Solution to Buoyancy Motor No. 3 by Ben Mitch.
Inventor's assumption: (1) The buoyancy of the bubble is sufficient to
raise the weight.
What goes wrong:
The buoyancy of the bubble is equal to the mass of liquid it displaces minus the mass of the bubble itself. For generality, we will assume that the bubble is not vacuum-filled. Let's say the density of the air in the bubble is p1 and that of the liquid is p2, and that the volume of the bubble is V. Then we'll call the mass of the weight m and the distance that the rod has moved through the hole in the drum d.
The inventor's assumption that the bubble is sufficiently buoyant to lift the weight may be written as:
The inventor's proposal is where the error creeps in. 'Clearly' the torque in (2) is greater than the torque in (4). But is it? The mass looks big and heavy yes, and the torque, T, due to the mass is indeed greater in (2) than in (4), but we've forgotten the torque due to the water that has moved from where the bubble isn't in fig. (2) to where the bubble isn't in fig. (4) (or, almost as opaquely, the mass of water that has swapped places with the bubble from (2) to (4)). The change in torque is equal to the sum of that caused by the shift of the water in one direction and that caused by the shift of the air in the other, and looks familiar
We can then write the inventor's proposal that the torque decreases as
where md is the change in torque due to the shift of the mass of the weight. Cancel out the d's from this last equation and we've just contradicted ourselves. The inventor's assumption and proposal cannot co-exist according to Newton, so he concurs with Stevin. We can have one or the other - a bubble that buoys up a mass which remains pointing upwards or a mass that falls to the bottom under its own weight and stays there - but we can't have both.
My interest in this problem is the 'almost but not quite'-ness which is common to PM proposals. The disc on near-perfect bearings is commonly chosen as a basis because it's dangerously close to being a PM machine just as is. Add a small mass here and a lever there and it's not difficult to confuse Newtonian analysis (until it's done on paper). I remember reading somewhere that we can only keep track of five or six pieces of information at once, so an inventor who adds enough bells and whistles is sure to confuse me at least temporarily, and that's why we who don't have the time to waste refuting these things are tempted to rely on Stevin since that only requires us to establish one fact - has the system changed from time A to time B?
In this case, the intent (or mistake) is that we forget the movement of water and how it affects the torque on the drum's axis. With constantly being taught at school to ignore negligible quantities it is common practice for us to break things down into their simplest form. In this case, we opt for a mass at a distance from an axis, and ignore the drum and it's contents as homogeneous and thus having no effect on the torque, forgetting that negligible quantities are only relatively negligible. The negligible charge on a single electron passing through a lightbulb is certainly not negligible when directing the chemistry of gunpowder. I wonder if there's a mistake being made here, or perhaps teaching practice differs in the US? Anyway, progress marches on and I'm waiting patiently for someone to discover a way of transferring energy between universes and the PM people to explain to us gently and encouragingly that that's what they meant all along, weren't we paying attention?
Give me strength for that day. Thanks, I quite enjoyed un-rusting my mechanics.
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