The Flipping Linkage.
The inventor is correct that this system is in static equilibrium at the extremes of its motion. The geometry forces that conclusion. When the center weight of mass 2M moves a distance h from lowest to highest position, the other two outer weights (each of mass M) move down an equal distance h. So the gravitational potential energy at the upper and lower extremes of motion is the same, whatever happens in between, if no additional energy is supplied to the system. The only energy you could get out of this device is the small energy you supplied with that initial upward "mysterious force" on the blue mass.
This energy argument will likely not satisfy the inventor, for most who design perpetual motion and free energy devices do not accept the principle of energy conservation. Analysis of forces and torques give us a better idea of what going on in this mechanical device.
The blue mass is supported against gravity by the two slotted beams. In the ideal case, each beam takes half the load. So, looking only at the left beam, we have two equal masses on a uniform balanced beam. The clockwise torque on the beam due to the left mass is its mass multiplied by its lever arm. The lever arm is the horizontal distance from the mass to the vertical line through the axle of that beam. As the red mass swings down, its torque increases until the beam is horizontal, then decreases until it reaches its lowest point. The torque on the blue mass is constrained to move in a straight vertical line, in the ideal case. (Irregularities in the two beams could alter that, but won't compromise our analysis.)
The torques on the beam's ends are initially balanced, but if given a slight nudge when the blue mass is at its lowest point, the red mass swings downward in an arc. Therefore the counter-clockqies torque on the left end increases in size, while the clockwise torque on the other end is constant. Therefore, the beam will rotate counter-clockwise as the inventor claims, and will continue to do so until it reaches its lowest point.
Now the blue mass is at the top of its range of motion. Nudging the system won't reverse the process, for the torque on the left end of the beam is always greater than on the right. This position, with the blue mass at the top is one of stable equilibirum. When the blue mass at the bottom, the equilibrium was unstable.
To repeat the motion, the inventor inverts the entire device by rotation about its horizontal axis (the dotted line). Of course, this requires some work done by an external agent, for to move any mass from rest requires work, and an equal amount of work is required to bring it to a stop. Granted, this isn't a large amount of work. But even in the idealized case of zero friction, that work is required to keep this device going.
Someone is sure to suggest that we should give it an initial nudge and a torque to set it into rotation about the horizontal axis, and then let it alone, rotating frictionlessly forever, with the beams in something like simple harmonic motion back and and forth between the extreams of their motion. We wish you luck with the mathematics of that complex situation. In any case, as soon as you extract from it the energy you put into it (very soon), it will have come to a stop.
The inventor refers to the Newton-Leibniz formula. This is not a law of physics. It is the fundamental theorem of calculus, and so broadly applicable that it isn't at all clear how, specifically, he is using it here. The quoted numbers suggest to me that he's using some unorthodox definition of efficiency, and perhaps, like so many inventors of unworkable devices, has no clear idea of its proper definition, or the correct way to experimentally measure it.
A related question: The blue mass is loosely held in place by a pin or bolt that passes through the slots of the two beams. Construct a free-body force diagram for the blue mass, identifying the source of each force acting on it. Neglect friction. Discuss how this relates to the motion of the blue mass.
Latest revision May 22, 2017.
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