[LB7 (OP)]

The blue curved line (the blue circle I drew for example) is composed of a lot of small blue spheres, for example, for a radius of 1 m the number of blue spheres is 10000. I take a big number of spheres to simplify the calculations. The size of the blue spheres is constant. The blue spheres don’t rotate around themselves at start but can rotate. There is friction between the blue spheres, the force of friction is F it is a constant value, it is the black arrows between the blue spheres, the value of F is the same between the blue spheres. The red arrow in the center of each blue sphere is the sum of forces from friction. The orange arrows are the sum of forces from the lateral spheres because the previous sphere pushes the next, there is more and more pressure because it is the addition of the red forces.  At start, the radius of the blue circle is 1 m so there is 10000 blue sphere, at the end for example the radius of the blue circle is R=2 m and the number of the blue spheres is 20000. The center of the blue circle A1 is fixed on the ground. For each sphere there is one violet segment, one end of each segment is fixed on A1 the other end is fixed on the center of the blue sphere. If there is N spheres, there is N violet segments. I can increase the length of the violet segments because I increase the radius of the circle. The violet segments can rotate freely around A1, it is the next sphere that stops the sphere to go further and it is for that I have the orange forces. I didn’t draw the violet segments for the circle at the left. The dot A moves in horizontal translation to the right. The energy to move the dot A is 0 because the force F1 is always vertical (in this view). I move in the new blue spheres where the pressure from the previous spheres is maximal, just before the dot A, the pressure is 2piF. The radius of the blue circle will increase, the violet segments and the blue spheres rotate clockwise, so like the blue spheres don’t rotate around themselves at start: they rotate counterclockwise and I have the forces like I drew. I lost an energy to enter the new blue spheres. I win an energy from the friction. I win an energy because the violet segments increase. The spheres keep constant their orientation constant, there are several methods for that: on each violet segment I add a motor that control the orientation of the sphere (the spheres don't rotate in a fixed referential) for example, in that case the energy lost by the motor is recover from the rotation of the violet segments. Another method: the spheres have a mass with a big inertia.

Infinitisemal calculation: with a radius of 1m, I add only one sphere of diameter 2r. I need the energy 2r*2*pi*F to enter the sphere. The angle the spheres rotates clockwise is 2r/1*(1/(1+2r)) = 2r if r very small in comparison to 1, the friction wins 2piF*2r, the same energy I need to enter the sphere. The perimeter of the blue circle increases of 2r so the radius of the blue circle increases of 2r/(2pi) so I win from the length of the violet segments: 2pi²F*2r/(2pi) = 2rpiF. The sum of energy is 2rpiF. The efficiency is (2rpiF+2r2piF)/(2r2piF) = 1.5 or 2/3 in the contrary direction.

It is possible to think in 2d with disks instead of spheres.


sw1.png (134.83 kB . 1626x812 - viewed 2508 times)


sw2.png (82.05 kB . 860x683 - viewed 2453 times)