Integral over a rotating ellipsoid

[Silviu]

Homework Statement

Calculate $\int x^2 dV$ over an ellipsoid with semi-axes a, b and c along x, y and z. rotating around the z axis with an angular speed $\omega$.

Homework Equations

The Attempt at a Solution

I managed to calculate this in the case when it is not rotating and I got $\frac{4\pi}{15}a^3bc$. But I am not sure how to do it now I expected to get the same result as before times $cos^2(\omega t)$ but the result seems to be $cos^2(\omega t) \int x^2 dV + sin^2(\omega t)\int y^2 dV$. Can someone help me with this? Thank you!

 

[Charles Link]

This problem is new to me too, but to start with, the ellipsoid apparently has the equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$. For rotation about the z-axis, the standard two dimensional rotation transformation for angle $\theta=\omega t$ should apply. You rotate the axes with such a transformation by angle $\theta$, and you get the equation of the rotated ellipsoid in the form $A(x')^2+B(y')^2 +C x'y'+z^2=1$. You then need to integrate $\int (x')^2 \, dV$. $\\$ Editing: Suggestion instead: Keep the ellipsoid fixed and rotate the $x^2$ term in the integral $\int x^2 \, dV$ using the 2-D rotation transformation. The $x^2$ term will become a mix of $x^2$, $y^2$ and $xy$ terms. I think the $xy$ integral over the ellipsoid will cancel from symmetry. Do you agree? And yes, I agree with your last statement in the OP. And since you solved it for the unrotated case, you should be able to write down the answer by inspection of the form you got for the unrotated case.

 

[Charles Link]

@Silviu See my edited post above.