For those who don’t know, I recently returned to academia by accepting a visiting professor position in the Physics Department at Simpson College. I’m very excited about this move, and it already seems I have a lot of great students. However, I had forgotten just how much work the beginning of the semester entails.
Since I’m super busy and currently teaching students about estimation, I thought it’d be a good opportunity to revisit one of my earlier posts from the original Diary of Numbers blog.1 This post is one of my favorites, partly because it’s a great question, and partly because it’s from MythBuster, geek superstar, and certifiable awesome person Adam Savage:
Today’s question comes courtesy of MythBuster and geek superhero Adam Savage. Adam was nice enough to contribute not one, but two questions:
How many licks does it take 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, 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!” but it’s never a good idea to offend geek superheroes, especially ones who are nice enough to write you back and have a propensity for blowing things up.
If we want to end up in Beijing and pass through the center of the Earth, we’ll need to start at the opposite end of the Earth and travel a distance equal to the Earth’s diameter (~1.3×10⁴ km). Beijing's longitude and latitude are 39.55º N and 116.25º E. The exact opposite position on the globe would be 39.55º S, 63.75º W. By plugging these coordinates into Google Maps, you can see that this would place 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, let’s say we dig a giant hole through the Earth, and you jump inside. Would you make it all the way to China? The answer is almost certainly no. The gravitational force decreases as you get closer to the center because, at the center, there is an equal amount of mass pulling on you from all directions. Once you pass the center, gravity starts pulling in the opposite direction. After passing the center, you’ll be digging up instead of down. (This might make the second half of the trip easier, as 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?2
In addition to these fascinating physics questions, there’s also a practical question: how long would it 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, near the center, there is extreme heat, with temperatures reaching as high as those on the surface of the Sun. And then, of course, there are the mole people. Putting all these issues aside, we can make a very cartoonish estimate in the following way.
If we only want one person going through the hole, it will only need to be about 1.0 m² wide. Using the diameter of the Earth, we can compute the total volume of dirt that needs to be dug:
volume = (area) · (height) = (1.0 m2) · (1.3×104 km) = 1.3×107 m3.
"If we can dig one shovelful (~1.0 liter) every five seconds, or equivalently, 0.2 L/s, we can calculate 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 had started back in the time of Christ and never took a break, you’d still be digging today.
Adam, thank you so much for taking the time to contribute such a great question!
What do you think? Should I do more estimation problems on the blog? Are there other geek superheroes I should ask to contribute questions? Let me know in the comments, and don’t forget to like and share!
Yes, I know, I’m cheating a bit by reposting. I may do this a few times during the semester just to keep my sanity. However, I’m ok with it, because (1) it was a fun post the first time, (2) it’s been 14 years, about the same time George Lucas waited to re-release Return of the Jedi, and (3) it’s Adam FREAKING Savage!
Yes, if the mass is symmetrically distributed.