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Sum of torque

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EOW:
Blue liquid can turn around red point. I suppose red disk like a mass M, this mass attrack liquid like gravity can do in water on Earth. I suppose pressure linear with distance like on Earth. There is buoyancy in liquid. I compute sum of torque :

With:

R1 = 1
R2 = 2

In this case L1 = 1 and L2 = sqrt( 2²-1² ) = sqrt(3)

T1:

integrate from 0 to 1 of (1+x)*(1-x) this give 0.666

T2:

integrate from 0 to sqrt(3) of x*(sqrt(3)-x)/sqrt(3) this give 1/2

Sum of torque is not 0, the system turn alone and give energy I think. Gravity can be change by electrostatic or magnetism forces, small balls can replace liquid.

EOW:
Oops, there is an error, the integration for T2, the sum is zero.


It's possible to use 3 disks like second image showing, they turn at different rotationnal velocity X and Y but in the same direction without friction. Red disks don't turn around itself. At time t, friction is on, this accelerate one disk and decellerate others but if inertia are differents work from torques are different, so the sum of energy is not constant (take in account friction = energy = temperature).

The Work for torque = Torque * angle of rotation ( W=T*Theta )

Theta = 1/2 * a * t² with a the rotationnal acceleration

Second law of Newton : Torque = I * a with I the inertia

Angle of rotation = 1/2 * T/I * t²

Work of torque = 1/2 * T²/I * t²

W1 = 1/2 * T²/I1 * t² with I1 the inertia of grey disk
W2 = -1/2 * T²/I2 * t² with I2 the inertia of red disks

It's logical that W1 is not equal to W2 (in value) if I1 is not equal to I2, no ?

EOW:
Even with linear velocity there is a problem. E=1/2mV². Two objects move at +V1 and +V2. At a moment Object1 give +F to Object2, and Object2 give -F to Object1 without friction. Object1 move faster than Object2. The sum of energy is:

S=1/2m1(V1+F/m1*t)²+1/2m2(V2-F/m2*t)² - (1/2m1V1²+1/2m2V2² )

With data:

m1=10
m2=10
F=100
V1=10
V2=15
t=1

S=500 J
the sum of energy is positive

EOW:
3 disks (no gear just friction). Before start, blue disk, red disk and grey disks turn without friction. Note, red disk turn at w2 around itself and at w1 around blue point. A t=0, friction1 and friction2 are ON. This give a torque to grey and blue disks, energy of blue and grey disks increase. Red disk increase its rotational velocity around blue point (w1 increase). Rotationnal velocity of red disk around its center of gravity slow down SO the energy increase too (energy is 1/2I(w1-w2). All energies increase here. And there is energy from friction too. Friction 1 and 2 are choose for have a torque on red disk and only a torque. For this, friction must change all along the distance. It's just a technical problem. Note this works only fex second, until w2 is stable.

EOW:
Note about w2, w3 and w1. absolute value of w2 must be greater than 0. Absolute value of w2 must be greater than (abs( ((R+r)/r)*w1) +abs(w3)  ). R=radius of big disk. r=radius of small disk. This works only for a transitional regime due to these conditions.

 I think big disk (disk that turn at w1) must be lower than small disk (disk that have w2) for have a difference of energy (w1-w2) increasing when w2 increase like image shows

better with 2 diameters for  disk 2 (second image). Remember energy come from red/yellow disk, it gives energy to grey disk and to blue disk. The goal is to increase energy of red/yellow disk in the same time.

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