Since there seems to be some interest in qualitative reasoning, here's
another riddle for you.
Suppose I take an ordinary uniform solid cube and skewer it with an axis
that runs through an arbitrary point on one face, perhaps (X=1, Y=0.33,
Z=pi/4), thence through the center and out the opposite
face. Question: what can you say about the moment of inertia of the cube
as it rotates about this axis?
Hint #1: There is something very important you can say even without
writing down the exact solution. This is the whole point of the question.
Hint #2: Given the results of hint #1, you should be able to write down
the exact answer using nothing more than a pencil and a 3"x5" piece of
paper. If you are tempted to use something more than that, you've missed
the point.
Hint #3: What is a vector? Is it some arbitrary collection of 3
numbers? Is a tensor some arbitrary collection of 9 numbers? Or is it
more special than that? Does it have some geometric and physical significance?
(The following hint is optional. If it helps you, fine; otherwise don't
worry about it.)
Hint #4: Why might jsd bring such a skewered cube to class on the day he
explains the Wigner-Eckart theorem?