Perhaps the main conflict if the earth were a cube would be that students would become much more frustrated trying to estimate the gravitational landing field . For a consistent cube with side length L and concentration rho , the gravitational military force on mass m at berth ( x , y , z ) is given by
with
where alpha , genus Beta , and gamma are + /-1 . This is the 10 - component of the force . By symmetricalness , the y and z components are given by swap y for x and z for x in the above equation . ( It ’s just a subject of integrating Newton ’s jurisprudence of gravity over the block , but it produce a little messy . )
If we want a cuboidal earth , with the same mass as spherical land , we get L = 9340 kilometer . Let ’s define aside the concern that such a planet might not be stable for very prospicient , and take a face at its property .
The graphs below show line of products of force in blue , and equipotential lines in purple for two different slice through the cube , as shown . The black square ( rectangle ) bespeak the bounds of the cube .
First , notice that at the surface of the regular hexahedron , the lines of force are not in ecumenical perpendicular to the faces of the cube . If you were standing on the cube , you would therefore comprehend that you were stand on a slope . The apparent angle of this slope is leave by the angle of the gravitational force relative to the face of the cube . The graphical record below shows this perceive angle as a function of positioning over one face of the cube .
The apparent incline is high near the corners of the regular hexahedron , all the mode up to about 50 degree from the horizontal . Note that if you were standing on a fount of the cube , you would not see the control surface as testify in the bod above . The open would appear altogether flat , because it is . If you were accustomed to live on a orbicular world ( as we are ) , I believe you would find like the entire face of the cube you were standing on were splash at a fixed slant , and this angle would count on where on the face you were standing .
Second , we can take a look at the equipotential line on the graphs above . These lines , shown in purple , are line along which the gravitational potential energy is changeless . If the ground ’s water were divided equally among the cheek of the cube , it would lie with its open along an equipotential line . If we take the total volume of the body of water on earth ( ~1.4e9 km^3 ) , I estimate that the six ocean would sit on each human face , extending about L/5 from the center of the boldness , and have a height in the heart and soul of about 0.03L. This corresponds , perhaps , to the equipotential line closemouthed to the nerve of the regular hexahedron in the graphs above . That is , each sea would be a lens frame with a round of golf , pretty square off , border . These lense SHAPE would be roughly 4000 km across , and at the center , would offer about 300 km up from the face of the square block .
The equipotential lines also indicate a origin along which atmospheric pressure would be unceasing . From the right graphical record above , we can see that the atmospheric imperativeness at one of the corners is the same as the pressure about 0.22L ( 2200 km ) above the heart of the face . see as how on globose earth , the boundary between the atmosphere and space is considered to be somewhere around 100 kilometer , the same aura on cuboid dry land would leave the corners of the cube quite far outside the atmosphere . In fact , if we were to choose a particular insistency as the boundary between atmosphere and outer space , this boundary would look similar to the lens shape described above for the oceans , but somewhat larger .
Another belongings of the cubical earth ’s gravitative voltage is that the violence act on you would minify as you move from the center of a aspect to a corner . The graph below shows the total force represent on a 100 kg soul as a occasion of spot on a face of the cube , divided by the gravitational acceleration guanine at the surface of globose earth . By dividing the force by g , we can see your “ evident mass ” – the mass that would feel the same on spherical ground .
Note that even at the heart and soul of the grimace , you are lighter than on spherical earth , and your weight decreases importantly as you travel to the corners .
A final full stop of interest is to bet at how the gravitational force scales with distance from the regular hexahedron . On orbicular earth , soberness decreases according to inverse square of distance from the center of the dry land ( F~1 / r^2 ) . The behavior is more complicated for cubical ground . The graphical record below shows the tycoon of r by which the gravitative force changes , along a bank line extending vertical from the center of one grimace .
At large distances ( > 1 or 2L ) , the military force falls off as 1 / r^2 . At this aloofness , the particular anatomy does n’t really matter . But at close distance , the military force falls off more slowly , approaching 1 / radius near the surface .
What about equipotential surfaces, extent of oceans and atmospheres, and escape velocity?
I was unable to analytically find an manifestation for the gravitative potential of a cube , so I just had Mathematica numerically integrate the gravitative field along a way from the origin to the compass point of pursuit . From this , we can retrieve quantitative value of the potency . Also , the equipotential line can be evaluated numerically by solving a differential par to see lines that are always perpendicular to the gravitative study .
The figure below usher two equipotential crinkle , where the vertical bloc extend up from the inwardness of a fount , and the horizontal axis put out to the heart of an edge . This close to the surface , the equipotential surface are quite rotary ( see below ) , so the equipotential open that curb this seam can be gauge by revolving the billet about the erect axis of rotation . From this surface of gyration , we can forecast the loudness contained between the cube and this surface . The purple line in the figure shows the approximate surface of the sea , if the earth ’s water were divided equally among the faces . If the aura is such that pressure is 1 standard atmosphere at the surface of the ocean , and we assume that pressure decrease exponentially ( a middling good estimation ) , then the edge between “ atmosphere ” and “ distance ” would be approximately at the blue equipotential pipeline . ( This edge is chosen to be at the same pressure as at 100 km above the spherical earth ’s control surface . ) Interestingly , we see that the large legal age of the cube ’s surface is outside the standard atmosphere . Since humans can survive only at about 10 km above the world ’s control surface , this think that the habitable estate on the block would be a minute ring around the oceans , about 10 km wide of the mark .
It is interesting to take a looking at what the equipotential surfaces look like . The figure below indicate 4 equipotential surfaces outside the third power at increasing values of the potential . The leftmost public figure shows the shape of the oceans , and the second to left figure of speech shows the boundary of the standard atmosphere , as described above .
We can now easily work out the radius of curvature of the equipotential surface at any head . The chassis below shows the wheel spoke of curvature R of the equipotential surface at a point a aloofness x from the center of the cube , go towards the center of a face . For equivalence , the radius of curvature in a spherically symmetric potential is render , with the finical economic value of the solid ground ’s radius indicated .
We see that when the equipotential surface is far from the aerofoil of the cube , the curve is about the same as if the quite a little were spherically symmetric . Near the surface ( where we see the kink ) , the radius of curve increases , meaning that the equipotential surface is drop out , reflecting the shape of the block .
In peculiar , the radius of curvature at the top of the sea is about 8830 kilometer , ( For the surface of the oceans , we can assume that the surface is spherical , so the spoke of curvature is the same everywhere . ) mention that the top of the sea is about 4770 km from the meat of the cube , so this is quite near the twist above . This spoke of curvature is larger than the radius of curvature of the spherical solid ground ( 6400 km ) . This results in a horizon that would be slimly farther away , as see from the same height above the piss . ( At 10 m above the surface of the ocean , the apparent horizon is 13 kilometer away on the cube and 11 kilometre forth on earth . ) An interesting observation is that if you were sailing in a ship from the center of the ocean towards the shore , the first land to come in to sight would be a very remote edge or apex of the cube . A vertex could add up into prospect in as little as 130 km from the center of the sea . The shore would not hail into aspect until it was over the horizon , about 1300 km from the kernel of the sea ( and 13 km from shoring ) .
With the gravitative potential , we can also easily count the escape velocity – the velocity you would need to give to an object somewhere on the block so that it will move away from the cube forever ( neither descend back down nor go into cranial orbit ) . This is just the speed where energizing energy is equal to the variety in potential energy from the surface to infinity . At a vertex , escape speed is about 6050 m / s. At the center of a human face , it is about 7470 m / s. This can be compare to escape velocity on the surface of the solid ground , about 11000 m / s. If the cube is rotating with a full point of 24 hours , the speed of any degree on the block will be much less than escape velocity ( which also the caseful for spherical earth ) . In the best case scenario , if the third power is rotating about an axis through the middle of two diametric border , then farthest out point at one of the peak is moving at about 600 m/s .
Image : Shutterstock / Anton Balazh
This postwas republished with permission fromJesse Berezovsky , a Professor of Physics at Case Western Reserve University . It originally seem onQuora .
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