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Digging a hole through the earth

  1. Feb 5, 2010 #1
    have examined how g varies with distance from the Earth's surface. but how to change g if, instead dig down to the center of the earth?
    if we do Start in the center of the Earth (see picture) how will the value of g varies from there to 2r height above the Earth's surface. Suppose that Earth's density is constant throughout the Earth's volume.

    http://www.pluggakuten.se/wiki/images/5/5a/Martin.jpg [Broken]

    i do sow:
    [tex]M = \rho \cd V = \rho \fr{4\pi r^3}{3}\\ F_1=m\cdot g\\ F_2=G\frac{mM}{r^2}\\ F_1=F_2\rightarrow \\ g=G\frac{M}{r^2}\rightarrow g=G\frac{\fr{\rho 4\pi r^3}{3}}{r^2 }\rightarrow \\ g=G\frac{\rho 4\pi r}{3}[/tex]

    but its WRONG,
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Feb 5, 2010 #2

    HallsofIvy

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    Re: gravitational

    Why do you say it's wrong? It looks correct to me. The gravitational force due to any mass above you is canceled by a corresponding mass in the opposite direction so you only use the mass below you which is [itex](4/3)\pi r^3 \rho[/itex]. Since we can treat that mass as if it were all at the center of the earth the acceleration is [itex]g= (4/3)\pi r^3 \rho G/ r^2= (4/3)\pi\rho G r[/itex], exactly what you have.
     
    Last edited by a moderator: May 4, 2017
  4. Feb 5, 2010 #3
    Re: gravitational

    but what is r? in this formula [itex]g= (4/3)\pi r^3 \rho G/ r^2= (4/3)\pi\rho G r[/itex]
    r must be distance between celestial bodies, but in the formula are the r=radius.
     
  5. Feb 5, 2010 #4
    Re: gravitational

    Now I'm mixing up all sorts of things in each other. What is meant by the change of g instead dig down to the center of the earth?
    it means that we have different masses for the Earth?like;
    http://www.pluggakuten.se/wiki/images/6/6b/3.JPG [Broken]

    But what happens to gravity when earth curvature is half instead, see picture below;
    http://www.pluggakuten.se/wiki/images/7/75/2.JPG [Broken]

    does I thinking right? sorry, my english is bad
     
    Last edited by a moderator: May 4, 2017
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