- #1

- 421

- 0

Hi,

I take a big number of disks to composed a circle of a radius of 1 m, the blue curved line is in fact several very small disks:

I take a big number of disks to simplify the calculations, and I take the size of the disks very small in comparison of the radius of the circle. The center A1 of the circle is fixed to the ground. I take each disk from the center of the circle A1 with a segment. The disks keep constant their orientation in the space. There is friction between the disks, I count the energy from the friction. I insert a disk to rotate clockwise all the disks, the disk is inserted before the dot A, like each disk is taken from the center and like there is friction between the disks and like the segment that taken the disks rotate clockwise the forces are like that:

It is the dot A that blocks the force F1. The dot A will move in horizontal translation to the right.

Like I increase the radius of the circle, I need to increase the lengths of the segments that taken the disks.

I drew a limited number of disks.

Black arrows: forces (same value F) from friction between the disks.

Red arrows on the center of each disk: the sum of forces of friction on each center of the disk (c0..c10), F1 is the algebric sum of all these forces.

The dot A receives all the sum of the red forces (F1=2piF)

The violet segments can rotate freely around A1, only the dot A blocks the rotation of the violet segments around A1.

Each disk can rotate around itself (from c0 to c10) but keep constant their orientation.

The orange arrows are the forces due to the lateral forces from the last disks (it is like pressure).

I suppose the value of the force of friction between the disks like a constant to simplify the calculations. So, F1 is constant in value.

For the example of a radius of 1 m, the energy needed to insert the new disk is : ##2*r*2*\pi*F##. The angle of rotation of the disks is ##2r##, so the energy from friction is ##2*\pi*F*2*r## the same energy I need to enter the disk. I win an energy to increase the length of the segments that taken the disks, the perimeter is increased of ##2*r##, so the radius is increased of ##2*r/(2*\pi)##, so the energy recovered is ##2*r*\pi*F##. I don't find the sum at 0. Maybe when I increase the radius of the circle the friction is not so much and the angle of rotation lower than I calculate ?

I took a pure geometric example to simplify the calculations and to keep constant the orientation of the disks in a fixed referential there are a lot of method.

Have you an idea where my mistake is ?

I take a big number of disks to composed a circle of a radius of 1 m, the blue curved line is in fact several very small disks:

I take a big number of disks to simplify the calculations, and I take the size of the disks very small in comparison of the radius of the circle. The center A1 of the circle is fixed to the ground. I take each disk from the center of the circle A1 with a segment. The disks keep constant their orientation in the space. There is friction between the disks, I count the energy from the friction. I insert a disk to rotate clockwise all the disks, the disk is inserted before the dot A, like each disk is taken from the center and like there is friction between the disks and like the segment that taken the disks rotate clockwise the forces are like that:

It is the dot A that blocks the force F1. The dot A will move in horizontal translation to the right.

Like I increase the radius of the circle, I need to increase the lengths of the segments that taken the disks.

I drew a limited number of disks.

Black arrows: forces (same value F) from friction between the disks.

Red arrows on the center of each disk: the sum of forces of friction on each center of the disk (c0..c10), F1 is the algebric sum of all these forces.

The dot A receives all the sum of the red forces (F1=2piF)

The violet segments can rotate freely around A1, only the dot A blocks the rotation of the violet segments around A1.

Each disk can rotate around itself (from c0 to c10) but keep constant their orientation.

The orange arrows are the forces due to the lateral forces from the last disks (it is like pressure).

I suppose the value of the force of friction between the disks like a constant to simplify the calculations. So, F1 is constant in value.

For the example of a radius of 1 m, the energy needed to insert the new disk is : ##2*r*2*\pi*F##. The angle of rotation of the disks is ##2r##, so the energy from friction is ##2*\pi*F*2*r## the same energy I need to enter the disk. I win an energy to increase the length of the segments that taken the disks, the perimeter is increased of ##2*r##, so the radius is increased of ##2*r/(2*\pi)##, so the energy recovered is ##2*r*\pi*F##. I don't find the sum at 0. Maybe when I increase the radius of the circle the friction is not so much and the angle of rotation lower than I calculate ?

I took a pure geometric example to simplify the calculations and to keep constant the orientation of the disks in a fixed referential there are a lot of method.

Have you an idea where my mistake is ?

Last edited: