Sum of torque

[EOW]

I verified my tangential forces, they are correct.

Note, it's very important do not block the semicircle on the main object. Each semicircle must give it's force on the gray axis to have the sum of forces at 0 on the gray axis. So the device must accelerate more and more, it wins only a potential energy. This energy can be recover later not in same time.

And it's pretty good with the Neother's theorem

I added an image with the forces:

F1: force from pressure on the black semicircle to the black center
F2: force from pressure on the red semicircle to the red center
F3: force from pressure on the green semicircle to the green center

Like each semicircle can turn around the gray axis, forces on the red/green/black axis are reported to the gray axis
F1' : from F1
F2' : from F2
F3' : from F3

The springs gives a force on the gray axis:
Fs1 : for attract all balls of the bigger semidisk
Fs2: for attract all balls of the smaller semidisks

I added the external force Fext because I want the red semicircle follows the others semicircles.

The sum of forces on the gray axis is 0 (x or y)

At d=0 the torque is 0 but with d different of 0 there is a counterclockwise torque.

I don't need radial forces, but tangential forces are:

F1t = 1/28
F2t = 1/6
F3t = 3/20
Fs1t = 0.02648
Fs2t = 0.007433
Fext = 1/6

Tangential force on the gray axis  = 0

3 images to show each axis alone.

It's not only a sum of torque different of 0 but when the forces F1, F2, F3 apply their force on axis, these forces are applied to the gray axis, so the sum of force is at start (tangential forces):

F1=+1/28
F2=-1/6
F3=+3/20
On gray axis=-0.019

The sum is at 0 but the forces F1-F2+F3 go to the gray center so the sum is:
On gray axis: -0.019 +1/28-1/6+3/20 = 0
But the forces F1-F2+F3 is not 0 it is 0.019

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If I come back with only one semicircle with straight red line (look before) it's strange too, the sum of force on the gray axis is 0 but the forces are not the same so at distance the sum of torque is not 0. If d=10 then the red torque is integrate((0.5-1/(2-x))*(x-10) dx from x=0 to 1) = 1.79 but the black dot torque is 9.5 * 1/6 = 1.583 there is a difference. There is no force on the red axis.

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The last device, the torque is -0.25+0.0568 +1/6*1.5-1/6*0.5 different of 0

[EOW]

Note: axes don't interact with balls ! Each basic heart shape is composed with 3 bodies (3 semicircles). Axes are in another plane.

I take the basic shape I drawn before (like a heart shape). But there is nothing inside the heart shape. I imagine I have a lot of sizes, bigger, smaller, like that I can full all the blue disk with basics heart shapes. Each basic heart shape gives a clockwise torque. It's possible to reduce the size of the heart shape so small that it can full all the blue disk. So at final the springs attracted balls from the white center with the law 1/d^2 (or another law) and there are very few balls if balls are very small. In fact, I need only one thickness of ball and if the size of ball is near 0, the volume attracted by springs is near 0, so I can have no torque on the white center from springs.

In fact, I can dot like before and place balls inside the heart shapes IF I take a lot of heart shapes with different sizes, like that I attract all around the device balls everywhere. And like that I'm sure of my results. The torque is counterclockwise in this  case.

I plot the difference of torque, it's logical that it is always in the same direction but like that I'm sure all basic heart shapes give a clockwise torque (nothing in it).

[EOW]

I drawn the side view: balls don't touch the axes.

Note that each semicircle receives the forces only from the balls not from another semicircle.

I need to give to the red semicircle a torque in the good direction but I recover more from 2 others semicircles.

My integral at start was good. For example, the big circle (main device) is at 20, so I used 1/20 in the equation. I plot the result of the integral with plot3d, function of 'd' and function of 'r'. 'd' is the distance from the white center (main device) and 'r' is the radius of the semicircle. A basic heart shape is composed of 3 semicircles, 2 with the radius 'r' and one with the radius '2r' like you can see in the equation of the integral.

[dieter]

It's just amazing how productive you are. Maybe you should really try it in some physics simulation software, as Gabrielle suggested.

[EOW]

I don't have a simulator for now. Maybe it's possible to simulate only one heart shape to show if I'm right with the torque, someone ?

I think it's not a problem of sum of torque, but sum of forces. The sum of torques is always at 0 but the sum of tangential forces of 3 dots (green, red, black) is not 0, with one shape the springs compensate this difference but if I place a lot of sizes I don't have anything to attract (very few balls) so the main device can be a sector of a disk full with heart shapes. The forces from pressure on the walls 1 and 2 are canceled by the buoyancy force that each heart shape has on it. But the sum of tangential forces from the difference of green/red/black forces can't be canceled.