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Thread: Some questions I have for a video I'm making on the history of science

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    Some questions I have for a video I'm making on the history of science

    They're not particularly connected, but I didn't want to flood the forum.

    I'm confused about Archimedes's crown. Both crowns were the same volume, so presumably the experiment measured different mass. But wouldn't the volume be displaced, rather than the mass?

    What exactly does the Planck length measure? Wikipedia defines a Planck length as the distance light travels in a Planck time and a Planck time as the time it takes light to travel one Planck length.

    At what latitude does the earth experience sixth months of day and six months of night?

    this paper
    https://arxiv.org/PS_cache/arxiv/pdf...910.1589v1.pdf

    claims to be describing a multiverse. But it is a multiverse or simply a single universe with multiple mutually inaccessible regions?
    "Occam" is the name of the alien race that will enslave us all eventually. And they've got razors for hands. I don't know if that's true but it seems like the simplest answer."

    Stephen Colbert.

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    Quote Originally Posted by parallaxicality View Post
    They're not particularly connected, but I didn't want to flood the forum.

    I'm confused about Archimedes's crown. Both crowns were the same volume, so presumably the experiment measured different mass. But wouldn't the volume be displaced, rather than the mass?
    The story as it's usually told is that there was one crown, and an equal weight of gold. Precisely because the density of a counterfeit crown would be less than that of gold, it would displace a greater volume.

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    Six months of day and night is only at 90 degrees North or South (the rotational poles)
    Forming opinions as we speak

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    Quote Originally Posted by parallaxicality View Post
    What exactly does the Planck length measure? Wikipedia defines a Planck length as the distance light travels in a Planck time and a Planck time as the time it takes light to travel one Planck length.
    There are several ways to think about the Planck length. The first is simply that it is the only length you can make out of the fundamental constants of our best theories, gravity (G), quantum mechanics (h), and relativity (c). So it is the only unit of length that is built into the laws of physics (as we know them, anyway). Before there was quantum mechanics, we had no way to create a unit of length from our fundamental constants, so one might have imagined using that fact to anticipate much of quantum mechanics (though of course you'd always need the measure h to quantify the theory). Remarkable questions emerge as soon as you have a Planck length-- like why is the scale of the universe so many Planck lengths? (The observable universe is some 1062 Planck lengths, or 10186 Planck volumes. Can you imagine telling a small child, "here, take these legos and build something with them. Oh by the way, be sure to use at least 10186 of them.") We know the universe is big, but now we have a way to say how big that comes completely from the laws of physics themselves, so why is it that big? Again with remarkable prescience, that observation could have led us to anticipate the Big Bang answer: it is big because it has been expanding like crazy for a very long time.

    Another way to think of the Planck scale is to use it to bring out a problematic inconsistency between general relativity and quantum mechanics. In quantum mechanics, in order to give meaning to a small size scale we have to be able to localize a particle to that scale, which requires uncertainty in its momentum. To achieve that kind of uncertainty requires a minimum amount of momentum, and hence a minimum amount of energy for the particle, which has to be localized into the same scale as the particle is. But localizing that much energy on that small of a scale has implications for the local spacetime curvature. The Planck length is the scale that, in order to localize a particle to that scale, requires so much spacetime curvature that it would be a mini black hole. But general relativity pictures spacetime as locally smooth and continuous, so it can't be ripped up into little black holes everywhere on the Planck scale. Hence, it seems that the Planck scale is in some sense physically meaningless-- and how does one build a universe of 1061 Planck lengths by using building blocks that are themselves meaningless?

    There is a widespread hope that this defect will be repaired by a theory that unifies gravity and quantum mechanics. I'm not sure this needs to be regarded as a defect, however-- who said the universe is required to make sense? Of course the goal of science is to find whatever sense it can, so we will search for sense, but there was never any guarantee we would always find it. It has been said that if our minds were simple enough for us to understand, then our minds would be too simple to do the understanding. The universe has to be able to contain our minds.
    Last edited by Ken G; 2019-Mar-01 at 02:48 PM.

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    Quote Originally Posted by antoniseb View Post
    Six months of day and night is only at 90 degrees North or South (the rotational poles)
    That's the simple and correct answer ...

    ... unless you want to be an annoying, nit-picking jerk. In which case you can say "due to the refraction of light in the Earth's atmosphere, the Sun appears to be above the horizon when it is truly below the horizon; and that means that no place on Earth has a night which is half a year long."

    (Yes, there are times when I can be an annoying, nit-picking jerk)

    (as regular readers will know)

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    And if you want to be an even more nit-picking jerk, you add that the finite extent of the Sun also allows it to peak above the horizon longer than the idealized case. This even applies to the Moon with no atmosphere-- the Sun illuminates a bit more than half the Moon at all times, and it also means that the terminator between light and dark is not perfectly sharp.

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    I was walking on an elevated path in our recent clear weather with a low sun, watching my shadow get very long and gradually diffuse until it nearly disappeared because of the finite size of the sun and perhaps some bending due to temperature differences near me. It was rather a contemplative moment, at sufficient distance my local effect vanishes into the noise.
    sicut vis videre esto
    When we realize that patterns don't exist in the universe, they are a template that we hold to the universe to make sense of it, it all makes a lot more sense.
    Originally Posted by Ken G

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    Quote Originally Posted by profloater View Post
    It was rather a contemplative moment, at sufficient distance my local effect vanishes into the noise.
    Or, the illusion that the effect of you is somehow separate from the rest of reality became difficult to maintain.

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    Quote Originally Posted by StupendousMan View Post
    That's the simple and correct answer ...

    ... unless you want to be an annoying, nit-picking jerk. In which case you can say "due to the refraction of light in the Earth's atmosphere, the Sun appears to be above the horizon when it is truly below the horizon; and that means that no place on Earth has a night which is half a year long."

    ...
    I suspect that the South Pole might get close since it is in the dark during aphelion, so it gets a couple extra days for the refraction, and crossing of the full disk below the horizon.
    Forming opinions as we speak

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    Iíve had the OP about Archimedes and the crown buzzing around trying to get it straight in my head for a few days, so hereís my thoughts, hope they help.

    So you have a crown, say 20cm tall. How do you tell if itís solid gold? Well first step is obviously to weigh it - say you find itís 1kg. Ok, but doesnít help directly - after all, is it 1kg because thatís what a 20cm solid gold crown weighs, or would a 20cm solid gold crown weigh 1.1kg but itís been doped.

    The thing is the volume - but volumes of crowns are difficult to measure.

    One visualisation technique thats helpful is to take the comparison to extremes - suppose the crown weíre given is just 100g - how can we prove itís just cheap plastic ?

    What Archimedes realised (as Geo Kaplan said) is that cheap plastic has a different density than gold - ie for a given volume, gold is heavier than that same volume of plastic.

    To say the same thing in another way : For a given weight, a piece of gold will be smaller (ie less volume) than the same weight of plastic !

    So the question now reduces to how to measure volume - and the answer to that is the water overflowing from the bathtub.

    So, what Archimedes did was to weigh the crown (1kg as above), and then get a nugget of that same weight of pure gold.

    He dropped the crown into the water, and measured the overflow - ie the crownís volume.

    Dropping the nugget into the water would then reveal the answer. If the crown WAS pure gold, then the nugget would have the same volume, and so produce exactly the same amount of overflow.

    However, if the crown was our cheap plastic, then itís weight would be small, so the gold nugget would be small, and so the amount of overflow would be smaller than that produced from the crown.

    As I say, Iíve had that buzzing in my head for a couple of days. Archimedes figured all that out in the space of a few seconds getting into a relaxing bath. Such is the nature of genius !


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    There is more, to use his insight we weigh the crown in air and then in water, fully submerged. The second weight is less. His insight was that the weight of water displaced was the measure of the volume of the crown. You can calculate the density, hence the gold purity from those two weights and then you could check the volume of the spilled water as a physics class will do to verify that technique.

    In a way that is two steps of Eureka genius, the displaced volume and the weight of that displaced volume which lowers precisely the weight of the submerged crown. Many peole today still have trouble with the second stage!
    sicut vis videre esto
    When we realize that patterns don't exist in the universe, they are a template that we hold to the universe to make sense of it, it all makes a lot more sense.
    Originally Posted by Ken G

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    Quote Originally Posted by Ufonaut99 View Post
    Archimedes figured all that out in the space of a few seconds getting into a relaxing bath. Such is the nature of genius !
    The moment of insight was just a few seconds, perhaps. But I've always imagined that Archimedes had been spending all his waking hours for quite a while trying to figure out the answer to this dilemma: how do I calculate the volume of an irregular object? He knew how to calculate the volume of a simple shape, like a cube or a sphere, but he just couldn't figure out an answer for something like a crown. So, frustrated with getting nowhere, he went to relax in the bath, and then "suddenly" found the answer. My point is, I think that the spending a good deal of time kicking the problem around is probably just as important as the flash of insight; that you don't get the latter without the former.
    Conserve energy. Commute with the Hamiltonian.

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    Interestingly, Galileo weighed-in on the crown story, perhaps because Archimedes, apparently, never gave an account of this feat, but a Roman historian, Vitruvius, did. Galileo thought Archimedes may have simply scaled the crown with an equal amount of gold then, while still on the scale, placed each in jar of water to reveal if one was more buoyant than the other.
    We know time flies, we just can't see its wings.

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    We can imagine the inspiration of the bath because the water line rises, or a full bath spills, and we have to ask the key question why does it rise and by how much? Maybe there was inspiration in the bath or maybe it is a later embelishment but it remains a classic example of taking an everyday observation and going the extra level of thinking.
    sicut vis videre esto
    When we realize that patterns don't exist in the universe, they are a template that we hold to the universe to make sense of it, it all makes a lot more sense.
    Originally Posted by Ken G

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