Sum of torque

[Gabriele]

I think any solid on which we apply a an equal force for each point of its surface,will obtain only equilibrium

[Gabriele]

i dont know

[EOW]

My forces are correct for one device, I found the sum of forces at 0 and the sum of torques at 0 too. The difference of tangential force from the green/red/black force is canceled by the springs because the shape is asymmetrical. But I can take a sector of disk full with heart shapes with different sizes, in theory I can full the device and like that there is very few balls inside the sector and the springs can't compensate the sum of tangential forces. The buoyancy force is compensated by the forces on the straight lines (1) and (2). I calculated the indefinite integral and I plotted in 3d with the parameter 'd' and 'r' to show there is always the difference in the same direction. Have you any idea what can cancel the tangential forces ?

And I can take any shape, like half disk (straight line and center of the semicircle), or another shape, I need only to fill the sector with all the basic shape with different sizes. The sum of torque is 0 but not the sum of forces.

I need to take small semidisks, very small to prevent the asymmetrical forces from fa/fb. For example, if the sector has a radius of 10 I take for the semidisks a radius from 1e-10 to 1e-6 not more.

I calculated the difference of force when I reduce the size of the semidisk, if I divide by 2 the radius of the disk, the force is divided by 4 but like the surface is multiply by 4 I don't lost anything in the force:

#include <stdio.h>
#include <math.h>

#define N 100

int main()
{


double sum=0, sum2=0,r,d=2,l;
int i;

l=1.0;
r=l/(double)N/2.0;


for(i=0;i<N;i++)
{
  l=1+1/(double)N*(double)i;
  sum+=r*(-sqrt(l*l+4.0*r*l+4.0*r*r)/(r*l+r*r)+l/(r*l+r*r)+2.0/d);
  //   printf("\nl=%f",l);
   
  sum2+=-log(l+2*r)+log(l)+2*r/d;
}

printf("\nr=%f\ns=%f\ns2=%f\ndiff=%f\n",r,sum,sum2,sum-sum2);

return 0;
}

And for a big precision, it's possible to see the force is divided always by 4 when the radius is divided by 2 but the surface is 4 times higher:
#include <stdio.h>
#include <math.h>

#include <gmp.h>
#include <mpfr.h>

#define DG 400
#define N 1000

int main()
{

mpfr_t   s1,s2,r,d,l,t1,t2,t3,t4,diff;
mpfr_init2 (s1, DG);
mpfr_init2 (s2, DG);
mpfr_init2 (r, DG);
mpfr_init2 (d, DG);
mpfr_init2 (l, DG);
mpfr_init2 (t1, DG);
mpfr_init2 (t2, DG);
mpfr_init2 (t3, DG);
mpfr_init2 (t4, DG);
mpfr_init2 (diff, DG);

mpfr_set_d(s1, 0.0, MPFR_RNDD);
mpfr_set_d(s2, 0.0, MPFR_RNDD);
mpfr_set_d(d, 2.0, MPFR_RNDD);
mpfr_set_d(l, 1.0, MPFR_RNDD);

mpfr_div_d(t1, l, N, MPFR_RNDD);
mpfr_div_d(r, t1, 2.0, MPFR_RNDD);

//long double sum=0.0, sum2=0.0,r,d=2.0,l;
int i;

//l=1.0;
//r=l/(long double)N/2.0;


for(i=0;i<N;i++)
{
  //l=1+1.0/(long double)N*(long double)i;
  mpfr_set_d(t1, 1.0, MPFR_RNDD);
  mpfr_div_d(t1, t1, N, MPFR_RNDD);
  mpfr_mul_d(t1, t1, i, MPFR_RNDD);
  mpfr_add_d(l, t1, 1.0, MPFR_RNDD);

  //sum+=r*(-sqrtl(l*l+4.0*r*l+4.0*r*r)/(r*l+r*r)+l/(r*l+r*r)+2.0/d);
  //   printf("\nl=%f",l);
  mpfr_mul(t2, r, r, MPFR_RNDD); 
  mpfr_mul(t1, r, l, MPFR_RNDD); 
  mpfr_add(t2, t1, t2, MPFR_RNDD);

  mpfr_mul_d(t1, r, 4.0, MPFR_RNDD);     
  mpfr_mul(t3, t1, r, MPFR_RNDD);     

  mpfr_mul_d(t1, r, 4.0, MPFR_RNDD);     
  mpfr_mul(t4, t1, l, MPFR_RNDD);     
 
  mpfr_mul(t1, l, l, MPFR_RNDD);     

  mpfr_add(t1, t1, t4, MPFR_RNDD);     
  mpfr_add(t1, t1, t3, MPFR_RNDD);     
  mpfr_sqrt(t1, t1, MPFR_RNDD);     
  mpfr_div(t1, t1, t2, MPFR_RNDD);     

  mpfr_div(t3, l, t2, MPFR_RNDD);     

  mpfr_d_div(t4, 2.0, d, MPFR_RNDD);     

  mpfr_add(t2, t3, t4, MPFR_RNDD);     
  mpfr_sub(t1, t2, t1, MPFR_RNDD);     
  mpfr_mul(t1, t1, r, MPFR_RNDD);
  mpfr_add(s1, s1, t1, MPFR_RNDD);     
   
  //sum2+=-logl(l+2*r)+logl(l)+2*r/d;
  mpfr_mul_d(t1, r, 2.0, MPFR_RNDD);
  mpfr_add(t1, t1, l, MPFR_RNDD);
  mpfr_log(t1, t1, MPFR_RNDD);
  mpfr_sub(s2, s2, t1, MPFR_RNDD);

  mpfr_log(t1, l, MPFR_RNDD);
  mpfr_add(s2, s2, t1, MPFR_RNDD);

  mpfr_div(t1, r, d, MPFR_RNDD);
  mpfr_mul_d(t1, t1, 2.0, MPFR_RNDD);
  mpfr_add(s2, s2, t1, MPFR_RNDD);


}

mpfr_sub(diff, s1, s2, MPFR_RNDD);

//printf("\nr=%Lf\ns=%Lf\ns2=%Lf\ndiff=%Lf\n",r,sum,sum2,sum-sum2);
printf("\n");
mpfr_out_str (stdout, 10, 0, s1, MPFR_RNDD);
printf("\n");
mpfr_out_str (stdout, 10, 0, s2, MPFR_RNDD);
printf("\n");
mpfr_out_str (stdout, 10, 0, diff, MPFR_RNDD);
printf("\n");


return 0;
}

[EOW]

Maybe like that it's easier to the sum of forces: there are the forces from the springs and there are the forces from the axis, all at left.

[dieter]

Have you found a physics simulation software yet?

Do you believe a physics simulation software could accurately and correctly simulate it?

Maybe the linux community has some opensource tools for free.