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Math Help - Center of mass of a solid

  1. #1
    Newbie
    Joined
    Feb 2011
    Posts
    11

    Question Center of mass of a solid

    Hi,

    I'm confused on how to calculate the center of mass of a solid that's located out of one sphere and into another. The problem goes on like this:

    Determine the center of mass of a solid located out of a sphere, of radius 1, centered at the origin, and inside a sphere, of radius 1, centered at point \left(0,0,1\right).

    All I get from my textbook is the mass formula:
    m=\int\int\int_{E} \rho(x,y,z)\mathrm{d}V

    And then if the center of mass is actually the same thing as a center of inertia, I would have to calculate the moments related to each plan of coordinates.

    How can I do that when I've got two spheres one above the other. I believe their equations are respectively:
    x^2+y^2+z^2=1
    and
    x^2+y^2+(z-1)^2=1

    As plotted below.

    Can you please give me a hint? I'm blocked there.

    Thanks a lot,

    Bazinga
    Attached Thumbnails Attached Thumbnails Center of mass of a solid-figure.png  
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  2. #2
    MHF Contributor
    Joined
    Nov 2008
    From
    France
    Posts
    1,458
    Hi,

    The center of mass G of a solid is given by the formula

    \int\int\int_{E} \rho(x,y,z) \vec{OM}\mathrm{d}V = \left(\int\int\int_{E} \rho(x,y,z) \mathrm{d}V\right) \vec{OG}

    If the solid is homogeneous, \rho(x,y,z) is constant (independent with respect to x, y and z) and therefore can be pulled out of the integral

    \int\int\int_{E}\vec{OM}\mathrm{d}V = \left(\int\int\int_{E} \mathrm{d}V\right) \vec{OG}

    In your example, as far as I have understood, the solid is axysimmetric with respect to the z axis therefore G is on the z axis

    The projection of the formula on the z axis gives :

    \int\int\int_{E} z r \mathrm{d}r \mathrm{d}\theta \mathrm{d}z = \left(\int\int\int_{E} r \mathrm{d}r \mathrm{d}\theta \mathrm{d}z) z_G
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