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dynamics

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Find the mass center of a cylinder with height L and a hemisphere glued to the top with radius R. L=2R. The cylinder is standing up on its bottom end with hemisphere on top. Im so lost on this and cant use the formula to work.

  • dynamics -

    I assume radius of circular cylinder is r
    Volume of cylinder = (2 r* pi r^2) = 2 pi r^3
    moment of this cylinder about base = r(2 pi r^3) = 2 pi r^4

    Volume of hemisphere = (1/2) (4/3) pi r^2 = (2/3) pi r^3

    now to find cg of hemi
    moment of hemi about base of hemi:
    hemi goes from d = 0 to d = r where d is height of slice above top of cyllinder
    the radius at d = sqrt(r^2-d^2)
    the area at d = pi(r^2-d^2)
    the moment at d = pi(d r^2 - d^3)
    integrate over d from d = 0 to d = r
    pi (r^4/2 -r^4/4) = pi r^4/4
    so hemi cg above base of hemi = (r^4/4)/(2 r^3/3) = r/6
    so the cg of the hemi is r/6 above base of hemi
    whioh is
    2 r+ r/6 = 13 r/6 above the ground
    so moment of hemi above ground = (13 r/6)(2/3 pi r^3) = (13/9) pi r^4
    now
    total volume = 2 pi r^3 + (2/3) pi r^3 = (8/3) pi r^3
    total moment = 2 pi r^4 + (13/9 ) pi r^4
    = (31/9) pi r^4
    so
    cg above ground = (31/9)(3/8)r
    =(31/24) r
    check my arithmetic !

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