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Math Help - Find center of gravity.

  1. #1
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    Question Find center of gravity.

    Find the mass and the center of gravity of a cylindrical solid of height h and radius a , assuming that the density at each point is proportional to the distance between the point and the base of the solid.

    Note : the top of the cylinder is z=h.
    the bottom of the cylinder is x^2+y^2 =a .

    Thank you very much.
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  2. #2
    Eater of Worlds
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    Since the density is proportional to the distance z from the base, the density function has the form \delta(x,y,z)=kz, where k is some unknown positive constant of proportionality.

    M=\int_{-a}^{a}\int_{-\sqrt{a^{2}-x^{2}}}^{\sqrt{a^{2}-x^{2}}}\int_{0}^{h}kz \;\ dzdydx

    The constant k does not affect the center of gravity.


    Now, \overline{z}=\frac{1}{M}\int_{-a}^{a}\int_{-\sqrt{a^{2}-x^{2}}}^{\sqrt{a^{2}-x^{2}}}\int_{0}^{h}z(kz) \;\ dzdydx


    The center of gravity is on the z-axis, because \overline{x}=\overline{y}=0
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by kittycat View Post
    Find the mass and the center of gravity of a cylindrical solid of height h and radius a , assuming that the density at each point is proportional to the distance between the point and the base of the solid.

    Note : the top of the cylinder is z=h.
    the bottom of the cylinder is x^2+y^2 =a .

    Thank you very much.
    let the cylindrical solid be such that its base lies on the xy-plane, centered at the origin, and it extends vertically upwards until it hits the plane z = h. since the density at a point is proportional to the (vertical) distance of the point to the xy-plane, we have \rho (x,y,z) = kz for some constant k. use that \rho (x,y,z) in the formulas given here. i think you've proven that you need no help setting up and evaluating the integrals (i would probably use cylindrical coordinates to evaluate the integrals).

    (again by symmetry,  \bar {x} = \bar {y} = 0 )




    EDIT: thanks for beating me to it, galactus! ........well, at least i can take your answer as a confirmation i did the right thing
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  4. #4
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    It's symmetric about the x-axis and the y-axis.
    The density function is z, which is independent of both x- and y-axis.
    You've only the z-axis to worry about.
    Last edited by TKHunny; August 31st 2007 at 06:49 PM. Reason: Talk about slow on the draw.
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