[jimmy1843]

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?

[ebaines]

The center of mass is at the intersection of its semi-major and semi-minor axes.

[jimmy1843]

But where in the z axis? Remember this is half-ellipsoid.

[ebaines]

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.

[jimmy1843]

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?

[ebaines]

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

Not quite - the formula is:



Note the 'z' term that you left out. It results in the numerator becoming

Am I on the right track?

Clearly the value for must lie somewhere between 0 and z. With the correction above you should get it now.

[jimmy1843]

Not quite - the formula is:



Note the 'z' term that you left out. It results in the numerator becoming




Clearly the value for must lie somewhere between 0 and z. With the corection above you should get it now.

Oh yes! Sorry I missed it. My oversight. I think the answer should be:
= 3/8c
I hope this time I'm correct (?)
Thanks.

[ebaines]

I think the answer should be:
= 3/8c
I hope this time I'm correct (?)
Thanks.

Yes - I agree with your answer - nicely done!