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So is it 42 or not?

  1. Mar 26, 2015 #1
    There is a web site (Gizmodo) now claiming that the answer to the question: how long it would take to fall to the center of the Earth is only 38 minutes and 11 sec.

    First is it right and second does it really matter?
     
  2. jcsd
  3. Mar 26, 2015 #2

    wabbit

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    If it matters to you, you can compute it :). To me it sure doesn't, but it seems broadly the right order of magnitude at least if you assume a hole to the center.
     
  4. Mar 26, 2015 #3

    Doug Huffman

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  5. Mar 26, 2015 #4

    phinds

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    The logic behind his new figure seems impeccable.
     
  6. Mar 26, 2015 #5

    PeterDonis

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    The answer looks reasonable to me (I have not done a detailed calculation), but the way the article frames the whole issue is bogus. The article correctly notes that the "standard" answer to this question, 42 minutes, assumes that the Earth is a sphere of constant density. (It's notable, btw, that the article, after going on and on about how the constant density assumption is wrong, says nothing at all about how the sphere assumption is also wrong. More on that below.) However, nobody ever actually believed that the Earth was a sphere of constant density. The reason that assumption is made when this problem is assigned to undergraduates is that it makes the calculation tractable: you can get a closed-form solution. For a sphere whose density varies with depth (and even more so for an oblate spheroid whose density varies in complicated ways), there is no closed-form solution; you have to compute the answer numerically, and that's beyond the scope of your average undergraduate physics class.

    So the title claim of the article, that "scientists can't agree" on this question, is, to put it bluntly, bullshit. Scientists never claimed that the uniform density answer of 42 minutes was the "right" answer, just that it was an approximate answer that was easy for undergraduates to calculate. What's more, the article writer seems to think that the answer of 38 minutes and 11 seconds is somehow "right"; but that's not really true either, because, as I noted above, the calculation still assumes that the Earth is a sphere (at least, as far as I can tell from the abstract of the paper, it does), whereas the actual Earth is an oblate spheroid. The actual time it would take to fall to the center of the actual Earth will depend on where on the surface you start from. (And this is all the more true when you include the effect of the Earth's rotation, which is also ignored in the article and is not taken into account in the paper, as far as I can tell.)

    And for extra fun, the article writer not only puts a bogus interpretation on what he does write about, but actually misses the most important point raised in the paper! From the abstract:

    "The time taken to fall along a straight line between any two points is no longer independent of distance but interpolates between 42 min for short trips and 38 min for long trips."

    Here's what that means: for a uniform density sphere, it takes 42 minutes to fall through a tunnel from the surface to the deepest point, not just for a tunnel through the center, but for a tunnel between any two points on the surface. In other words, if the Earth were of uniform density and we dug a straight tunnel from New York to London (which obviously will go nowhere near the center of the Earth), it would take 42 minutes to free-fall through that tunnel from either end to its deepest point. (Yes, this would not be a "fall" straight down; think of something like a rail car running on frictionless rails with no engine, so the only motive power is gravity.) But, for the actual Earth, this time varies with the depth of the tunnel: it approaches 42 minutes for "short" tunnels (tunnels whose deepest point is near the surface), and approaches 38 minutes 11 seconds for "long" tunnels (whose deepest point approaches the center of the Earth).

    (If you read the fine print, it also turns out that these tunnels, as they get deeper, are also no longer straight; their deepest point is deeper than a straight tunnel between the same two surface points would be.)
     
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