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I corrected the comparison about the factor (t1-t0) and (t1-t0)², the text in red color. I'm looking for another mistake. No external gravity. No friction. No external motor. Rotational velocity of the disk at start: w0 The distance of the center of mass of the screw-thread from the center of the disk at start: d The mass of the screw-thread: m The mass of the disk: md = 0 Radius of the screw-thread: R The inertia of the thread in its main own axis (when the thread rotates around itself) : Ir The inertia of the thread in axis y: Iy At start, time t = 0 s The force of the device composed of several springs is constant: F The center of mass of the screw-thread moves away the center of the disk at the speed √3/2*2πR for each turn of the screw-thread on the disk. The rotational velocity of the screw-thread relatively to the disk is w(t)*cos(30°). So, the center of mass moves at v(t)=√3/2*2πR*w(t)*cos(30°)/2/π = √3/2*R*w(t)*cos(30°) = 3/4*R*w(t) Like there is no external motor the rotational velocity of the disk is a function of time: w(t)=d/(d+v(t)*t)*w0 The screw-thread is turning at ws(t)=√3/2*w(t) on the disk When I attach at time ‘t0’ the spring the angular velocities are changed and I need to take in accound the inertia of the screw-thread. At time = t1, I stop to count the energies. With the spring the angular velocity of the disk is changed: wr(t)=w(t)-(√3/2+1/2)RF/Ir*(t-t0) For a delta time t1-t0, the angle is : wr(t)*(t1-t0) = (w(t)-(√3/2+1/2)RF/Ir*(t1-t0) ) * (t1-t0) The major part of the angle is changed by w(t) * (t1-t0) not by (√3/2+1/2)RF/Ir*(t1-t0)² And the angular velocity of the thread is also changed by the spring in: wsr(t)=√3/2*wr(t)-1/2*RF/Iy*(t-t0) For a delta time t1-t0, the angle is : wsr(t)*(t1-t0) = (√3/2*wr(t)-1/2*RF/Iy*(t1-t0)) * (t1-t0) The major part of the angle is changed by wr(t)* (t1-t0) not by 1/2*RF/Iy*(t1-t0)² The energy won by the spring is : +(wsr(t1)+wrs(t0))/2*(t1-t0)*2*RF = +( √3/2*wr(t1)-1/2*RF/Iy*(t1-t0) + √3/2*wr(t0)-1/2*RF/Iy*(t0-t0) ) /2 *(t1-t0) *2*RF = +( √3/2*(w(t1)-(√3/2+1/2)RF/Ir*(t1-t0))-1/2*RF/Iy*(t1-t0) + √3/2*(w(t0)-(√3/2+1/2)RF/Ir*(t0-t0))-1/2*RF/Iy*(t0-t0) ) /2 *(t1-t0) *2*RF The energy lost from the torque of the force F on the disk, at distance R from the center of the disk is : -(wr(t1)+wr(t0))/2*(t1-t0)*√3/2*RF = -(w(t1)-(√3/2+1/2)RF/Ir*(t1-t0)+w(t0)-(√3/2+1/2)RF/Ir*(t0-t0))/2*(t1-t0)*√3/2*RF The energy lost from the force F on the axis ‘y’ on the disk is: -(wr(t1)+wr(t0))/2*(t1-t0)*1/2*RF = -(w(t1)-(√3/2+1/2)RF/Ir*(t1-t0) + w(t0)-(√3/2+1/2)RF/Ir*(t0-t0) )/2*(t1-t0)*1/2*RF The energy lost by the center of mass is: -3/4*RF*(wsr(t1)+wsr(t0))/2*(t1-t0) = -3/4*RF*(√3/2*wr(t1)-1/2*RF/Iy*(t1-t0) + √3/2*wr(t0)-1/2*RF/Iy*(t0-t0))/2*(t1-t0) = -3/4*RF*(√3/2*(w(t1)-(√3/2+1/2)RF/Ir*(t1-t0))-1/2*RF/Iy*(t1-t0) + √3/2*(w(t0)-(√3/2+1/2)RF/Ir*(t0-t0))-1/2*RF/Iy*(t0-t0))/2*(t1-t0) For a small angle from the time t0 I attached the spring, (t1-t0) near equal to 0 but -(√3/2+1/2)RF/Ir*(t1-t0)² is near 0 at square and for -1/2*RF/Iy*(t1-t0)² it is the same, I can approximate the sum of energy. For example if t1-t0=0.01 then (t1-t0)²=0.0001. The sum of energy is well at: - 0.28*RF*(t1-t0)*(wr(t1)+wr(t0))/2 In fact, it is logical because the angular velocity for the disk is w(t)=w0-k*t with 'k' a constant and the angle of rotation is w(t)*t = (w0-k*t)*t and if t is small the factor kt² is negligible. So, one work I calculated is wrong.

It is what I'm doing, I found a little mistake but it is more for the expression of the angular velocity of the disk due to the movement of the center of mass: No external gravity. No friction. No external motor. Rotational velocity of the disk at start: w0 The distance of the center of mass of the screw-thread from the center of the disk at start: d The mass of the screw-thread: m The mass of the disk: md = 0 Radius of the screw-thread: R The inertia of the thread in its main own axis (when the thread rotates around itself) : Ir The inertia of the thread in axis y: Iy At start, time t = 0 s The force of the device composed of several springs is constant: F The center of mass of the screw-thread moves away the center of the disk at the speed √3/2*2πR for each turn of the screw-thread on the disk. The rotational velocity of the screw-thread relatively to the disk is w(t)*cos(30°). So, the center of mass moves at v(t)=√3/2*2πR*w(t)*cos(30°)/2/π = √3/2*R*w(t)*cos(30°) = 3/4*R*w(t) Like there is no external motor the rotational velocity of the disk is a function of time: w(t)=d/(d+v(t)*t)*w0 The screw-thread is turning at ws(t)=√3/2*w(t) on the disk When I attach at time ‘t0’ the spring the angular velocities are changed and I need to take in accound the inertia of the screw-thread. At time = t1, I stop to count the energies. With the spring the angular velocity of the disk is changed: wr(t)=w(t)-(√3/2+1/2)RF/Ir*(t-t0) For a delta time t1-t0, the angle is : wr(t)*(t1-t0) = (w(t)-(√3/2+1/2)RF/Ir*(t1-t0) ) * (t1-t0) The major part of the angle is changed by w(t) * (t1-t0) not by (√3/2+1/2)RF/Ir*(t1-t0)² And the angular velocity of the thread is also changed by the spring in: wsr(t)=√3/2*wr(t)-1/2*RF/Iy*(t-t0) For a delta time t1-t0, the angle is : wsr(t)*(t1-t0) = (√3/2*wr(t)-1/2*RF/Iy*(t1-t0)) * (t1-t0) The major part of the angle is changed by wr(t)* (t1-t0) not by 1/2*RF/Iy*(t1-t0)² The energy won by the spring is : +(wsr(t1)+wrs(t0))/2*(t1-t0)*2*RF = +( √3/2*wr(t1)-1/2*RF/Iy*(t1-t0) + √3/2*wr(t0)-1/2*RF/Iy*(t0-t0) ) /2 *(t1-t0) *2*RF = +( √3/2*(w(t1)-(√3/2+1/2)RF/Ir*(t1-t0))-1/2*RF/Iy*(t1-t0) + √3/2*(w(t0)-(√3/2+1/2)RF/Ir*(t0-t0))-1/2*RF/Iy*(t0-t0) ) /2 *(t1-t0) *2*RF The energy lost from the torque of the force F on the disk, at distance R from the center of the disk is : -(wr(t1)+wr(t0))/2*(t1-t0)*√3/2*RF = -(w(t1)-(√3/2+1/2)RF/Ir*(t1-t0)+w(t0)-(√3/2+1/2)RF/Ir*(t0-t0))/2*(t1-t0)*√3/2*RF The energy lost from the force F on the axis ‘y’ on the disk is: -(wr(t1)+wr(t0))/2*(t1-t0)*1/2*RF = -(w(t1)-(√3/2+1/2)RF/Ir*(t1-t0) + w(t0)-(√3/2+1/2)RF/Ir*(t0-t0) )/2*(t1-t0)*1/2*RF The energy lost by the center of mass is: -3/4*RF*(wsr(t1)+wsr(t0))/2*(t1-t0) = -3/4*RF*(√3/2*wr(t1)-1/2*RF/Iy*(t1-t0) + √3/2*wr(t0)-1/2*RF/Iy*(t0-t0))/2*(t1-t0) = -3/4*RF*(√3/2*(w(t1)-(√3/2+1/2)RF/Ir*(t1-t0))-1/2*RF/Iy*(t1-t0) + √3/2*(w(t0)-(√3/2+1/2)RF/Ir*(t0-t0))-1/2*RF/Iy*(t0-t0))/2*(t1-t0) For a small angle from the time t0 I attached the spring, (t1-t0) near equal to 0 but -(√3/2+1/2)RF/Ir*(t1-t0)² is 10 times smaller and -1/2*RF/Iy*(t1-t0)² is 10 times smaller, I can approximate the sum of energy. The sum of energy is well at: - 0.28*RF*(t1-t0)*(wr(t1)+wr(t0))/2

No external gravity. No friction. No external motor. Rotational velocity of the disk at start: w0 The distance of the center of mass of the screw-thread from the center of the disk at start: d The mass of the screw-thread: m The mass of the disk: md = 0 Radius of the screw-thread: R The inertia of the thread in its main own axis (when the thread rotates around itself) : Ir The inertia of the thread in axis y: Iy At start, time t = 0 s The force of the device composed of several springs is constant: F The center of mass of the screw-thread moves away the center of the disk at the speed √3/2*2πR for each turn of the screw-thread on the disk. The rotational velocity of the screw-thread relatively to the disk is w(t)*cos(30°). So, the center of mass moves at v(t)=√3/2*2πR*w(t)*cos(30°)/2/π = √3/2*R*w(t)*cos(30°) = 3/4*R*w(t) Like there is no external motor the rotational velocity of the disk is a function of time: w(t)=w0-d/(d+v(t)*t) So the screw-thread is turning at ws(t)=√3/2*w(t) on the disk When I attach at time ‘t0’ the spring the angular velocities are changed and I need to take in accound the inertia of the screw-thread. At time = t1, I stop to count the energies. With the spring the angular velocity of the disk is changed: wr(t)=w(t)-(√3/2+1/2)RF/Ir*(t-t0) For a delta time t1-t0, the angle is : wr(t)*(t1-t0) = (w(t)-(√3/2+1/2)RF/Ir*(t1-t0) ) * (t1-t0) The major part of the angle is changed by w(t) not by (√3/2+1/2)RF/Ir*(t-t0) And the angular velocity of the thread is also changed by the spring in: wsr(t)=√3/2*wr(t)-1/2*RF/Iy*(t-t0) For a delta time t1-t0, the angle is : wsr(t)*(t1-t0) = (√3/2*wr(t)-1/2*RF/Iy*(t1-t0)) * (t1-t0) The major part of the angle is changed by wr(t) not by 1/2*RF/Iy*(t-t0) The energy won by the spring is : +(wsr(t1)+wrs(t0))/2*(t1-t0)*2*RF = +( √3/2*wr(t1)-1/2*RF/Iy*(t1-t0) + √3/2*wr(t0)-1/2*RF/Iy*(t0-t0) ) /2 *(t1-t0) *2*RF = +( √3/2*(w(t1)-(√3/2+1/2)RF/Ir*(t1-t0))-1/2*RF/Iy*(t1-t0) + √3/2*(w(t0)-(√3/2+1/2)RF/Ir*(t0-t0))-1/2*RF/Iy*(t0-t0) ) /2 *(t1-t0) *2*RF The energy lost from the torque of the force F on the disk, at distance R from the center of the disk is : -(wr(t1)+wr(t0))/2*(t1-t0)*√3/2*RF = -(w(t1)-(√3/2+1/2)RF/Ir*(t1-t0)+w(t0)-(√3/2+1/2)RF/Ir*(t0-t0))/2*(t1-t0)*√3/2*RF The energy lost from the force F on the axis ‘y’ on the disk is: -(wr(t1)+wr(t0))/2*(t1-t0)*1/2*RF = -(w(t1)-(√3/2+1/2)RF/Ir*(t1-t0) + w(t0)-(√3/2+1/2)RF/Ir*(t0-t0) )/2*(t1-t0)*1/2*RF The energy lost by the center of mass is: -3/4*RF*(wsr(t1)+wsr(t0))/2*(t1-t0) = -3/4*RF*(√3/2*wr(t1)-1/2*RF/Iy*(t1-t0) + √3/2*wr(t0)-1/2*RF/Iy*(t0-t0))/2*(t1-t0) = -3/4*RF*(√3/2*(w(t1)-(√3/2+1/2)RF/Ir*(t1-t0))-1/2*RF/Iy*(t1-t0) + √3/2*(w(t0)-(√3/2+1/2)RF/Ir*(t0-t0))-1/2*RF/Iy*(t0-t0))/2*(t1-t0) For a small angle from the time t0 I attached the spring, (t1-t0) near equal to 0 but -(√3/2+1/2)RF/Ir*(t1-t0)² is 10 times smaller and -1/2*RF/Iy*(t1-t0)² is 10 times smaller too, the energies are counted from t0 to t1, the energies are: The sum of energy is well at: - 0.28*RF*(t1-t0)*(wr(t1)+wr(t0))/2

Ok, I done calculations without any motor. No external gravity. No friction. No external motor. Rotational velocity of the disk at start: w0 The distance of the center of mass of the screw-thread from the center of the disk at start: d The mass of the screw-thread: m The mass of the disk: md = 0 Radius of the screw-thread: R The inertia of the thread in its main own axis (when the thread rotates around itself) : Ir The inertia of the thread in axis y: Iy At start, time t = 0 s The force of the device (I drew a spring but it is a composition of springs) composed of several springs is constant: F The center of mass of the screw-thread moves away the center of the disk at the speed √3/2*2πR for each turn of the screw-thread on the disk. The rotational velocity of the screw-thread relatively to the disk is w(t)*cos(30°). So, the center of mass moves at v(t)=√3/2*2πR*w(t)*cos(30°)/2/π = √3/2*R*w(t)*cos(30°) = 3/4*R*w(t) Like there is no external motor the rotational velocity of the disk is a function of time: w(t)=w0-d/(d+v(t)*t) So the screw-thread is turning at ws(t)=√3/2*w(t) on the disk When I attach at time ‘t0’ the spring the angular velocities are changed and I need to take in accound the inertia of the screw-thread. At time = t1, I stop to count the energies. With the spring the angular velocity of the disk is changed: wr(t)=w(t)-(√3/2+1/2)RF/Ir*(t-t0) And the thread is also changed in: wsr(t)=√3/2*wr(t)-1/2*RF/Iy*(t-t0) For a small angle from the time t0 I attached the spring, (t1-t0) near equal to 0 so -(√3/2+1/2)RF/Ir*(t1-t0) near to 0 and -1/2*RF/Iy*(t1-t0) near to 0, the energies are counted from t0 to t1, the energies are: The energy won by the spring is : +(wsr(t1)+wrs(t0))/2*2*RF=√3/2*(wr(t1)+wr(t0))/2*2*RF The energy lost from the torque of the force F on the disk, at distance R from the center of the disk is : -(wr(t1)+wr(t0))/2*√3/2*RF The energy lost from the force F on the axis ‘y’ on the disk is: -(wr(t1)+wr(t0))/2*1/2*RF The energy lost by the center of mass is: -3/4*RF*(wsr(t1)+wsr(t0))/2 = - 3/4*√3/2*RF*(wr(t1)+wr(t0))/2 The sum of energy is well at: - 0.28*RF*(wr(t1)+wr(t0))/2

Why ? for a small angle of study δ=1e-10rd, the distance increased by the spring is sqrt(3)*δ*R, it is 3.46e-11 m, the force is constant for that small difference of length. And it is possible in theory to have a "device" like a spring but with a constant force. No, for a small angle of rotation δ. And it is possible to have a device composed of several springs to have a constant force:

For example, with a bolt of 1000 kg with a radius for the bolt at 0.2 m, the moment of inertia could be at I=20 (it is an example), a spring with a force of 1N, with w=10rd/s and a study for δ=1e-10 rd. The time of study is dt=δ/w=1e-11 s. The torque is T=0.2 Nm. The angular acceleration will be T/I=0.01 rd/s². The angular velocity at the end of 1e-11s will be 1e-13 rd/s compared with w = 10 rd/s, I supposed it is constant or don't change the result enough to have the sum at 0.

Of course, sorry for that miss. The angular velocity I supposed constant ? I thought I could only do the sum of energy without the mass IF I compare with the same device but without the spring and for a small angle of study. So, like I'm not able to study the transient sum of energy, I take a motor, I will count the energy I give to the motor to let constant 'w'. Before I attached the spring, I supposed all the parameters of the device like constant, I mean the bolt doesn't rotate around itself (no friction), the center of mass of the bolt rotates at 'w', the disk rotates at 'w' and the center of mass of the bolt is moving more and more far away from the center of the disk. The sum of energies of the closed device :{external power + motor + bolt + disk } is constant. The motor needs an energy but the center of mass will increase its kinetic energy. At time=0, I add the spring: The motor need to give an extra energy because there is the spring: b) The force F from the spring is applied in the axis 'x' with an angle of 30° on the disk, then the force that works is √3/2F, so in δ, the disk lost -√3/2F * δR = - √3/2*δRF. c) The force F from the spring is applied in the axis 'y' with an angle of 60° so the force that works is F/2 (the helix applies a torque on the disk, the torque depends of F on 'y'), so in δ the disk lost F/2 * δR = - 1/2*δRF I win the potential energy from the length of the spring: a) The lead of the helix is 2πR√3 so the spring increases its length of 2*2πR in one turn of the helix, so the energy won by the spring is √3/2*2*δRF = + √3*δRF. I took the step of the helix like that to keep constant the angle Ω And I lost the energy that the spring recovers from the movement of the center of mass (farther from the center of the disk): d)/ The center of mass of the helix changes its radius, the difference of length is δR*sqrt(3)/2 for a δ angle of the helix but the helix rotates at cos(30°) of the rotational velocity of the disk and the force is at 30° so the energy lost is - 3/4*sqrt(3)/2*δRF. I lost that energy because I can’t recover it with another device. For a small angle of rotation, the bolt doesn't have time to rotate around itself because its angular velocity is 0 rd/s when I attach the spring. So the center of mass will move like no spring. I can take an angle of study of 1e-10 rd, with a mass of the bolt at 1000 kg and the force of the spring at 1 N.

Do you mean potential energy from gravity ? There is no external gravity. The sum of energy of the closed device: disk + bolt is conserved (there is no friction). The center of mass of the bolt moves more and more far away from the center but this will change the angular velocity of the disk. It is possible to add a motor to the disk and keep constant the rotational velocity of the disk. The energy from the motor goes to the kinetic energy of the bolt. Again, the sum of energy is conserved in the close device (even composed of a motor if I take in account the energy I give to the motor). My work, suppose the energy is conserved without the spring. And for a small angle of rotation I have the energies due to the presence of the spring: a) The lead of the helix is 2πR√3 so the spring increases its length of 2*2πR in one turn of the helix, so the energy won by the spring is √3/2*2*δRF = + √3*δRF. I took the step of the helix like that to keep constant the angle Ω b)/ The force F from the spring is applied in the axis 'x' with an angle of 30° on the disk, then the force that works is √3/2F, so in δ, the disk lost -√3/2F * δR = - √3/2*δRF. c) The force F from the spring is applied in the axis 'y' with an angle of 60° so the force that works is F/2 (the helix applies a torque on the disk, the torque depends of F on 'y'), so in δ the disk lost F/2 * δR = - 1/2*δRF . d)/ The center of mass of the helix changes its radius, the difference of length is δR*sqrt(3)/2 for a δ angle of the helix but the helix rotates at cos(30°) of the rotational velocity of the disk and the force is at 30° so the energy lost is - 3/4*sqrt(3)/2*δRF. I lost that energy because I can’t recover it with another device. The sum of energy for an angle δ is (√3/2 - 1/2 -3/4*√3/2) * δRF = - 0.28 * δRF

I mean around itself. For a small angle and because the bolt has a mass. The center of gravity of the bolt rotate like the disk but the bolt doesn't rotate around itself. The disk rotates around itself at w. The bolt rotates with the disk but the bolt doesn't rotate around itself at start. The bolt inside the disk is fixed to the disk, the disk rotates clockwise at w, so if you are on the disk you see the bolt rotates counterclockwise. No friction, correct. I added the consequences about the energy because I add the spring. Without the spring is conserved, I add the spring and I calculate the energies. I gave only the differences.

The spring will give a clockwise torque to the bolt and it will start to rotate. I gave the calculation for an angle of 2pi but it is possible to study for a small angle, in that case the thread don't have time to rotate (around itself). The bolt will rotate relatively to the disk and move. The center of gravity of the bolt will move more and more far away from the center of the disk. In my calculations I forgot the work lost because the center of gravity moves it is 2piRF * 3/4*sqrt(3)/2, because in one turn of the disk, the bolt rotates only at w*cos(30°) and the angle of the force is at 30°. In one turn of the bolt the center of gravity moves of 2pi*R*sqrt(3)/2. Even, with that work, the sum is not at 0.

The disk rotates but the bolt doesn't rotate around itself so if you are on the disk you see the bolt rotates counterclockwise. Without friction nor the spring it is always like that.

Because there is a nut inside the disk. The disk rotates at 'w' when I start to count the energy. The bolt doesn't turn around itself when I set on the nut, the bolt is at -w*cos(30°) relatively to the disk when the spring is attached and the nut is set on. I give only the energies in comparison with the device without the spring (the energy is conserved with or without the spring). When I add the spring there is only the energy wins by the length of the spring and the energies lost by the torques from the force F on each axis 'x' and ''y'. What do you need ?

I redraw the start position without transparency and corrected the images:

The device is a disk in rotation around the orange axis. The disk rotates only, don't move in vertical translation. There is a hole inside the disk (hole at 30° relatively to the orange axis). In that hole I fixed a controllable nut. Controllable nut because I want to control the time the nut works, I can set OFF/ON the nut, it is simply walls that moves. I passed through the nut a screw thread. That's all. I drew 2 views, the left view is a side view and the right a top view. In the left view, I drew the thread by transparency to see the thread. Before start, the screw thread doens't rotate around itself. The device-nut is OFF, the spring is not attached on the thread. After, I rotate the disk at w=10rd/s for example, I need time for that. I set ON the device-nut and I attached in the same time the spring to the thread, and I count the energies. It it easier to take a controllable nut, because I know the rotational velocity of the disk (it is not an acceleration). The controllable nut is only retractable walls, it is just to simplify the calculations.

Nothing relative to my old post. I try to find my mistake. It is a classical simple mechanics device. Maybe the rotational velocity of the helix ?