Tiny ant sits on solid uniform brick 4" x 12" x 16", which floats in space?

Question

Initally the brick is spinning around the shortest axis, and its kinetic energy is Ko = 1Joule.

How much work must the ant perform to realign the axis of rotation along the longest axis of the brick?

Answer 1

Let I be the brick's moment of inertia (around its current axis) and let ω be its angular speed.

Brick's initial energy due to rotation:

E = Iω²/2 = 1 Joule

Brick's angular momentum:

P = Iω

Furthermore, you can see from the above formulae that there's a relationship between P and E:

E = P²/(2·I) [we'll use this later].

Now, when the rotational axis gets realligned, the brick will have a different moment of inertia, which let's call I′ .

Presumably, since this is being done by the ants (and not by external torques), the final angular momentum P′ will be the same as the initial angular momentum P. Therefore the final energy E′ is given by:

E′ = P′² / (2·I′)
= P² / (2·I′)

But since P² = 2·I·E we have:
E′ = (2·I·E) / (2·I′) = E(I / I′)

In other words, the ratio of the energies is the inverse of the ratio of the moments of inertia.

According to formula from Wikipedia:

I = (m/12)((12 in)² + (16 in)²)
I′ = (m/12)((4 in)² + (12 in)²)

so (I / I′) = 2.5

I'll let you work out the remaining math :-)

Answer 2

I know of no brick that size.
You must mean a concrete block.
Kinetic energy? You must mean rotational energy.

Answer 3

what?!?!?