Revising, this is what I'm stuck on: inertia tensors, rotating rigid bodies about axis other than its axis of symmety,...
I think it'd help a lot to see a worked example and I can't find anything on the web. Here is my example:
Considering a cylinder of unit density occupying the region $x²+y² \leq a²$, $0 \leq z \leq h$, what is its angular momentum about the axis $x=z$ , $y=0$ ? Can you explain to me how to get there? This is very general so I think I'll be able to generalize it when I understand this one!
Any help would be amazing! Thank you
(I had posted this previously but it was left unanswered, I know it's meant to be really easy but I just can't get my head round it and it is very important, you'd help me a lot, even with very little explanation, thanks)
When a mass is tied to an axis, the farthest it is from the axis, the harder it is to set it in motion. This is measured by the moment of inertia, proportional to the mass and to the square of the distance to the axis.
Considering your cylinder, you have to integrate the squared distance over the volume of the cylinder: $$\iiint\rho r^2\ dx\ dy\ dz,$$ where $\rho$ is the uniform volumic mass, $\frac MV$.
The distance from $P=(x, y, z)$ to the axis can be found by considering a unit vector $U=\frac1{\sqrt2}(1, 0, 1)$ along the axis, and computing the projection of $P$ as $Q=(P.U).U=(\frac{x+z}2, 0, \frac{x+z}2)$. Then $r^2={PQ}^2$.
Another approach is to use precomputed moments from a table and use formulas for axis translation and rotation. http://en.wikipedia.org/wiki/Moment_of_inertia