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I calculate the energy in/out from start to end. I start at 45° and I stop at 45.001° for example. The conservation of the energy is always true, even for a small movement, even for a small time.The force on the circle is at 45° relatively to the horizontal so I need to correct it with the cosine function. The movement of the circle is horizontal, the force is at 45°, the work is F*lg*cos(45°). I don't understand why there is a problem there.
Years ago, my intuition sees that difference of length but I didn't think at the elastic to measure the distance. You can make the experience to see the trajectories. Here, I have 2 different examples, where the sum of energy is the same, not equal to 0, the example with the elastic is a direct measure.
No, there is no mistake in the drawing. I have the exact movement with the friction. The dot B is ALWAYS at the dot of contact between the circle and the red wall.
To have the sum of energy at 0 (the energy needed to move the circle equal to the energy from the heating or the elastic) the distance must be d1. But I found d2.
The 'missing' energy has been converted from kinetic energy to potential energy in the elastic.
I see, the problem comes from the fixed dot, I can't use a fixed dot, so attach the elastic between the circle and the red wall and I change the elastic at each step:
energy recovered from the friction
But I don't know if the calculations are correct.
But it is worst than I think: the method of the needle is an amplificator of the energy, it destroys more energy.
The energy to move the circle is lg*cos(44.5°)*F = d1*F. The distance of friction is d2, so the energy from heating is d2*F. The energy is not conserved. With a line and a circle ... I like !!!!
The only way to prove your idea is to do the math, your drawings are not enough.
If there was no friction then d1 would equal d2. But there was friction so d2 is shorter than d1. So the friction contribution is d1- d2, not d2.