Q: If you could drill a tunnel through the whole planet and then jumped down this tunnel, how would you fall?

Physicist: This is a beautiful question, in a small part because it’s an interesting thought experiment with some clever math, but mostly because of all the reasons it couldn’t be done and wouldn’t work.  Right off the bat; clearly a hole can’t be drilled through the Earth.  By the time you’ve gotten no more than 30 miles down (less than 0.4% of the way through) you’ll find your tunnel filling will magma, which tends to gunk up drill bits (also melt everything).

Jumping into a hole drilled through the Earth. What’s the worst that could happen?

But!  Assuming that wasn’t an issue, and you’ve got a tube through the Earth (made of unobtainium or something), you still have to contend with the air in the tube.  In addition to air-resistance, which on its own would drag you to a stop near the core, just having air in the tube would be really really fatal.  The lower you are, the more air is above you, and the higher the pressure.  The highest air pressure we see on the surface of the Earth is a little under 16 psi.  But keep in mind that we only have about 100 km of real atmosphere above us, and most of that is pretty thin.  If the air in the tube were to increase in pressure and temperature the way the atmosphere does, then you’d only have to drop around 50 km before the pressure in the tube was as high as the bottom of the ocean.

Even worse, a big pile of air (like the atmosphere) is hotter at the bottom than at the top (hence all the snow on top of mountains).  Temperature varies by about 10°C per km or 30 °F per mile.  So, by the time you’ve fallen about 20 miles you’re really on fire a lot.  After a few hundred miles (still a long way from the core) you can expect the air to be a ludicrously hot sorta-gas-sorta-fluid, eventually becoming a solid plug.

But!  Assuming that there’s no air in the tube, you’re still in trouble.  If the Earth is rotating, then in short order you’d be ground against the walls of the tunnel, and would either be pulverized or would slow down and slide to rest near the center of the Earth.  This is an effect of “coriolis forces” which show up whenever you try to describe things moving around on spinning things (like planets).  To describe it accurately requires the use of angular momentum, but you can picture it pretty well in terms of “higher things move faster”.  Because the Earth is turning, how fast you’re moving is proportional to your altitude.  Normally this isn’t noticeable.  For example, the top of a ten story building is moving about 0.001 mph faster than the ground (ever notice that?), so an object nudged off of the roof can expect to land about 1 millimeter off-target.  But over large changes in altitude (and falling through the Earth counts) the effect is very noticeable: about halfway to the center of the Earth you’ll find that you’re moving sideways about 1,500 mph faster than the walls of your tube, which is unhealthy.

The farther from the center you are, the faster you’re moving.

But!  Assuming that you’ve got some kind of a super-tube, that the inside of that tube is a vacuum, and that the Earth isn’t turning (and that there’s nothing else to worry about, like building up static electricity or some other unforeseen problem), then you would be free to fall all the way to the far side of the Earth.  Once you got there, you would fall right through the Earth again, oscillating back and forth sinusoidally exactly like a bouncing spring or a clock pendulum.  It would take you about 42 minutes to make the trip from one side of the Earth to the other.

The clever math behind calculating how an object would fall through the Earth:  As you fall all of the layers farther from the center than you cancel out, so you always seem to be falling as though you were on the the surface of a shrinking planet.

What follows is interesting mostly to physics/engineering majors and to almost no one else.

It turns out that spherically symmetric things, which includes things like the Earth, have a cute property: the gravity at any point only depends on the amount of matter below you, and not at all on the amount of matter above you.  There are a couple of ways to show this, but since it was done before (with pictures!), take it as read.  So, as you fall in all of the layers above you can be ignored (as far as gravity is concerned), and it “feels” as though you’re always falling right next to the surface of a progressively smaller and smaller planet.  This, by the way, is just another reason why the exact center of the Earth is in free-fall.

The force of gravity is F = -\frac{GMm}{r^2}, where M is the big mass, and m is the smaller, falling mass.  But, since you only have to consider the mass below you, then if the Earth has a fixed density (it doesn’t, but if it did) then you could say M = \rho \frac{4}{3}\pi r^3, where ρ is the density.  So, as you’re falling F = -\left(\frac{Gm}{r^2}\right)\left(\rho \frac{4}{3}\pi r^3\right) = -\left(\frac{4}{3}G\rho \pi\right) mr.

Holy crap!  This is the (in)famous spring equation, F = – kx!  Physicists get very excited when they see this because it’s one of, like, 3 questions that can be exactly answered (seriously).  In this case that answer is r(t) = R\cos{\left(t\sqrt{\frac{4}{3}G\rho \pi} \right)}, where R is the radius of the Earth, and t is how long you’ve been falling.  Cosine, it’s worth pointing out, is sinusoidal.

Interesting fun-fact: the time it takes to oscillate back-and-forth through a planet is dependent only on the density of that planet and not on the size!

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45 Responses to Q: If you could drill a tunnel through the whole planet and then jumped down this tunnel, how would you fall?

  1. Andrew says:

    Period is not dependent on size if you don’t take into account relativistic effects, speed of light limitations for one thing.

    I wonder though, even if you had a planet the same density of earth but say, a light hour in radius (so impossible to have a period of oscillation of less than 2 hours), the period of oscillation in the frame of reference of the falling dude might still be equal to 1 hour and 24 mins. Not sure how to do the math to work that one out though.

  2. Will says:

    Presumably the time it takes to oscillate would depend on the force of gravity the falling object is subjected to, which explains why density is important and size isn’t.

  3. Locutus says:

    Will: size is important in determining density, though.

    If you were drilling down through the Earth, then, like seen on Core (at least I think Core was the movie name), is there any material available that could withstand the pressure (no matter its economic or production feasibility), and would drilling even be possible?

    Love Long and Prosper

  4. Viktor says:

    Hi,

    Why exactly is it that you would fall, as you say, “sinusoidally” and not keep going all the way up to the other side everytime? I always assumed that friction was responsible for this bouncing spring-effect – but as we sort of assumed all friction away here, I assume now that my first assumption was a misassumption.

    Wikipedia-link to the answer maybe?

    Thank you!

  5. Will says:

    Size isn’t actually important in determining density. A neutron star is quite small yet is absurdly dense, and of course black holes are infinitely small yet still incredibly dense, while nebulae can be gargantuan but are mostly empty space. Material is more important than size; size alone tells you very little in fact.

    There are materials that could withstand the pressure near the earth’s core (Diamond springs to mind) but I don’t think you could actually construct a vehicle out of them that could safely carry people around. A solid ball of diamond might be able to resist the pressure, but stick an air pocket in there and i’d doubt it’d last long before collapsing. Of course if the diamond was thick enough and the air pocket small enough it could withstand the pressure, but a ‘ship’ with walls of diamond hundreds of meters thick isn’t exactly what you’d call remotely practical.

  6. The Physicist The Physicist says:

    @Viktor: You would keep going all the way to the other side. In general, you can tell if there’s friction around because it slows everything down, but it doesn’t reverse the direction of things.

  7. Bobb Dobbs says:

    I’m not sure about air pressure in the tube. It seems to be a more complex calculation because the force of gravity decreases (is neutralized) as you go toward the center of the earth. Plus it would be no problem capping the ends of the tube at the surface and allowing only the desired mass of air inside. In that case, since air is free to move about and equalize itself — would there be a pressure gradient with height or would the entire tube have the same pressure?

  8. The Physicist The Physicist says:

    There would be a change in pressure with height even if the ends of the tube were capped.
    The “10°C per km” thing is based on standard Earth gravity after the air has has a chance to equilibrate. In practice, you’ll often see “inversion layers” caused by mixing and air movement. In this the calculation is tricky because the ideal gas law fails at higher pressures and the gravity decreases as you get lower. However, for the first hundred miles or so, the pressure and temperature increase in a nice reasonable extension-of-the-atmosphere kind of way.

  9. Chris Clarke says:

    Pretty sure it is 42 minutes for a one-way trip and not 84 minutes as above which would be the period. Also, angular momentum depends on latitude so you don’t need to stop the planet from spinning, just choose your tunnel to go down the rotational axis. And it doesn’t need to go through the centre of the earth (see gravity trains).

  10. Jesse says:

    I remember this problem from freshman physics. A related problem is if you drill a straight hole not necessarily through the center, but from LAX to NYC for example, the force is still Hooke’s law, maybe not so surprising.

    What was surprising to me is that the period is exactly the same no matter which two points you are connecting! So it’s the same time to go from north pole to south pole as from LAX to NYC on your “frictionless gravity train”.

    At least you don’t have to dig as deep to connect LAX to NYC. Still deep enough to get melted though. Bummer.

  11. radley says:

    lets say you drill a small hole hollow size of quarter made of tantalum the drill head made of diamond graphene and then by remote open a hole in the drill head after drilling 1,700 miles into the mantel whats going to happen ? i don’t think it would coast more than 90 billion would it destroy the planet ? and could you make power out of it or other scientific benefits or even turn into laser for laser telescope or shot an asteroid atom smasher ?????????????? comments please just curios

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  13. Simon says:

    If the time it takes to oscillate back and forth through a planet is dependent on its density rather than size, how long would it take to fall through a black hole?

  14. The Physicist The Physicist says:

    One of the assumptions of this post is that gravity behaves in the way that Newton described it (at least approximately).
    Black holes work very differently. Attempting to “fall through” a black hole would be a short, one-way trip.

  15. Kay Gee says:

    If you could drill a tunnel through the whole planet and then jumped down this tunnel, how would you fall?

    The simple answer is: Like a rock!

  16. Ashiq Rahman says:

    well… someone mentioned abt a planet of abt a light year across in diameter… if that was the case, then wouldn’t it be possible that the falling bofy would stick to the surface of the hole… as there would be too much gravity??

  17. Ashiq Rahman says:

    if there is a vacuum in the hole, then the horizontal component during falling might affect the fall, wouldn’t it?

  18. Cal says:

    With all the “ifs” aside, why wouldn’t you stop falling at the center of the earth? Isn’t all gravity pulling towards the center? I don’t understand why it says you would continue all the way to the other side (momentum?)

    To me it seems once you’ve “fallen” past the center, why would you continue all the way to the other side before “bouncing” back? The way I imagine it, once you reach the center, momentum may take you slightly past the center, but gravity would bring you back towards it, decreasing every time, like a ball bouncing less every time, until it stops and you end your journey just sort of floating in the exact middle.

  19. The Physicist The Physicist says:

    Momentum is right. To slow down and stop you’d need to be losing energy, but without friction or something you always return to the same height.

  20. Mindbeenblown says:

    ^^Hmm, would gravity not be enough to pull you back? Once you passed the center wouldn’t gravity be pulling in the opposite direction of your momentum? You’d essentially be falling/moving in the direction of the sky after the center.

  21. Craig says:

    What would happen if the tube was build with an outer sleeve and an inner sleeve with a cooling agent always cooling and flowing between the two sleeves? Also instead of the capsule being hollow and having air inside, could it be a solid magnetic sphere that was molded around whatever the sphere was carrying.

    The last thing I am wondering is could a magnetic field that would be emitted from the walls of the tube be created to compensate for the speed, moving sideways “about 1,500 mph faster than the walls of your tube”, which in combination with building the tunnel on a curve and then back again would keep the capsule from being pulverized in to the side of the tunnel?

  22. The Physicist The Physicist says:

    Cooling the contents of the tube would increase the density, pull in more air from the atmosphere outside and ultimately increase the pressure. I suspect (guess) that any half-way reasonable attempt to cool most of the tube to survivable temperatures would result in a plug of solid, frozen air in the middle.
    You could use a bunch of magnets to keep from hitting the walls, but smooth walls and a luge board would probably work just as well. Curving the tunnel so that you could fall though without touching the walls is an option that would remove the whole “wall problem” completely, but it would be a one-way trip. The tube would be curved the wrong way for the return trip.

  23. Adam Kelly says:

    @Mindbeenblown ….. I think the reason you return back to the same height is because when you pass the centre and it starts pulling back on you, there is still a decent section of the earth ahead of you pulling you forward at the same time.

  24. Bob says:

    WAIT! Physicist, I heard from another statistic that it would take 42 minutes if you jumped in from one side of the Earth, to reach the other side. So your statistic (1 hour 24 minutes) would be how long it would take for you to jump into the tube, reach the other side of the Earth, then to go back and reach your starting point.

  25. Bob says:

    Sorry, for that comment, I didn’t want to be rude- I just wanted to correct the facts.

  26. The Physicist The Physicist says:

    @Bob
    Don’t worry about it! If I get something wrong it should be corrected. I’ll double check this.

  27. Trey says:

    @AdamKelly It clearly states gravitational pull comes from the center, so once you pass the center of the earth on your decent, the earth’s gravitational pull, comes from it’s core. Therefore, you would not be getting pulled by surface and core gravity.If that were the case, aslong as 1 pull is less and the difference is below x amount of g’s, you would go back to the stronger pull, which would be the center uncontestably.

    @Will Everything has a burning temperature, including diamonds Size and density go together because of gravity. Forgot the equation, I am no physicist, but weight is determined by gravity, and gravity is determined on the density and size of an object..

    @Physicist I love all of your what ifs in your modeled theory, well done. I need to get grants to go back to school, my mind has been going nonstop lately.

  28. The Physicist The Physicist says:

    @Bob
    You’re right!

  29. Phil says:

    I would do this if it was possible.

  30. Andojar says:

    Unfortunately, the article contains a basic error, the equation of motion is:

    [tex]\frac{d^2x}{dt^2}=+kx[/tex]

    and not:

    [tex]\frac{d^2x}{dt^2}=-kx[/tex]

    You are just propagating an urban legend (and an obvious error) with the above article. The motion is dampened, not harmonic.

  31. prince.thenextking says:

    i think you are missing a point in air pressure as far as i know air pressure will increase as you will get more and more closer to the crust and it will be from both the sides .
    and when you ‘ll reach the crust your speed will be so slow that you ‘ll nt be able to move in any direction coz of pressure equiliberium from both the sides.

  32. Arturo.D says:

    It’s a standard calculation that gravity (G force) pulls an object at 32 feet per second/per second. But that’s ONLY on this planet (Other planets have a different factor depending on the mass of that planet) and only if the ENTIRE planet is below the subject. Once you enter the imagery hole the entire planet is no longer below you at an ever decreasing amount because as you fall down the hole you are leaving more and more earth (mater) above you, now the force of gravity (G force) will gradually decrease as you reach an equilibrium in the center of the earth. The calculation to determine the amount of time to fall through the earth as mentioned above assumes that the rate of fall is consistent, it’s not, and the calculation does not take this ever changing rate of fall into account.
    The most interesting thing is at what pressure would the hole (filled with air) would the air condense into a liquid assuming the air (gas mixture) stays at a constant 20.9% Oxygen and 79.1% Nitrogen and its at a constant 72°F?

  33. C says:

    If the Earth is close to a perfect sphere you might negate spin by going thorough the poles and then the tunnel spins in a nonharmful way. But the other effects mentioned and a good deal of other problems makes it an engineering feat we don’t currently have the knowledge to overcome.

  34. 60 Watts says:

    If you managed to stop in a vacuum tube at the centre of the earth, what effect would the gravitational force from the planet be? Would you be pulled to the centre of your body or would all the forces cancel out?

  35. The Physicist The Physicist says:

    They would cancel out.

  36. Ratheesh says:

    Earth attracts every object towards it center isn’t it ?then how come a falling object go past the center towards the other end.

  37. Kay Gee says:

    Theoretically, I believe it is because the velocity of the mass that is accelerated towards the center would decelerated past the center and then would eventually find equilibrium at the center after a series of decreasing oscillations back and forth (up and down).

  38. The Physicist The Physicist says:

    @Ratheesh
    All of the matter on the Earth pulls on you, it’s just that here on the surface of the Earth the combined effect points down. Closer to the center there’s about the same amount of matter in every direction, and thus no net force in any direction.

  39. Monkey says:

    many of you are confusing velocity with acceleration. As you get to the center of the earth your acceleration due to gravity will go to zero, but your velocity(speed and direction) will continue to increase(remember nothing is slowing you down if we take away friction), until you reach the midpoint. The rate at which your speed increases(acceleration) goes to zero, not your speed. As soon as you pass it you are still going very fast but the acceleration due to gravity is now pulling the opposite direction from the way you are traveling. This slows you down but only a little because this “reverse” acceleration is small at the center, but you so slow down just not much. the further “up” the other side you go this acceleration gets stronger and you slow faster and faster. and it all works out that you will exit the other side at the same rate and height at witch you entered. So as long as you had something to grab on to, you would not keep bouncing back and forth for ever if that was not your goal.

  40. Monkey says:

    so->do
    witch->which

  41. kk says:

    So as you fall through to the other side of the earth, is there a surface and atmosphere at the other end like we have at our end?

  42. rktgrl says:

    Will: Of course size is important in determining density. Specifically volume, which is described by size. The only reason anything is dense, like a neutron star for example is because the matter within is confined to such a small volume. You even contradicted yourself in your answer. You said a neutron star is incredibly dense even though it is “small.” Isn’t small a reference to size?

  43. Richard Benish says:

    The most exceptionally scientific attitude among these comments, in my opinion, is represented by Phil’s remark:

    “I would do this if it was possible.”

    Happily, using smaller bodies of matter it is quite possible, either in an orbiting satellite or an Earth-based laboratory. Unhappily, nobody has yet bothered to do it.

    Note that the first one to propose the experiment was Galileo, in 1632. Newton’s equations are demonstrably accurate for falling motion over the surfaces of large gravitating bodies. But they have not yet been tested for gravity-induced motion inside gravitating bodies.

    According to the ideals of science, we are supposed to remain unconvinced by mere predictions until they are verified by empirical evidence. Therefore, in my opinion, we are way overdue to provide that evidence (and satisfy the spirit of Galileo) by doing the experiment. For additional arguments to this effect, see:

    http://vixra.org/abs/1407.0041

  44. Léon Grenier says:

    I like the idea, very original questions. And a lot to analyze. About air pressure being 16 psi on the surface, and the pressure in the tunnel. The pressure on the surface of the earth would then fall under 16 psi to balance itself depending of the size of the whole. And since its original pressure or outer Crost pressure is 16 psi, it is impossible for it to get gratter than its origins. Take a compressor for example if 50 psi is in The tank and you open a valve to a hose of 20 feet by 1/4 inche long the psi will drop at 45 to fill the void not increase. And yes you probably would day of all the effect but is not the object would stick to the side like you do in a rotary rollercoaster before reaching the center of the earth.

  45. Léon Grenier says:

    Also I’d you dig a tunnel like that, you probably would kill earth balance as well as everything on it. The tunnel would allow the core’s heath to move outward much faster which would propell you back out by the way. By doing so the core would cool down and it’s magnetic field would rise one strength expodentially, tearing apart all carbon base life on it except lobsters, cause the are exoskeleton. Lol just joking. Than the earth, rock and other materials removed to create the tunnel would most likely upset the earth trajectory around the sun as well as its frequency pattern which could lead our planet to collide with another. The orbit of the moon would also change as well as tides. And I could keep going….

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