What is the centroid (or center of mass) of a homogenous half-ellipsoid in terms of its semi-axes (a, b, c)? Is it different from that of a half-ellipse?
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What is the centroid (or center of mass) of a homogenous half-ellipsoid in terms of its semi-axes (a, b, c)? Is it different from that of a half-ellipse?
The center of mass is at the intersection of its semi-major and semi-minor axes.
But where in the z axis? Remember this is half-ellipsoid.
Sorry - I missed that. You have values for a, b, c, for the equation
Right? The center of mass along the z-axis can be found from
where V = volume of the half elipsoid, which is, and A(z) is the cross-sectional area as a function of z. The cross-section is an ellipse - you will need to come up with equations for the lengths of the semi-major an semi-minor axes as a function of z, use that to find an expression for A as a function of z (hint - the area of an ellipse is
times the lengths of the semi-major and semi-minor axes), and put that into the above integral. It works out pretty nicely - post back with what you get.
In order to findan expression for A as a function of z, we consider:
pi*x*y=A(z)
We use:
and
solving for x and y in terms of z, we obtain:
x = a*sqrt(1 - z^2/c^2)
y = b*sqrt(1 - z^2/c^2)
putting these two in the integral, we come up with
Substituting x and y:
After simplifying and integrating we get:
Am I on the right track?
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