Discussion of the Bob Schadewald Gravity Engine.
Bob's deception is so absurdly simple that everyone who has offered public commentary on this has, so far as I know, missed it. The wheel does gain kinetic energy on the downward half cycle; we can't deny that. Bob then says that the eccentric mass then has more than enough kinetic energy for it to rise to the top again against the force due to gravity (which will be slightly smaller while the mass rises). That's also true. But does the wheel therefore gain kinetic energy over the "long haul" of successive cycles, providing excess energy we can tap? Bob has cleverly led us to suppose it will. But if his logic and assumptions are followed to their conclusion, we see that it will not.
The only energy we could possibly extract from this system would be that kinetic energy the ball attains during its first fall to the floor, slightly less than mgh.
More detailed analysis.
We will begin with an energy analysis, just to see what's going on. Suppose that we do include some mechanism for extracting that extra amount of energy at the end of each cycle. This returns the wheel to zero speed at the top, so it begins from rest each cycle. But during each cycle the kinetic energy gain at the top is mh(gf - gi) = mhQ where Q = gf - gi is the small change in g during one cycle. Q is a negative quantity, and is just equal to the change in the potential energy at the top of the cycle due to the decrease of g.
By the time n cycles are completed we have extracted nQ = mgh of energy, the same amount we gave it initially by lifting the weight to the top to get the wheel started. During this time the speed at the bottom gradually decreases on subsequent cycles till it is zero when gravity "runs out". We can get no more energy from this machine than we put in initially.
There's a related puzzle in which we don't "steal energy" from the machine. We also assume there are no dissipative forces (like friction) removing energy from the machine. Then when gravity reaches zero the final kinetic energy of the wheel must be exactly what it was at the end of the first half-cycle. Also, it must then be moving at constant speed at all positions. That will be the same speed it had at the end of the first half-cycle.
Why did Bob say "the wheel may pick up speed at the top"? That will happen if we don't extract any energy from the machine, or don't extract it at a sufficient rate to ensure that the speed is zero at the top at the end of each cycle.
Also, if g happens to reach zero when the eccentric mass is anywhere but at the bottom, the mass will retain whatever speed it had at that point, and that includes the point at the top. This could be considerable if g decreases so rapidly it reaches zero during the first few cycles. But the speed at the top will never exceed the speed the mass had when it first reached bottom.
It's interesting to note that in his Gravity Engine spoof, Bob Schadewald never lies to the reader. He lets the reader make the incorrect inferences. The closest he came to making an incorrect statement was "With every revolution the wheel speeds up." Well, it does, at the top, but never enough to cause its kinetic energy at any point to exceed the kinetic energy it had attained when it first reached bottom.
We have used only conservation of energy (kinetic and potential) in this discussion. The same result should be obtainable by a strictly kinematic analysis using Newton's laws.