Aerich said:
What's e=mc^2 got to do with the energy released by chemical reactions? There is no mass-energy conversion going on during ordinary chemical processes such as burning a piece of wood. The box will weigh precisely the same before the fire starts and after it self-extinguishes.
If there was mass conversion going on during burning, campfires would be a bit more exciting than they actually are.

Mass conversion happens during what people commonly refer to as nuclear reactions, such as atomic bombs and nuclear power plants. (OK, technically all forms of matter are constantly undergoing nuclear decay at some low rate, but this rate is not affected by whether burning is going on, and the energy released by burning has nothing to do with mass conversion.)
Sorry, there is, or if my memory is incorrect about such measurements being performed, there should be a mass difference between the starting materials and the cooled down result material, which equals via E=MC'2, the released heat energy and the change in potential energy the material might have due to now floating around or so.
So its easier to believe, some further explanation(not exactly correct):
Atoms are a core of protons and neutrons surrounded by a "cloud" of electrons.
Nuclear reactions are the change of the core of atoms, e.g. if one proton goes from one atom to the next you've got a nuclear reaction. There will be energy released or neccessary in this process mainly due to the proton having in "his new" atom a different potential energy since the strong field its no in is different than in "his old" atom. Also energy levels might change in the "old" atom.
Chemical reactions now are the change os the electron "cloud" of the atom. So if one electron goes from one atom to the next you have a chemical reaction. Now why can there be some energy released?
Well, same as before those 2 atoms have different electromagnetic fields and around his new atom the electron will therefore have a different potential energy and any difference in potential energies is released as heat or whatever.
If you compare the two descriptions, you will see that the main differences are proton/electron and strong/electromagnetic force.
If you have a mass change with the one reaction you ought to have a mass change with the other rection as well or something is wrong.
As the electromagnetic binding energies of the electrons are a factor 1000 smaller than binding energies inside atom core, the mass change will be a 1000 times smaller and therefore far harder to detect.
E of 1 kg of matter is 9 * 10'16 Joule, if i assume chemical reactions to release roughly the same energy as the human body can get out of butter(3600 kj per 100 g), which will not be more far off that a factor 1000, the mass change due to chemical reactions will be roughly 36000 /(9*10'13 )kg= 0,0000000004 kg = 4* 10'(-10)kg.
Thats pretty small, therefore it would take very precise measurements to detect the mass change.
But this also means that any machine that does not take anything in from the outside, but releases energy would either have to lose mass or is a huge scientific breakthrough.
As for whether it is worthwhile to weigh the box to detect mass conversion, frankly, if the fellow has worked out how to put THAT in a small box with a nice easy to use 12V DC output, he's going to be getting a lot more money than Randi's paltry $1M. You may safely assume this is not the case.
Yes, very likely(think i said that), but if there is a mass change according to the released energy, then it cannot be perpetua mobile, as it surely stops working when reaching mass 0 and is likely something that does not contradict any physical law, probably just a new invention or a old modified.
But if a lot of energy is released without according mass loss and that is measured, then it would be either braking a known law or at least be a brand new fantastic device, that deserves the nobel prize.
I'm no authority on orbital mechanics, but I believe that no energy is drawn from the rotation of a planet during "slingshot" maneuvers. It's more like using the planet's gravity well as a pivot to change the direction of the spacecraft's velocity vector.
Its not only change of direction, but also speed, otherwise it would not be interesting. Voayager probes for example had a very tight schedule to start, otherwise it could not have gone from planet to planet always getting a little speed and have enough speed to leave solar system in the end.
And the moon is slowing the rotation speed of earth(i think dinosaurs had a 18-20 hour day), so a object moving past earth can slow rotation and that way gain movement energy.
Carn
Just realized slowing of earths rotation might be caused by earth circling round the sun or comets that pass earth from time to time, maybe all three together, but there is nothing else to slow rotation of earth around its axis.