## Thursday, February 18, 2010

### Adam Savage and Digging Holes to China

Today’s question comes courtesy of MythBuster and geek superhero Adam Savage. Adam was nice enough to write not one, but two questions:

How many licks to get to the center of a Tootsie Pop?
Where do you have to dig a hole through the center of the Earth to reach China?

Since I’ve already addressed the first one in How Many Licks?, I’ll focus on the latter. First off, there are a couple of ways to interpret this question. Adam specifically asked “where” you have to dig a hole. The snarky answer is, of course, “Through the center of the Earth, dummy,” but it’s never a good idea to offend geek superheroes, especially ones that are nice enough to write you back and have a propensity for blowing things up. But if we want to end up in Beijing and pass through the center of the Earth, then we’ll have to start at the opposite end of the Earth and travel a distance equal the Earth’s diameter (~1.3×104 km). China’s longitude and latitude are 39.55º N and 116.25º E. The exact opposite position on the globe would be given by 39.55º S and 63.75º W. By plugging these coordinates into Google Maps, you can see that this would put us on the east side of Argentina.

I’d like to expand on Adam’s original question because there’s a lot of interesting physics in this problem. First off, let’s say we build a giant elevator shaft through the Earth and you jump inside. Would you make it all the way to China? The answer is almost certainly no. The net gravitational force gets smaller and smaller the closer you get to the center since you’re getting closer to having an equal amount of mass pulling on you from both sides. Once you pass the center, gravity start pulling in the opposite direct so that eventually you’ll be digging up instead of down. (This might make the second half of the trip easier so long as you can avoid the falling debris.) But what happens exactly in the middle? Will you float there with no net gravitational force on you?

In addition to these interesting physics questions, there’s also a practical question: how long it would take? There are obviously many technical difficulties. First, there’s the issue of pressure. Much like deep-sea diving, as you travel further into the Earth, the pressure increases to levels where life cannot survive. Second, there is extreme heat with temperatures reaching as high as temperatures on the surface of the Sun. Then, of course, there are the mole people. Putting all these issues aside, we can make a very cartoon-y estimate in the following way.

If we only want one person going through the hole, the hole will only need to be about 1.0 m2 wide. Using the diameter of the Earth, we can compute the total volume of dirt to be dug,

volume = (area) · (height)
= (1.0 m2) · (1.3×104 km)
= 1.3×107 m3.

If we can dig one shovel full (~1.0 L) every five seconds, or, equivalently, 0.2 L/s, we can compute how long it would take to dig a hole to China,

time = (volume) / (volume per second)
= (1.3×107 m3) / (0.2 L/s)
= 2100 years.

If you started back in the time of Christ and never took a break, you’d still be digging today.

Adam, thanks so much for taking the time to contribute a great question!