This is a very clear explanation. With math.
I was hoping I wouldnât be the only one who canât even figure out what the diagram is supposed to represent.
I see two arrows, three circles, and some dots, and from this Iâm supposed to get âone coin rolls around another coin of equal sizeâ ?!
ETA: Wikipedia has a GIF.
Exactly⊠Iâm genuinely amazed that this isnât intuitively obvious to everybody⊠maybe itâs just because I work with 3D graphics that my brain has become accustomed to this sort of thing.
This very Calvin & Hobbes strip has inspired a lot of brain racking over a similar problem Iâve been ruminating for years. Literally.
Are there any mathematicians in the house?
Serious question.
Fucking circumference, how does it work?
â Insane Coin Posse
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This riddle (itâs not a paradox) works by getting the reader to confuse rotational and linear movement. Take the case where the coin rolls along the flat surface. Instead, think of the coin as rolling along the surface of an infinitely larger coin. The red line will be 2piD in length.
Now think of the coin rolling around an infinitely smaller coin (i.e. a point). The red line would have a length of zero.
All other cased fall in between these extremes. Only when the coins are identical size does the red line equal pi*D, which is 1/2 the circumference of the coin. Itâs a trick, not a paradox. There is nothing paradoxical about it.
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SRSLY, youâre in luck. In this bbs, you canât (or can) swing Schrödingerâs cat without hitting one.
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Well, but in the buttom diagram the red line doesnât follow one point on the coin either, it follows a line that the coin rolls around, which every point on the outside edge of the coin touches. That doesnât really have to do with the problem, I donât think, since neither actually follows a point on the coin around (otherwise the second one would show a semi-circle).
But youâll get different distances by choosing different points on the coin as well, even if you stick with the same one all the way around. In this example, think about the distance traveled by the top edge of the coin compared to the bottom edge of it. The bottom goes much further. If you spin the coin without moving it than the points on the outside travel 2 Pi r but the centre wonât move at all.
This is just playing with our ill-defined ideas of what a thing is and applying intuitive ideas to a place where they tend to break. A coin going around a circular track has an âinside edgeâ even which exact point that is on the coin changes as the coin moves.
I think the issue is once you start picking a point which is not the inside track, it becomes obvious that your intuitive idea of âinside trackâ doesnât actually work. Watch the top edge of the coin, it goes up, as the coin rolls a quarter turn, comes back down, goes back up, comes back down. If they put that line on the diagram then it would remind us that rolling things move all around.
Donât worry, it took me a few minutes to figure out what they were talking about.
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With his clarification, @Nelsie is right too. Like you say, the rolling coin travels 2ÏR and makes a full rotation. But it only rolls ÏR along the semicircle and so makes a half rotation relative to that track. Which works out, because the other half rotation is that the semicircular track itself turns around.
Iâd thought he was mistaken because we werenât using the same language. For instance I thought his âlineâ meant the straight line, but it seems to have been the semicircle being followed. Which is why I figured I should post a spoiler; sorry to @anon50609448 and anyone else who was avoiding one.
By the way, in case anyone does want to know, the notation for that is [spoiler]âŠ[/spoiler].
In case anyone is interested in what happens to a particular point on the coin, the page Jorpho links has an animated gif of that too. It also appears on the article for cardioid, which discusses some properties of its path.
That should say, âno matter what the coin rolls the length of the circumferenceâ.
Hereâs the issue. Forgive the long-windedness.
What Calvinâs father says is true about the album, that the edge has further to travel to make a full rotation than the centre so therefore has to travel faster to make that rotation.
This is easily demonstrable to anyone whoâs played âWhipâ as a kid, where you make a long chain holding hands and the edge kid gets spun off due to speed.
I was wondering about extrapolating this to planets and the effect that this may have.
If I understand correctly, according to Einsteinâs equations, the faster an object of mass is traveling, the more mass it accumulates.
And that the greater the mass of an object, the stronger the gravitational pull. This would suggest to me that the gravity at the equator of a spinning body would be far stronger than the gravity at its poles. This would be less pronounced for smaller planets like Earth and Mars, and even less so when topography is taken into account (taller mountains = higher gravity in that locale), but would become far more pronounced when applied to large bodies like the Sun. A point that made me think further along these lines was the recent discovery about the near-perfect spherical shape of the Sun, and how the juryâs still out on why. It didnât match theoretical predictions that the Sun should be fatter in the middle.
Could it be the higher gravity pulling the equator toward the centre?
This, in turn, got me to thinking about accretion discs and their formation. Iâve read a fair bit on it but havenât seen any reference to âEquatorial Gravityâ, only the inwards spiraling and the loss of angular momentum of matter.
Is it possible that part of their formation is also down to a higher gravitational plane at the equator than at the poles?
This led me then to think about the formation of spiral galaxies and their similar disc-like appearance using the same logic.
Using this as a base, I thought that we cannot simply use the size of a body, or a galaxy and the number of bodies within it to calculate its mass, but that we would have to use the rotational speed to calculate the gravity along the equatorial plane, which to my mind would account for a lot more mass than simply using the former.
Where would this leave dark matter in the difference in the calculations?
I really donât know.
If I were an Astrophysicist or Mathematician I could at least do some sums on the topic.
I would love to hear some educated opinions on this.
I was going to go with âround is funnyâ, butâŠ
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Well, yes and no. Mass can mean either invariant mass, which is an inherent property of a particle and doesnât change, or relativistic mass, which is in a general sense a measure of its resistance to forces. I think these days physicists usually stick to the former, since the latter is just minvariant + Ekinetic/c2.
In any case, itâs true that when you see someone traveling faster relative to you, you observe their relativistic mass as increasing. But do note itâs not really an accumulation, because relative to them, youâre the one traveling faster, and so your relativistic mass is what increases. Trying to apply gravity and rotation in these situations can be messy, and to truly work it out properly, you need to use general relativity considering how spacetime bends and such.
Hereâs the main point, though: when you have a rotating frame, the increased centrifugal force at the equator is always going to completely overwhelm any such increases. In fact nothing in the solar system turns nearly fast enough to care about such relativistic effects; even for the sun, the equatorial speed is something like 1/30000000th the speed of light, so the changes will be correspondingly negligible.
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Thanks for the reply, you seem like a likely candidate for answers. If I can ask then, if the relativistic mass is not truly an accumulation, then how does that square with the proposed speed limit of light, where an object with mass requires increased acceleration to theoretically come close to this speed due to the mass increase, and would require infinite energy to propel it thusly?
Or am I missing something else?
As @chenille said, centrifugal force completely overwhelms any relativistic effects of the rotation of the sun. If you want to get strong relativistic effects due to rotational energy of macroscopic objects, youâre gonna have to look at massive rapidly spinning pulsars or black holes. And even with black holes, you get interesting distributions of gravitational effects.
One interesting theoretical model shows that if you spin up a black hole to mind-boggling angular velocities, its event horizon squishes out into an oblate spheroid, then mashes into a bit of a doughnut shape, then eventually shrinks to the point where the event horizon would be within the singularity itself, creating a ânaked singularityâ. This violates the black hole censorship conjecture, and we donât know if itâs actually true, but it would be interesting to see if it actually.
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I like to think of it like this: The more energy you add to a particle, the more likely it is to just spit out more matter of equal energy to what you put in.
This is why at the LHC, when two incredibly fast protons collide, it doesnât usually make something like an uber-proton-dealie with a ton of mass-energy (not for very long at least), but instead you get tons of electrons and muons and gamma-rays and other particle zoo animals, because entropy favors the creation of particle pairs over incredibly massive particles that are unstable. If you put enough energy into a photon of light, itâs gonna split into an electron and a positron.
So that increased mass from trying to accelerate the particle to c is kinda like ratcheting up the probability that the particle just shatters into other stuff moving more slowly.
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Hmm, that kinda makes sense. Though Iâd thought that the Particle Zoo was attained by the sheer force of smashing alone. Further reading required on my part.
Regarding naked singularities, if Loop quantum gravity turns out to be correct, then it would appear they actually could exist in nature.
So much to know, so little time, yet how far weâve come.
Well, it does work. Iâm just cautioning against viewing it as any actual change in the object, rather than just in your observations. In its frame of reference, you are the one who would be moving, and so itâs your energy that would be approaching infinity.
Another way to look at the same thing is to forget about relativistic mass, stick with the invariant kind, and just notice things like kinetic energy change more than Newtonian physics recognizes:
Ek = minvc2/(1 - v2/c2)1/2 - minvc2
Ek = 1/2 minvv2 + 3/8 minvv4/c2 + 5/16 minvv6/c4 + âŠ
This simplifies to Ek â 1/2 mv2 when the speed is small, but it goes to infinity when you approach v = c. Personally I find that a little easier to appreciate. You do still end up with lots of cases where different frames of references get the same result in different ways, though.
For instance, cosmic ray particles like muons can make it through the atmosphere even though their half-life is shorter than the trip. In our frame that happens because theyâre moving quickly and so their clock is slower. In their frame itâs our clock that is slower, but also our rulers are shorter, and in particular the atmosphere is much shorter to travel through. Such is relativity.
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This has probably already been answered, so sorry to those that have âŠ
The misdirection is/are the arrows pointing to the circumference of the circles. The key distance traveled is actually indicated by the center point of the blue arrowed circle. In the flat route the center moves the same distance as the circumference of the circle. In the hop over route the center point moves half the distance of the circumference of a circle with a diameter equal to that of the blue and red circles (two blue circles). Half of twice as long = as long.
These kinds of relativistic ideas I can grasp, and I can start to see how this applies to my initial premise (though through a glass darkly still). Thanks muchly for yours and @LDoBeâs responses. Iâm going to have to do a lot more reading on Invariant mass and its relation to relativistic mass to lift the mote from mine eyes.
EDIT:
A great explainer here too.
And another great site.
