5 ms·

Fun idea, but didn't everyone here have to work this out in their college physics class?

by te_platt 17y ago

scorxn 17y ago
Since Friday, Feb. 11, 1966 maybe ;)
- eru 17y ago
  I just wrote a simple python script to test whether this holds only when all of earth's mass is concentrated at its center, or in the slightly more realistic case of constant density. My numerical experiments point to the latter. Does anyone care to derive an analytic solution?
  - madcaptenor 17y ago
    The derivation in the original article (Paul W. Cooper, Through the Earth in Forty Minutes, Am. J. Phys. vol. 34 (1966) p. 68) relies on the assumption of constant density. (It's hard to say this for sure because some of the details are left out, but Cooper at least states he's making this assumption.) In the case where all the earth's mass is concentrated at the center, a point mass starting on the surface of the Earth would just go to the center and stay there. In the opposite extreme where all the mass is concentrated on the surface, (that is, the Earth is a hollow shell) it actually turns out that the gravitational acceleration at any point inside the shell is zero, so it wouldn't work in that case either. I suspect that for some reasonable class of spherically symmetric mass distributions (that is, the density only depends on the distance to the center of the Earth), tunnel systems like this are mathematically possible. But I'd be surprised to learn that there are mass distributions other than the uniform one for which the travel time doesn't depend on distance. But I'm not going to work this out because I have Real Work to do. (Now I wish I were teaching calculus so I would have an excuse to work out this problem...)
    - eru 17y ago
      > In the case where all the earth's mass is concentrated at the center, a point mass starting on the surface of the Earth would just go to the center and stay there. Why? That would violate conservation of energy, wouldn't it? In a naive model of a point mass you'd get a singularity at the center. But using standard techniques (e.g. numeric pertubation, or Lebesgue integration) one gets an objects that swings back and forth like in the other scenarios.
      - jibiki 17y ago
        Hmm, unless I'm going crazy here, we get the differential equation x'' = -(x^-2) x = (kt)^(2/3)(with k = (2/9)^(-3/2), not that it matters) seems to be a solution? I guess that doesn't help with the singularity, but neither does looking at energy (since you have infinite kinetic energy at the center and infinite potential energy everywhere else.)
        eru 17y ago
        That differential equation only holds for positive x. Try something like the following: x'' = - signum(x) * (x^-2)
      - madcaptenor 17y ago
        That's a good point. I don't know what I was thinking.
- madcaptenor 17y ago
  I didn't notice that at first, and when the Time article mentioned that this was in the current issue of a journal I actually went and looked at the May 2009 issue. Of course it wasn't there.
- te_platt 17y ago
  Doh!
  - electromagnetic 17y ago
    You forgot the apostrophe, (Annoyed Grunt)!

Fun idea, but didn't everyone here have to work this out in their college physics class?

scorxn 17y ago

Since Friday, Feb. 11, 1966 maybe ;)

eru 17y ago
I just wrote a simple python script to test whether this holds only when all of earth's mass is concentrated at its center, or in the slightly more realistic case of constant density. My numerical experiments point to the latter. Does anyone care to derive an analytic solution?
- madcaptenor 17y ago
  The derivation in the original article (Paul W. Cooper, Through the Earth in Forty Minutes, Am. J. Phys. vol. 34 (1966) p. 68) relies on the assumption of constant density. (It's hard to say this for sure because some of the details are left out, but Cooper at least states he's making this assumption.) In the case where all the earth's mass is concentrated at the center, a point mass starting on the surface of the Earth would just go to the center and stay there. In the opposite extreme where all the mass is concentrated on the surface, (that is, the Earth is a hollow shell) it actually turns out that the gravitational acceleration at any point inside the shell is zero, so it wouldn't work in that case either. I suspect that for some reasonable class of spherically symmetric mass distributions (that is, the density only depends on the distance to the center of the Earth), tunnel systems like this are mathematically possible. But I'd be surprised to learn that there are mass distributions other than the uniform one for which the travel time doesn't depend on distance. But I'm not going to work this out because I have Real Work to do. (Now I wish I were teaching calculus so I would have an excuse to work out this problem...)
  - eru 17y ago
    > In the case where all the earth's mass is concentrated at the center, a point mass starting on the surface of the Earth would just go to the center and stay there. Why? That would violate conservation of energy, wouldn't it? In a naive model of a point mass you'd get a singularity at the center. But using standard techniques (e.g. numeric pertubation, or Lebesgue integration) one gets an objects that swings back and forth like in the other scenarios.
    - jibiki 17y ago
      Hmm, unless I'm going crazy here, we get the differential equation x'' = -(x^-2) x = (kt)^(2/3)(with k = (2/9)^(-3/2), not that it matters) seems to be a solution? I guess that doesn't help with the singularity, but neither does looking at energy (since you have infinite kinetic energy at the center and infinite potential energy everywhere else.)
      - eru 17y ago
        That differential equation only holds for positive x. Try something like the following: x'' = - signum(x) * (x^-2)
    - madcaptenor 17y ago
      That's a good point. I don't know what I was thinking.
madcaptenor 17y ago
I didn't notice that at first, and when the Time article mentioned that this was in the current issue of a journal I actually went and looked at the May 2009 issue. Of course it wasn't there.
te_platt 17y ago
Doh!
- electromagnetic 17y ago
  You forgot the apostrophe, (Annoyed Grunt)!