In one of my courses this semester students were learning about the coriolis and centrifugal forces that things seem to experience on the earth. There’s a problem at the back of the chapter that asks when a dropped rock would hit the side of a well. A couple of students attempted it, but I noticed that they assumed the well was drilled straight toward the center of the earth. I encouraged them to think about that and figured I’d get a resubmission before the end of the term. I haven’t yet, and I started to realize today that it’s a pretty interesting problem. Namely, how do you drill a straight well?
I came up with three ways (four if you count straight to the center of the earth) and all of them give a different shape for the well:
- Slowly reel out a heavy plumb bob on a line. When you stop reeling, it’ll point out away from the axis of rotation of the earth due to the centrifugal force. As long as you do it slow enough, you can ignore the coriolis force. Have the well shape match the path the plumb bob takes during its (slow) descent.
- Slowly reel out a chain or rope that doesn’t have negligible weight but also doesn’t have a heavy plumb bob at the bottom. Have the well shape match the shape of the rope when you’ve got it as deep as you like it.
- Drop a rock and see which way it goes. Have the shape of the well match the trajectory. This one has the benefit that future rocks won’t hit the side!
So I set about trying to figure out each of those shapes. For those that don’t want the details, here’s a plot showing all three. Well 1 is in blue, well 2 is in red, well 3 is in green.
The easiest one is the green one. I just do a pretty straightforward integration of the equations of motion where:
Where the first term is gravity (note that g changes like a spring as you go deeper), the second is the centrifugal force, and the third is the coriolis force. Once you know the form of the force, doing an NDSolve command in Mathematica is pretty easy.
The other two wells were a little harder to calculate. For well 2 I modeled several (19) masses connected by inextensible strings. I started with them all anchored at the top and hanging straight toward the center of the earth. Then I turn on time with a lot of friction to see where they all settle down to. Here’s an animated gif of that process:
I actually used Lagrange multipliers for all the strings and simply solved for x,y, and z of every particle along with the tension forces necessary to keep their spacing correct.
For well 1 I only used one particle but I changed the constraint to be time dependent. Basically I just had the length between the ball and the anchor grow linearly.
Cool, huh? I wonder which is the better way to do it. What do you think? Here’s some starters for you:
- I’m in this class and I love this. I’ll redo that standard using all of these just so you’ll brag about me.
- I’m in this class and I’m pissed. Why couldn’t you post this a long time ago?
- This is cool. What happens with well 3 if you add air resistance?
- I’m confused. What do you mean when you say “g” changes like a spring as you go deeper?
- This is cool, but I don’t think you did _____ correctly. Here’s a post of mine that does it correctly.
- This is dumb. I build wells for a living and we don’t do this at all. Instead we . . .
- This is cool, but as usual you haven’t bothered to post your code, jerk.
- Having just redone all your work, I’m now convinced that you used a huge omega value (0.5) and a tiny earth (10 m radius) with g=9.8 at the surface, all at 45 degrees N latitude. You could have just put that in your post, you know. Jerk.