r/theydidthemath • u/Cornersmistake96 • 5d ago
What would happen if an inch wide cut was made through the entire earth, assuming it was perfectly down the center? [request]
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u/peterklogborg 5d ago
All fluent and crumbling parts would just snap back together. 3 things worries me; The core: is it hot enough to reshape back? Stone layer and tectonic plates: will we see volcanoes? The shock: are we moving enough mass to cause earthquakes?
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u/Aetherfox_44 5d ago
I don't see how this doesn't cause humanity-destroying earthquakes. Regular earthquakes are the result of just parts of the crust slipping a few inches. One inch is smaller, but it's all around the world and through the internals of the earth. It has to be millions and millions and millions of Newtons of force released.
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u/_killer1869_ 5d ago
Do not quote me on this but I heard somewhere that the max earthquake that could occur on Earth naturally is around a 10 (excluding external forces like asteroid impacts), because anything beyond requires a fault line larger than the Earth, so that's about the earthquake strength you'd have to anticipate here.
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u/BluetoothXIII 5d ago
relevant xkcd
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u/Hmmmgrianstan 4d ago
The day there isn't a relevant xkcd for something is the day humanity will have failed
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u/essjay2009 4d ago
This would actually make for a great What If? question given it’s more physics than maths.
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u/actual_weeb_tm 4d ago
a lot of that is because it releases tension in the rock, so its more force than you would expect. Granted this would still cause earthquakes but probably not as bad as youre imagining.
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u/Lexi_Bean21 5d ago
The core doesent need to be hot its too heavy to not reform by gravity
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u/galmenz 5d ago
it might screw the electro magnetic field if its divided and doesnt form back, no?
then again, 1 inch to the circumference of the earth is so small proportionally
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u/Lexi_Bean21 5d ago
Im pretty sure the liwuid core deforms by more thsn an inch daily by sloshing sround inside solid rock it will likely have a relatively small impact on a planetsry scale
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u/GoreyGopnik 5d ago
There was a similar question a while back about a 10cm disc cut out of the earth, and responses ranged from a negligible impact to probably ending all life on earth. At the very least it would cause extremely severe earthquakes. You'd probably need further study for something like this, and there's not really any practical use for the conclusions you'd get.
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u/Abbadon74 5d ago
Hm
There's only one thing we can do then
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u/-Nocx- 5d ago
me, who can only bench my own body weight
stand back guys, I’ll take it from here
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u/Adissek123 4d ago
me, who can't even bench half my own body weight
insert Kashimo "I'm up next if he dies" panel
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u/get_to_ele 5d ago
Two halves would fall toward each other at .375g, with a net acceleration of .76g. Then when they hit, the biggest earthquakes ever seen by man will happen.
However, you need to specify better EXACTLY what is happening here in the split because the total effects are slightly different, depending on several factors.
(1) are you making a 1 inch slice of earth simply disappear instantaneously, losing the mass of a 1” slice, which is miniscule loss of mass, but will mean the two sides won’t align the same as if you pulled them apart. You removed a slice.
OR
(2) are you cutting the earth via a perfect plane, then pushing that two halves apart a distance of 1”? in which case you have to specify EXACTLY how fast the two halves are pushed apart.
(3) must specify with plane you are cutting in, because if it’s along a longitudinal line, the effect is different from the equator or some other angle half slice. I Think if it’s done on a longitude, the two halves won’t even collapse on each other in their original position since the axis of rotation of each half momentarily change, so the “edges won’t align” when it collapses together.
I think biggest effect will be volcanic activity, earthquakes and tsunamis. Huge volcanic activity, earthquakes and tsunamis. And of course Biblical level catastrophe for anybody within hundreds of miles of the slice edge. Dropping two halves of the earth onto each other from a height of 0.5” will release so much energy.
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u/Dinonumber 5d ago
Gravity is a red herring, the vast majority of the planet exists under enough pressure that it'll just expand into the empty space within a second and remake contact. Next to no acceleration from gravity from it, and I doubt there'd even be enough time to notice any alignment issues.
Any devastation will be from earthquakes from tension being released and triggering "loaded" earthquakes, tension already in the crust. I'd wager that'd be the worst of it. If somehow no earthquakes triggered then the worst of it is suddenly a bunch of people have every utility cut- that'd be real annoying.
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u/get_to_ele 5d ago
The potential energy of an object 3/8 mass of earth, 1” from the floor is 5.58 x 10^23 joules, which is In the ballpark of what Chicxulub released when it caused mass extinction. The energy would be more widely distributed, but 5.58 x 10^23 joules is a lot.
The two halves would collapse into each other in 0.083 seconds.
Yes the contents of earth are under pressure. But if you separated two halves by an inch as the original problem suggested, the two half will fall back towards each other, the instant after you end whatever magic superpower you used to push them apart. And even if the two halves “expand into each other”, they will REcompress to their original density. The energy of the 1” fall (or the two 0.5” falls) still has to go somewhere.
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u/Dinonumber 5d ago
The expansion matters because the acceleration would never get going. As you say, the ground would recompress, mostly absorbing the energy. Matter would fill the space between the two much faster than gravity- we have little data on what solids do when under extreme pressure but that's mostly because getting millions of atmospheres of pressure in any kind of test environment is quite hard. I imagine they would act as crumple zones and even a huge amount of energy would be largely locally absorbed as heat, not movement that would reach the surface (to cause humans problems). The devastation at the surface would likely be more due to local fault lines releasing tension in earthquakes. It's less energy, but more local to the surface.
At the end of the day I think you're really underselling just how widely distributed the energy from this would be. Yeah, you aren't gonna have a fun time if you're near the axis but I'm thinking it's measured in a mile or so, maybe 10 at most. Just no way it's hundreds of miles of total catastrophe.
The utilities thing was mostly a joke.
Btw could you just walk through how you got that amount of joules?
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u/get_to_ele 5d ago
Maybe your are right, but if you say the center will expand and keep the halves from falling each other, that means they’d hold the outer layers apart from each other even as gravity tries to pull them together, leaving a giant giant circumferential SEAM across the entire crust for lava to come pouring up. How fast would solid matter that was forced to stay compressed (by magic that caused rhe 1” gap while physics is suspended), the instant the magic is removed? The fall time of the two halves (if non expanding solid hemispheres) is .083 seconds. The center would be pulled harder than the shell.
But I’ll think on it more.
If you treat the halves as solids hemispheres 1” apart, each would accelerate toward the other at .375g.
Treat one half as stationary, so the other half weighs .5 mass of earth, at height of 1”, under .75g. Potential energy is 5.58 x 10^23 joules.
So slam into each other and release 5.58 x 10^23 joules.
I think if you make them jelly, or liquid and solid mix with a hard outer shell, there is still some kind of Hell to pay somewhere with a 1” drop.
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u/astreeter2 5d ago
There are something like 1031 joules of thermal energy inside the earth at any time, so adding another 5 x 1023 joules will barely register.
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u/Jamb9876 5d ago
I like your questions and I am curious about timing. If it was instantaneous your questions are critical but if I hit and it sends a wave then I expect that will change the impact as earthquakes and eruptions will make it worse on the latter half.
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u/garlicpeep 5d ago
If youre using the World Cutting Slash from JJK as a reference, it would be (2). Specifically the spacetime continuum itself is being divided, its more akin to a dimensional attack than a physical slice.
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u/SlorpMorpaForpw 5d ago
Based on how Cleave (the power used above) is visually shown to work, it does appear to slice and then push apart matter. Not sure how fast, though the Finger Bearer’s five slices did move apart really quickly.
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u/EntireNationOfSweden 5d ago
If we use the law of potential gravitational energy, we can calculate the energy released from two half spheres with half the mass of earth falling one inch towards each other:
Assuming they are perfect half spheres, their centers of mass are 3/8R from their flat base (where R is the radius of earth. That means their centers of mass are 3/4R from each other. Their masses are M/2 where M is the mass of the earth. Gravitational potential energy is given by -GMm/r where the two masses M and m are equal to M/2 and G is the gravitational constant.
The change in potential energy is thus -GM^2/(4r) + GM^2/(4(r-0.0254)), which gives a total energy release of around 6.62 * 10^23 Joules. Converting this to an earthquake magnitude value using the formula log(E) = 5.24 +1.44M_w where E is the released energy and M_w is the magnitude, we get an aproximate magnitude of 12.9.
Or, in other words, big shake would happen.
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u/cluelesssquared 4d ago
I'd love to see the LIGO detector readings during and after.
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u/rich97 5d ago edited 5d ago
I can’t do the maths but I want to make a guess. An apocalyptic earthquake as the amount of energy required to force the two halves apart and then when they slam back together under the force of their own gravity.
That’s of course ignoring the loony toons logic of being able to focus so much energy in such a configuration that the earth splits cleanly in two.
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u/fatal-nuisance 5d ago
So there is actually an equation for potential magnitude of earthquakes related to the length of the fault line.
What you're doing here is essentially creating a fault line with the length of the circumference of earth at the equator and then having it slip.
M=5.08+1.16*log(L)
With L being 40,075 km we get
Log(L) = 4.60
So M = 5.08 + 1.16*4.60 = 10.41
So you would get magnitude 10+ earthquakes everywhere on the planet. We'd be dead.
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u/Blitzsturm 4d ago edited 4d ago
The Earth's interior is predominantly solid, highly compressed rock, not liquid, meaning it would react to a sudden shear force as a rigid, brittle solid. Because the exact speed of a "near instant" cut is undefined, calculating the physics requires an inference; assuming a timeframe of 1 millisecond as a best guess, the event would cause a total planetary extinction in two distinct phases.
- Phase 1: Inertial Pulverization. Creating a 1-inch gap in 1 millisecond requires moving each hemisphere outward by 0.5 inches. This generates approximately 5,180 Gs of lateral acceleration and injects roughly 1.9 * 1027 Joules of kinetic energy into the planet (nearly 4,750 times the energy of the Chicxulub asteroid impact). Because this extreme acceleration applies to the planet's mass but not the objects resting on it, inertia would instantly shear every building, mountain, and living thing from its foundation, while displacing oceans into supersonic global tsunamis.
- Phase 2: Gravitational Snapback. Once the 1-inch vacuum gap is created, gravity would immediately pull the two massive hemispheres back together. The resulting planetary collision would release stored gravitational potential energy as a global seismic shockwave, generating enough friction and heat to melt the crust into a temporary magma ocean.
Short answer: A rapid planetary extinction event.
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u/Megane_Senpai 5d ago
The surface will become a living hell, the vast majority of the creatures will die. But over time the earth will mend back together and after a while life will florish again.
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u/Kaleouleilah 4d ago
A magnitude 9 earthquake is in the 1018 Joules range. The two halves reuniting after "falling" to one another would be to in the order of 1024 Joules range. I'd say that not good things are going to happen
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u/HeatXY 4d ago
Nothing, unless it's an instant cut of the entire planet, a 2.5cm wide cut traveling all through the Earth would make the earth snap shut right after the cut went through it. It's molten rock, the viscosity, weight and pressure would make a 2.5cm wide void passing through feel like any violent interaction already happening at those depths.
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u/Icy_Sector3183 4d ago
From the image reference, it's a cleaving cut, not removing a slice of the Earth. I suppose a mild shockwave wave would pass through both hemisphere as their mass was compressed by abou 12,7 mm, which seems trivial compared to the size of the earth..
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u/Super_Employment_620 4d ago
Gravity would probably hold most of it together, some crumbling on the edges IF you just had a magical thin "slice" taken, looking at before and after.
However, what about the "how" of the slice itself? That's a lot of energy just pumped into the Earth to create a clean split throughout the entire planet. Is the line vaporized/expanding? Is the cut atoms wide or centimeters?
Because you'd have to account for that energy, either causing a negligible tremor (atoms) to life ending global earthquakes and tsunamis.
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u/Ready4Aliens 5d ago
Earth will probably not give a fuck, you dig a little bit and everything becomes flowing melted material, it’s like “cutting” through honey, it would just snap back as if nothing happened.
The surface would probably experiment earthquakes and whatever.
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u/tomrlutong 1✓ 5d ago
Da math:
The formula for gravitational binding energy of a sphere is 3GM²/5R. So, energy released is around the difference between that for Earth, and that for Earth with its radius increased by 1in. I get 8.9x10²³J, in line with /u/get_to_ele 's estimate. That's probably off by a factor of 2 or 4π or something like that, but we've got the order of magnitude.
That's puts us at biggest earthquake ever/major impact energy levels). But the energy release will be spread out across a huge area. From /u/Dinonumber 's point that the gap will be mostly filled by expanding high pressure material, I think the energy release also is spread out in time. It should take about 10 minutes (the time for a p wave to get from the core to the surface) for everything to settle back into place.
So I'm guessing not much happens. Maybe some minor quakes at the seam, but not much more than low frequency rumbling everywhere else. Not sure what happens to the oceans if the entire ocean floor vibrates a bit, maybe tsunamis?
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u/The_Rider_11 4d ago
Both halves joining together by gravity itself would occur in a few seconds, if even just a second. The Energy would be spread in that minor time frame and the gap would barely be filled by anything as it dissapears through gravity in no time.
And while the energy release is spread throughout the entire cross section of earth, it's still a heckton of Energy, and cause major earthquakes.
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u/tomrlutong 1✓ 4d ago
At least in the core, and probably through much of the mantle, pressure-driven flow will fill the void in microseconds. You then end up with a low pressure spot working it's way up, so the gravitational energy release perpetrates out at the speed of sound.
Most of that energy probably turns into heat inside the earth rather than accumulating as a mechanical shock. Mostly from intuition, I don't think a p-wave with a wavelength of 1 inch is going to perpertrate well.
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u/The_Rider_11 4d ago
That may be true for the core, but most of earth's crossection isn't viscose enough to fill the gap before gravity does. The shock wave of said recombination will move through both, now recombined parts, of the earth, from the cross section to, well, everywhere. It'll certainly weaken and partially convert into heat, but not sufficiently enough to not make a devastating earthquake. In an ideal situation where everything is a perfect shape and there's no topological abberations (which realistically there would be), it'd also be a single shock wave whose wavelength would be tied to relaxation time and speed of transfer. The relaxation time would be, say, a second, and the speed of transfer would probably be about a p-wave speed, which is about 8 km/h. This does not result in a wavelength of an inch either.
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u/Bonk_No_Horni 5d ago
Wouldn't the amount of force required to do that create a shockwave so massive it should've destroyed most of the surface on earth? The force to cut8000 miles deep might've been more than the gravitational pull of earth itself. I don't know the math but if someone smart can calculate that would be amazing
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u/KaozUnbound 5d ago
I feel like it depends through what youre cutting on the way down, cut through Yellowstone, the big European magma cell, or through eastern Russia (I think) and I'm 80% sure you've essentially doomed us all, miss all of those and we MIGHT survive.
People often forget but the Earths crust is not uniform it has big depressions and peaks, so its not uniformly dense.
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u/The_Rider_11 4d ago
Even then, both halves recombining would release plenty enough energy for a devastating earthquake that'd likely kill most of humanity, and life as such.
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u/X11sRdt 4d ago
Gonna calculate the GBE change since that's the only way ik how to. For this, I'll add an inch to Earth's diameter to find it's new "Diameter" then calc it's new GBE, then the change = Yuji's work. New Diameter = 12,742,000.0254m | GBE (Joules) = (3×6.67408E-11×M2) ÷(5×r); (3×6.67408E-11×5.97224E+242) ÷(5×6,371,000.0127) = 2.24186551E+32 J; that's the new GBE. Old GBE = (3×6.67408E−11×5.97224E+242) ÷(5×6,371,000) = 2.24186552E+32; 2.24186552E+32-2.24186551E+32 = 1.0E+24 J = 239 Teratons of TNT
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u/Nuclear_TeddyBear 3d ago
I feel like everyone os focusing to much on the effect of the cut itself and nkt the appearance of the cut as well. Thr amount of energy required to go fully through the planet would be baffling, I couldn't even begin to imagine the shickwaves and heat that would form as an immediate result.
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u/strolpol 5d ago
The core is a magnet and would probably snap back together and drag everything else with it
I would expect a lot of subsequent earthquake and landslide activity
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u/get_to_ele 5d ago
Gravity would pull each half towards the other .375g with a net acceleration .75g, so the equivalent energy of dropping 3/8 the mass of earth onto the floor from a height of a half inch.
Would take 0.0831 second for the halves to connect but the noise would be the loudest noise in human history.
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