because the average distance between two neurons will be a hundred times smaller... So there are further consequences.
What about Earth's acceleration? For me, it seems that a freely falling object without resistance has a speed of about 10 m/s after the first second. (v = g*t; g=9.81 m/s^2, t = 1 s). But since a dwarf second is only 0.01 of a human second, after 1 dwarf second, this object will have a speed of only 0.1 human meters per human second. But let's convert this speed from human meters per second to dwarf units. A human meter is 100 dwarf meters, a human second is 100 dwarf seconds. So the dwarf perceives speeds the same way we do! So Earth's acceleration seems to be 100 times smaller for the dwarf than for us... about 0.1 m/s^2. So yes, he perceives speeds as the same, and accelerations as 100 times smaller... Aha, yes, and that's why Earth's acceleration is 100 times smaller for him. So: imagine that I drop my human phone from the table and the dwarf drops his dwarf phone from the dwarf table. And let's compare these two situations. For me, my phone weighs a newton and falls with acceleration b because it is attracted by Earth with mass c. The dwarf phone, weighing a/100 dwarf newtons, falls with acceleration b/100 because it is attracted by Earth (assuming the dwarf, although he has his small dwarf phone, car, and house, still lives on our Earth. If he lived on a dwarf Earth, 100 times smaller than ours, some things would look different. But more on that in a moment); aha, so it is attracted by Earth with mass... exactly, how is it with masses? How do we assess the mass of a body? We assess inertial mass by looking at what acceleration a body will have after applying a certain force to it. So yes. If I push a piano with all my strength, it will gain acceleration d. If the dwarf pushes with all his strength (objectively 100^2 times less than mine) a dwarf piano (objectively having a mass 100^3 times smaller than the human one), I will assess this piano's acceleration as 100 times greater. But this acceleration will seem 100 times smaller to the dwarf. So in summary: if I push my piano with all my strength, and the dwarf pushes his, we both observe the same acceleration - so the dwarf's piano seems to have the same inertial mass to him as my piano does to me, even though its weight is 100 times smaller for the dwarf than my piano's weight is for me. So the dwarf assesses the specific mass (I mean the mass of a unit volume of a given substance) of various substances the same way I do. But as I mentioned, I am considering the case of a dwarf living on our human Earth, which seems to him to have a volume 100^3 times greater than mine - so the Earth's mass is (100^3)*c for him.
Returning to the example with the phone: the acceleration with which the phone falls does not depend on the phone's weight. But note that although this falling is caused by the attraction of a mass 100^3 times greater, the phone still falls with an acceleration 100 times smaller... So for the dwarf, the gravitational constant (big G in the formula F = G*((m1*m2)/d^2) ) is 100^4 times smaller than for me... Is that right? No! I didn't account for the fact that for the dwarf, the center of our shared Earth is 100 times farther away. So in total G is only 100^2 (ten thousand) times smaller.
And how would everything look if the dwarf lived on a dwarf Earth, 100 times (linearly) smaller than ours?
0^3 times greater, the phone still falls with an acceleration 100 times smaller... So for the dwarf, the gravitational constant (big G in the formula F = G*((m1*m2)/d^2) ) is 100^4 times smaller than for me... Is that right? No! I didn't account for the fact that for the dwarf, the center of our shared Earth is 100 times farther away. So in (...)

Kr**sztof Mn*** responded to this:
as follows:
~beginning~of~quote~
**** wrote: [long but nice]

Great fun, highly recommend!
I also tried, and got slightly different results. Everything depends on the assumptions made, models can vary. Let's start with length. It's obvious, at scale s, length l -> s*l. Let's assume a dwarf meter = 10cm. Forces - you rightly noticed that they depend on cross-sectional area. And this applies not only to muscle strength but also to bone strength, etc. So F~s^2, meaning 1 dwarf newton = 0.01N. Now time. Here everything depends on assumptions. The simplest: the path a nerve impulse has to travel is proportional to body length. In limbs, that's true, but in the brain? Who knows how many neurons a dwarf has?! The same number or 1000x fewer? But let's stick to the simplest version - t -> s*t, meaning 1 dwarf second = 0.1s. Now note - speed v -> v remains unchanged! A dwarf runs as fast as we do. Does that make sense? Probably, just look at how fast squirrels run. Acceleration a -> s^-1 *a (length/time^2), meaning 1 dwarf meter/s^2 = 100m/s^2. So when a dwarf falls from a table, he falls 100x faster than us! Similar with muscle acceleration. Now back to weight. w->s^3*w. Our dwarf meter = 0.1m, so 1dm^3 dwarf = 1cm^3, mass 1g = 1000kg. Finally, density - we assume it's the same as ours (s^3/s^3=1). Conclusion - a 1cm^3 dwarf balloon weighs 1mg!

That leaves time. This still needs consideration... Imagine a 100x scaled-down human world: cars, houses, phones, animals... For now, let's consider a dwarf phone. Assume it's a tiny phone with tiny components but functions similarly to our phones. So for a dwarf, using this phone means pressing tiny keys, which implies precision that surpasses human abilities. The dwarf must have exceptionally steady hands and great dexterity. Also, the dwarf's phone emits sounds. Does this sound higher-pitched because the dwarf hears better? Likely yes - sound waves have frequencies proportional to the size of the emitting source. So the dwarf hears in a higher range than we do. All in all, we see that dwarfs, though they live in a scaled-down world, experience reality differently in terms of physics and perception.

But let's look even deeper into the effects of this scaling... When it comes to movements, how fast can a dwarf move compared to us? If a dwarf runs with the same speed, relatively speaking, then a dwarf sprinting feels like us sprinting through our human world. But if the dwarf's muscles work faster, does this dwarf sprint become a blur to our human eyes?
And then there's aging... Because a dwarf's cells are smaller and perhaps his metabolism is faster, does he age at a different rate? Perhaps a dwarf, because he processes everything faster, ages quickly compared to us. So a dwarf's lifetime, although experienced fully by him, would seem like a fleeting moment to us.
Summarizing these thoughts - it all leads to a fascinating yet intricate insight into how beings of different sizes could perceive and interact with the world. This not only fuels imagination but also enhances our understanding of fundamental principles in biology and physics.

comments:
2016.01.19 09:19 gfedorynski

Jest taki film dla dzieci, Królestwo Zielonej Polany. I tam bohaterka zmniejsza się do wielkości pewnie gdzieś tak 1/10 człowieka. Dzięki temu może bardzo wysoko i daleko skakać. Czyli wszystko się zgadza.

2016.01.19 09:22 gfedorynski

Ile trwa orgazm krasnoludka?

back to homepage

RSS