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Acceleration Inside the Earth

  1. Jan 9, 2016 #1
    Here is an interesting phenomenon: If one were to drill a hole that went all the way to the center of the earth and drop an object through the hole, according to a model called PREM (Preliminary Reference Earth Model), the acceleration of the object would initially increase as it approached the outer core of the earth and then decrease as the gravitational pull from the mass above the object started canceling the gravitational pull from the mass below (see pic below).

    8743453a90cde1918630a1c58a8d37b99cc8b826.png



    Now Newton's law of gravitation predicts that the acceleration should decrease as an object moves down the hole because there is less mass below the object and more above, pulling in the other direction. Anyone know why this widely accepted PREM model predicts otherwise?


    Here is Newton's law of gravitation prediction where b297e5dd04aaa172eb7de68e20328516747e2691.png is the mass of the object, b55ca7a0aa88ab7d58f4fc035317fdac39b17861.png is the distance from the core, 0027034d8a10372a06deaf4f4084c01956587479.png is the density of the earth (assuming it is constant), 6e28ce12d49d39f160d5a0ef54077fc98e4b9d2b.png is the gravitational constant, and c7d457e388298246adb06c587bccd419ea67f7e8.png is the acceleration of the object.

    e4b716351056e7c410ad32685c80f373cc8dbb6b.png
    93f6384db22ec0397be241d8c9f60d0ffbe4a1f1.png
    Clearly, Newton's law predicts something very different from the PREM model.

    How could acceleration possibly increase according to the PREM model when there is less mass below the object as it is moving toward the core?
     
  2. jcsd
  3. Jan 9, 2016 #2

    Doc Al

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    Staff: Mentor

    Presumably the PREM model also uses Newton's law of gravity. What you are calling "Newton's law" is just the calculation done with the simplifying (but incorrect) assumption of uniform density. See the green line in the diagram.
     
  4. Jan 9, 2016 #3

    Vanadium 50

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    There's your mistake. It's not constant, and PREM takes that into account.
     
  5. Jan 9, 2016 #4
    Even if density were not constant and varied as a function of r, how could acceleration increase as one got closer to the core? The mass of the sphere contained by a radius ##r_1## must be smaller than the mass of a sphere contained by radius ##r_2## where ##r_2 > r_1## regardless of the density distribution of the earth right?
     
  6. Jan 9, 2016 #5

    DaveC426913

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    Gold Member

    If the outer layer are less dense than the inner layers then acceleration should increase as one nears the core.

    Consider a hypothetical extreme: a planet of 5000 miles radius where its outer 1000 miles are so rarified one could almost call it vacuum. As something falls through this first thousand miles, one could apply Newton's shell theorem, but what do you expect its acceleration will do? It is essentially still falling toward a massive body. Taking into account Newtons shell theorem, it will have little effect on net acceleration.
     
  7. Jan 9, 2016 #6
    I understand now. Thanks!
     
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