[FRIAM] just for fun

[Steven A Smith]

So... *IF* you could bore a perfectly straight hole thorugh the earth to 
your precise antipodal location (probably having to put one hell of a 
"well casing" in, probably of neutronium to withstand the temperatures 
and turbulences of the core?) what would the "orbit" of a falling body 
be?   Would a massive object (e.g.  metal sphere) simply fall to the 
center and then by it's momentum "rise" until it reached it's apogee 
somewhat short of the antipodal end of the casing, slowed somewhat by 
the atmosphere?

Is this a good problem for a second year differential calculus student?  
Or might there be some simplifying assumptions that could be made?


My rough attempt to estimate the behaviour/trajectory: (spoiler?)

The "boundary conditions" suggest that upon dropping the mass, gravity 
and air density would be very close to what we have at or just above the 
surface and the mass would achieve terminal velocity (122mph for a 
sky-belly-flopper, a bit more for a true sky-diver, and probably 
somewhat higher for an iron or steel sphere, for instance) long before 
air density nor the value of gravity changed appreciably.

Near the core, the air density would approach zero (my assumption of a 
spherical earth and that the gravitational attraction of the mass 
"outside" the radius of the current location of the sphere summing to 
zero) it seems likely that terminal velocity would rise to some point, 
but it seems very difficult to estimate.

Other assumptions include that the diameter of the sphere is small 
enough compared to the borehole that there would be no significant 
amount of compression of the column of air in front of the sphere, if it 
were a "tight fit" I suspect the ball would compress the column of air 
under it until that pressure's exerted force exceeded that of 
gravitational pull and would eventually "bounce" long before it got near 
the core.

I also thought of coriolis forces, but then realized that the trajectory 
has only an R, no theta nor phi component, so in principle the sphere 
would not experience any coriolis force. (nod to Nick's Swirlies)  On 
the other hand, since the sphere would nominally be in freefall, it's 
trajectory would be influenced by it's initial velocity (relative to the 
rotation of the earth), suggesting it would follow a spiral path toward 
the center of the  earth, suggesting that if we wanted a "bullet-train" 
that went straight through the earth, we would need to give it a 
*spiral* core?    Evacuating such a a tube would allow true orbital 
speeds.   The precision required to "drop" a bullet-train car "through* 
the earth seems excruciatingly difficult (as would be coming up with 
methods for the coring and the lining) seems insurmountable... but some 
form of magnetic levitation type "correction" along the way would seem 
possible if not easy.

I think tidal forces can safely be ignored?

I fondly remember when I first heard about the Freeman Dyson's "Dyson 
Sphere" and then Niven's _Ringworld_ and his followon _Integral Trees_ 
and Bob Forward's _Rocheworld_, realizing that there were alternative 
physics/engineering regimes not that far from our current experience, 
yet quite counter-intuitive to us.

'nuff for now,

  - Steve


On 8/6/17 5:30 PM, Gary Schiltz wrote:
> That's really cool, Gillian. If you click on Santa Fe, you get a nice 
> snarky response like one of the following for the Antipodes Location:
>
> You`re alone and the water is so cold.
> Incredible! There is no one around you, just fish.
> You`re in the water and all you need is a boat.
> Most likely the ocean. Watch out for sharks.
>
> So, if you decide to tunnel straight through the center of the earth, 
> you might want to try making a little course correction somewhere.
>
> Fortunately for me, most of Ecuador's antipodes location is on the 
> island of Sumatra, which happens to grow my favorite variety of coffee.
>
> On Sun, Aug 6, 2017 at 11:28 AM, Gillian Densmore 
> <gil.densmore at gmail.com <mailto:gil.densmore at gmail.com>> wrote:
>
>     https://www.antipodesmap.com/#about-antipodes
>     <https://www.antipodesmap.com/#about-antipodes>
>
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