John2020

I don't see the analogy with what I present in Fig.1-Upper. Make a drawing.

As i said I am addressing an ideal situation as exactly is presented in Fig.1-Upper. In my first post I mentioned the system is internally powered. We don't have to address motors and power in this analysis (it is not the purpose of this paper). As you see in Fig.1-Upper there is a couple F_A and F_A' (perpendicular to the screw) that applies upon the screw provided by the motor in the housing. I brought this subject for discussion to see if it may work in principle. Again, the rotational energy of the screw is assumed to be entirely converted to mass m_T kinetic energy. It is a theoretical study of an ideal system. That is all we need to know as starting point to address this ideal system. So, for me it remains the following statements:to be addressed: 1.Inertial frame: There is no Euler force 2.Rotating frame: There is Euler force 3.Inertial frame:The rotating frame is an intrinsic part of the inertial frame that means the whole is considered as a single Inertial frame 4.Rotating frame: Euler force is the cause behind the acceleration of mass m_T without causing a reaction upon the rest of the inertial frame 5.Inertial frame: Due to (3) and (4) the COM of the Inertial frame will change along with its momentum 6.Inertial frame: It will start to accelerated because of (5) Is there any objection on the above? The above statements address this ideal situation. An alternative construction that has no contact forces is the magnetic lead screw (contactless lead screw). See the YouTube video below:

If the above statement is related to real forces along the axis of rotation of the translation screw then, you are wrong because there is none. That would be evident only in Fig.1-Lower but not in Fig.1-Upper.

Ideally, we assume the momentum of the translation screw is entirely converted into drive nut (mass m_T) displacement. In this case the rotation of the device is negligible. This is what I also I assume (an ideal situation) in my paper.

The Newton pairs appear perpendicular to the acquired momentum of mass m_T since they follow the helix trajectory of the thread upon the translation screw. You overlook something here. The rotating frame is an intrinsic part of the Inertial frame therefore is significant for the COM. The rotating frame is an intrinsic part of the Inertial frame therefore is significant for the COM. Drive nut (mass m_T) is being affected by the accelerating rotation of the translation screw that implies a Fictitious force (Euler) is at play. It is relevant to COM because it is the cause behind a reactionless accelerating transfer of mass m_T. You cannot ignore the accelerating mass transfer caused by the reactionless Euler force that is seen from the Inertial as a "real" but still reactionless force. Therefore, you are not using the right kind of the Lagrangian formalism. You are the first who clearly acknowledges how a linear actuator works. The next step is to acknowledge the rotating frame is an intrinsic part of the rotating frame. Acknowledging this is equal to the statement "rotating frame is significant to COM" (since a reactionless accelerating mass transfer is taking place).

That would be possible when there wouldn't be an accelerating mass m_T. I am expecting the behavior you just mentioned to be negligible in Fig.1-Upper because the angular momentum of the device (Ideal Machine) is transferred to mass m_T. What is your view on the below: 1.Inertial frame: There is no Euler force 2.Rotating frame: There is Euler force 3.Inertial frame:The rotating frame is an intrinsic part of the inertial frame that means the whole is considered as a single Inertial frame 4.Rotating frame: Euler force is the cause behind the acceleration of mass m_T without causing a reaction upon the rest of the inertial frame 5.Inertial frame: Due to (3) and (4) the COM of the Inertial frame will change along with its momentum 6.Inertial frame: It will start to accelerated because of (5) Is there any objection on the above?

Yes. I mentioned this on my first post. See red colored text. The power supply and motor are enclosed in the housings that hold the translation screw. I agree. This is my answer: 1.Inertial frame: There is no Euler force 2.Rotating frame: There is Euler force 3.Inertial frame:The rotating frame is an intrinsic part of the inertial frame that means the whole is considered as a single Inertial frame 4.Rotating frame: Euler force is the cause behind the acceleration of mass m_T without causing a reaction upon the rest of the inertial frame 5.Inertial frame: Due to (3) and (4) the COM of the Inertial frame will change along with its momentum 6.Inertial frame: It will start to accelerated because of (5) Is there any objection on the above?

The topology of the threads of the translation screw is a helix that means the action-reaction pair is forced to follow the helix trajectory that implies the action-reaction pair is perpendicular to the acquired momentum of mass m_T.

If the translation screw wouldn't rotate, would mass m_T advance to the right just by pushing it with a real force from its left side? The answer is obviously NO. Because the rod is threaded that inhibits the appearance of real forces along the axis of rotation of the translation screw. I agree. Let's take it step by step again: 1.Inertial frame: There is no Euler force 2.Rotating frame: There is Euler force 3.Inertial frame:The rotating frame is an intrinsic part of the inertial frame that means the whole is considered as a single Inertial frame 4.Rotating frame: Euler force is the cause behind the acceleration of mass m_T without causing a reaction upon the rest of the inertial frame 5.Inertial frame: Due to (3) and (4) the COM of the Inertial frame will change along with its momentum 6.Inertial frame: It will start to accelerated because of (5) Is there any objection on the above?

No.As I mentioned above the action-reaction pair is perpendicular to the acquired momentum of mass m_T that implies mass m_T acceleration is attributed to an Euler Force. Just think the following: If the translation screw wouldn't rotate, would mass m_T advance to the right just by pushing it with a real force from its left side? The answer is obviously no. Because the rod is threaded that inhibit the appearance of real forces along the axis of rotation of the translation screw.

The forces you mention are in essence the real action-reaction collinear forces pair, that are perpendicular to the acquired momentum of mass m_T. There are no real forces along the axis of rotation of the translation screw, just the fictitious one (Euler force).

In the construction's skeleton (inertial frame of reference). I am sitting inside the skeleton (inertial frame of reference) but not on the rotating frame (translation screw).

The acceleration of mass m_T appears as "real" force from the inertial frame (skeleton), however it is is in essence fictitious and attributed to the rotating frame. In this regard, I would agree with swansont but we cannot ignore the mechanism just in the name of math (this is where you misinterpret how the construction works). The mechanism must be visible/justifiable when we describe the motion of the system with forces that means in the expression that shows the advancing of mass m_T, the fictitious force must be clearly identifiable (and not as a real force). A real force wouldn't be dω depended. Because that kind of Lagrangian you use assumes all internal happening are action-reaction pairs. This is not evident in our case. While mass m_T is being accelerated do you see any force to apply in the opposite direction upon the rest of the system?

It would be a contradiction if one attempts to describe the motion of the construction based on the rocket concept, which is wrong. As I explained the construction motion mechanism is not related to the rocket concept but to a reactionless internal mass transfer because of the rise of the Euler Force which is in essence, inertial (no counter part = no reaction). The above implies although the system maintains its mass, its effective inertia reduces (the effect is attributed to the redeployment of COM and acceleration of the system) while being accelerated. This is the new effect my work predicts, namely the reduction of inertia. Crucial question: Does the acceleration of mass m_T is attributed to a real or to a fictitious force (Euler Force)?

Well, here we have to clarify something that was not shared in my initial post because it goes a bit further. I would speak about it when we would have clarified and acknowledged how the construction in Fig.1-Upper accelerates by means of the Euler Inertial Force. From the moment you are asking about this now then I have to share more. See eq.37 and 38 in the paper and the justification about them which is the following: "However, the way Eq.(33) is formulated, it appears as a momentum exchange that points to a separation between the moving part (m_T) and the rest of the system, which is not valid. The mass m_T is an inseparable intrinsic part of the system. To agree with the construction’s integrity and the observations from an external inertial frame of reference, Eq.(33) may have a different and more general form." 1.delta(m) stays in the system and in our case corresponds to m_T. In order to use the internal mass transfer rate notion, we have to keep the relative velocity u_rel (du_T) constant but not zero. Then we have to correspond m_T -> dm/dt 2.Crucial point: The internal accelerated mass transfer is conducted without causing a reaction upon the rest of the system. This is justified from the fact that mass m_T is propelled by the Euler Inertial Force that by nature does not possess a reaction (in contrast to collinear forces. See Newton's 3rd law). As you may see in Fig.1-Lower we have the case where the contact collinear forces F_A and F_R does not let system to accelerate because any action (pushing mass m_T) will be counteracted by the F_R upon the rest of the system. 3.Due to (2) and since the mass m_T is an intrinsic part of the system (skeleton of the construction), this has as consequence the COM of the system to change and accelerate. 4.Linear momentum conservation always hold and applies for the COM which is intrinsic to the system, too. 5.Because of (3) and (4), the system will start to accelerate. No. What you are saying would happen when there were external torques in the system and the translation screw would extend beyond the housings (in order the construction to advance counterclockwise). In Fig.1-Upper, we have just internal torques that means the translation screw accelerates clockwise while mass m_T advances counterclockwise. But as you may see there are two linear guides that restrict a counterclockwise rotation of mass m_T. This restriction forces mass m_T to advance to the right (that is inherently counterclockwise). In other words, when one removes the linear guides and holds the drive nut (m_T) with his hand, when the linear actuator is powered and the translation screw accelerates clockwise, the construction will advance to the left while the drive nut (m_T) to the right (conservation of angular momentum).

I will address this question and that of Ghideon in a moment.I have to finish some uncompleted task and I will be back in an hour or so.

I will make you a simple question: What is the cause behind the acceleration of mass m_T, a real or a fictitious force? I am still waiting to verify the 6 points I shared above. Please show me what is invalid in the above 6 points ((a) to (f)).

I don't think you follow the entire conversation from the beginning as also you haven't look at Fig.1-Upper carefully. The frame is being accelerate by the couple F_A and F_A' whom are both not constant (see Fig.1-Upper as also the expressions I share on my first post) that implies a varying angular velocity dω/dt ≠ 0. I don't think you follow this discussion. I have more than once refer to this. In order to help you understand how the construction works, please verify the 6 statements below: a) There is an accelerating internal mass transfer due to the Euler force (it is clear there is no real force here) b) Due to (a), we have no reaction upon the rest of the system c) Mass m_T is an intrinsic part of the system (mass m) d) Due to (c) the reactionless internal mass transfer of mass m_T will inevitably cause a change in system's COM along with its momentum e) Conservation of linear momentum always holds f) Due to (c), (d) and (e) the system will acquire an acceleration (having the same direction as the momentum of mass m_T) because its COM has been changed due to the acceleration of mass m_T Is there any objections on the above? I address the concept my own way and leads to the 6 points conclusion I shared above. Do you confirm them or not? If not then tell us why? You are right in the inertial frame you don't have fictitious forces but just on the rotating frame. Here is the key to your understanding: The rotating frame is an intrinsic part of the inertial frame. The rotating frame causes the rise of the Euler force that is the cause behind the acceleration of mass m_T. From the moment the rotating frame is an intrinsic part of the inertial frame, the acceleration of mass m_T has effectively change the COM of the inertial frame along with its momentum.

Let's put aside the cosmological stuff because I cannot support through my work. They are just assumptions, OK. Consider, the telescopes cannot observe down to the big bang where there even GR is expected to break. So as long an angular velocity is so negligible even at a distance several thousands of light years from the center of the Universe, GR will still work. Again, let this subject for another thread and another paper (that derives, the mass, the radius (approx. 1E28m much larger from what is assumed today), the acceleration of the Universe, the quantum length (it is much smaller (approx. 1E-58 m) than planck length), the thermodynamic temperature of the Universe approx. 2.76 Kelvin just from the fundamental constants and the concept of this work, without using the hubble constant. If you read the paper, it doesn't claim that. When you read the extended version of the Lorentz transformations that apply just for the quasiparticle case, there you will see the speed of light in regards to spacetime is constant everywhere, except in the vicinity of the quasiparticle frame where there the speed of light (propagation speed of the em waves) reduces as the velocity of the quasiparticle increases. Consequently, we speak about a local reduction of the speed of light that presupposes the presence of the theoretical particle being propelled by means of Euler Forces. The reduction of the speed of light occurs within the quasiparticle frame. I use the definition of the quasiparticle just to distinguish the equations of the wider framework from those of the bare particle (Einstein's special relativity). In other words, the theoretical quasiparticle I mention in my work must have such a structure that can be propelled by means of Euler Forces. Bare particles have no inner structure e.g. electron therefore they cannot be propelled using internal forces. Forger about the cosmological assumptions I mentioned previously. From the moment they are not referred in my work, I cannot support them. Here I have some statements that indicate the construction in Fig.1-Upper will acquire an acceleration: a) There is an accelerating internal mass transfer due to the Euler force (it is clear there is no real force here) b) Due to (a), we have no reaction upon the rest of the system c) Mass m_T is an intrinsic part of the system (mass m) d) Due to (c) the reactionless internal mass transfer of mass m_T will inevitably cause a change in COM along with its momentum e) Conservation of linear momentum always holds f) Due to (c), (d) and (e) the system will acquire an acceleration (having the same direction as the momentum of mass m_T) because its COM has been changed due to the acceleration of mass m_T Is there any objections on the above? If not then, the Lagrangian you provided for the COM in a previous thread should show a non-null result if the Euler force is taken into account. You said the Lagrangian takes everything internal into account because it essentially assumes that all internal forces are collinear. In the construction in Fig.1-Upper, we have an Euler force that has no counter part (no reaction). So, if one uses the right kind of Lagrangian that considers the Euler force then it should be possible to demonstrate/prove a change in momentum of the COM. I didn't. Who said that? If you are referring to those cosmological assumptions above, just ignore them. I cannot support something that I am not addressing in my work. I can tell you about the Special Relativity where I address in my work as also the Lorentz transformations. However, this will require first to have no objections about the 6 points above. I have to go to sleep: It is 01.30 in the morning. Good Night everybody! You speak about the external observer being away from the construction in another frame of reference. When I wrote about the Inertial Frame above, I was speaking about the skeleton of the construction that has the rotating frame (translation screw) enclosed.

I was wondering why you stated the above. I just would like to mention, all of its content and ideas/concepts come from me (I am the only Author of this paper). Do you know Ricardo Carezani? I read his work (there is a book on the web), however we agree only in two things. We conclude the same momentum and total energy (in my case about the quasiparticle) if I set nr = 1. Ricardo Carezani made some assumptions in Lorentz transformations that resulted on these new relativistic equations. Here is how he derives the new relativistic energy (eq.65 in my paper when one sets nr = 1): http://www.autodynamics.org/frames-derivation/ In the older version of my paper, I had his work as reference but on the new one, I removed it. The reason was besides he does not speak about things related to circumventing Newton's 3rd law or about action-reaction symmetry breaking (see References in my work), my work is currently under review by a Springer Journal and I wanted to keep the references as clean as possible (to be just relevant to the subject I present). As I said, the Lagrangian will work if one takes into account the Euler Force. I was strictly following mainstream classical mechanics. Nothing dismissed. This is only you. I would suggest you to first read the observations from (a) to (f) I shared above and please tell me if you have any objections. Correct.

This is a wrong assumption from your part. We do not speak about real forces causing a change in momentum of the COM but fictitious one that possess no reaction. The fictitious force (Euler force) manifests in the rotating frame where the latter is enclosed in the inertial one. I dismiss nothing and nobody. I made it quite clear how the construction may acquire an acceleration. Now if this has further implications (as I show it in the relativity and Lorentz transformations section in my work), it is another story and another discussion. The device acts according to the current laws of physics, I didn't add something new and the result is obviously an acceleration of the system (see the math above). I stressed the only overlooked fact that an Inertial Force that possess no reaction, can be utilized for the internal mass transfer in a system that consists of a rotating frame (being the cause of the internal mass transfer) inside an inertial one. It needs just the proper understanding and no new physics. There are the following simple observations that indicate the construction in Fig.1-Upper will acquire an acceleration: a) There is an accelerating internal mass transfer due to the Euler force (it is clear there is no real force here) b) Due to (a), we have no reaction upon the rest of the system c) Mass m_T is an intrinsic part of the system (mass m) d) Due to (c) the reactionless internal mass transfer of mass m_T will inevitably cause a change in COM along with its momentum e) Conservation of linear momentum always holds f) Due to (c), (d) and (e) the system will acquire an acceleration (having the same direction as the momentum of mass m_T) because its COM has been changed due to the acceleration of mass m_T Is there any objections on the above?

As I mentioned to swansont, the construction consists of a rotating frame inside an inertial one. Please check again Fig.1-Upper. The COM is relevant for the inertial frame (having enclosed the rotating one). The rotating frame creates the Euler Force (Inertial Force) due to \(\frac{\mathrm{d} \vec{\omega}}{\mathrm{d}t} \neq \vec{0} \) being the cause of the acceleration of mass m_T (without creating a reaction upon the rest of the system). Consequently, the acceleration of mass m_T (being also an intrinsic part of the inertial frame) will cause a change in momentum of COM of the system as a whole (inertial frame). Yes. If Fig.1-Upper may work then it will work for quasiparticles (not for bare particles, I clearly distinguish this in my paper) too. For bare particles holds Einstein's special relativity (the wider framework I present there, it automatically reduces to Einstein's special relativity that means there is no contradiction with Einstein's special relativity). In case of the theoretical quasiparticle e.g. an electron entrapped in the nodes of a standing wave, we have a system and not a bare particle anymore. This means the quasiparticle may move by means of an Euler Force in essence (applying a frequency shift or \(\frac{\mathrm{d} \vec{\omega}}{\mathrm{d}t} \neq \vec{0}\) on the standing wave (seeing it classically) results in a shift in nodes position that leads to the redeployment of the COM of the quasiparticle as a whole. https://en.wikipedia.org/wiki/Fictitious_force#:~:text=A fictitious force (also called,accelerating or rotating reference frame. https://en.wikipedia.org/wiki/Euler_force#:~:text=In classical mechanics%2C the Euler,the reference frame's axes. I stress the fact an Euler Force is actually an Inertial Force that means it doesn't possess reaction. That is all! You should google either "Fictitious force" or "Euler force". Here we have (see Fig.1-Upper) a rotating frame inside an inertial one. We like or not, there are fictitious force on the rotating frame as seen while the observer is sitting on the inertial frame (non-rotating frame of the system). Your assertion sounds like "linear actuators do not exist". The today's application of a linear actuator (positioning system, lifting weight etc) is by design restricted to work with constant angular velocity. Of course a merry-go-round COM cannot start moving as a whole because there is no accelerating mass transfer on its surface being additionally an intrinsic part of the system (merry-go-round). In other words, no mass transfer (along with its acceleration), results in no change in momentum of COM. Please check Fig.1-Upper. What you indirectly assume, it has also occured to me. If what I shared above proves to be true then, there is a high probability the space (or the Universe) to have also an angular velocity that increases as we go towards the center of the Universe. The fact that we haven't measure any such effect is probably the angular velocity is extremely small. A further consequence of it (hypothetically based on the findings of this work) leads to a reduction of the speed of light towards the center of the Universe (there it will be exactly null). Check eq.64 as applies to the quasiparticle. There the propagation of the em wave (speed of light) may drop up to zero (the quasiparticle becomes essentially undetectable). a) When @joigus applies the Euler force inside the Lagrangian of COM then, he will have a non-zero result. b) I cannot present a new model of space, instead I presented a mechanical device that utilizes two frames in one, a rotating frame inside an inertial one. c) "About going deep". Well, I am not a physicist. A physicist being good in math may start more abstract than me and may propose a new model for space. This is not my strength. Instead I present my arguments using classical mechanics. I don't see a problem with this. If the device may work then, we have to revise our understanding about the concept of space. Starting in reverse, it is more difficult and more controversial from my point of view. The inertial frame does nothing in the entire concept. It is just being affected by the enclosed rotating frame as being the cause of the redeployment of the center of mass of the inertial frame. You are confusing with what an observer sees while being on board of a rotating frame. See here: https://en.wikipedia.org/wiki/Fictitious_force#/media/File:Corioliskraftanimation.gif The inertial frame in Fig.1-Upper is the non-rotating skeleton of the construction. The rotating frame is just the translation screw. An external observer watching from an inertial frame of reference (away from the construction) sees a non-rotating skeleton with an enclosed rotating frame (translation screw). It implies the external observer sees mass m_T as being propelled by a "real" force (but fictitious in nature), however its cause is in essence the Euler Force (which is fictitious).

The construction in Fig.1-Upper consists of a rotating frame inside an inertial one. The fictitious force (in our case just the Euler Inertial Force) manifests just in the rotating frame (translation screw). If you read carefully the description of Fig.1-Upper, shows the following: The accelerating rotation of the translation screw induces an Euler Inertial Force (since the motion of m_T is restricted along the axis of rotation of the translation screw). This has as result (we are still in the rotating frame) the acceleration of mass m_T without triggering a reaction (since the Euler force is not a real force). The acceleration of mass m_T (is an intrinsic part of the inertial frame), leads to the redeployment of system's (inertial frame) center of mass that eventually affects the momentum of the inertial frame (having the rotating frame enclosed) as a whole due to the conservation of linear momentum (in our case due to the redeployment of the center of mass). As you see we do not even touch real forces in the above description. All the job is done by the rotating frame (reactionless accelerating internal mass transfer) being enclosed in the inertial one (the reactionless accelerating internal mass transfer leads to the redeployment of the center of mass of the system (inertial frame)).

Dear moderator @Phi for All, Thanks to the valuable feedback of the members of this forum (especially thanks to @swansont, @joigus and @Ghideon) I realized where I was wrong (use of wrong definitions and unclear arguments) and I am asking for a third and final round (revised work) regarding the same discussion about the possibility of triggering motion in an isolated system through internal forces (Euler Inertial Forces). Again, I would be thankful to show me where I could possibly be wrong. The revised paper (abstract, first two pages and References. Check the new paper in my profile) is now named "Circumventing Newton's third law through Euler Inertial Forces". My research through the years upon this subject led me to the following obvious but mostly overlooked conclusions: 1. Newton’s 3rd law always holds 2. Utilizing collinear forces, one can never build a reactionless drive (see (1)) 3. Are there real forces that possess no reactions? No. 4. Are there fictitious forces that possess no reactions? Certainly, yes! 5. There are three type of fictitious force: Coriolis, Centrifugal, and the Euler force. 6. For these fictitious forces to manifest in a system, either the entire system must be a rotating frame or a mixture of an inertial with a rotating one. 7. The next step is what we are going to do with the acknowledged reactionless inertial force. What is the working principle that will make a system move through fictitious internal forces? 8. We may answer (7) as follows: When a body or system accelerates because of an external force, the center of mass moves along with it. Could we influence just the center of mass of a system to trigger its acceleration? 9. The answer to (8) requires two things: a) a reactionless inertial force (Euler force), b) an internal mass transfer that will lead to the redeployment of the center of mass and acceleration of the system. FIG. 1: Proof of concept. Internal forces in isolated systems. Upper: Ideal machine. Creation of an artificially directed (translation screw) and enhanced (mechanical advantage) Euler inertial force. The accelerating rotation of the translation screw (threaded rod) frame induces an Euler inertial force (F_I) that causes the acceleration of mass m_T, leading to the redeployment of the center of mass (cm) and system’s acceleration as a whole. Lower: Collinear internal forces. The action exerts a force F_A upon mass m_T that slides over the non-rotating unthreaded rod. At the same moment, a reaction exerts a force F_R upon the rest of the system resulting in no system’s acceleration because of Newton’s third law. FIG.1 - Upper is a typical linear actuator device. It consists of a translation screw, a drive nut (mass m_T) attached on the translation screw, two linear guides, a front, and a rear housing that, besides holding the translation screw, they have a motor and a power supply enclosed (hidden). In terms of physics, the linear actuator is classified as a rotating frame inside an inertial one. In general, a rotating frame induces three different types of inertial forces, as seen from an external inertial observer. So, \[\sum \vec{F}_\mathrm{Inertial} = -\vec{F}_\mathrm{Coriolis} -\vec{F}_\mathrm{Centrifugal} -\vec{F}_\mathrm{Euler}.\] The construction in FIG.1 - Upper is driven by a couple \(\vec{F}_{\mathrm{A}} \text{ and } \vec{F}_{\mathrm{A}^{\prime}}\), where the motion of mass \(m_{\mathrm{T}}\) is restricted along the axis of rotation of the translation screw. Consequently, the resulting motion of mass \(m_{\mathrm{T}}\) can be attributed just to the Euler force. Thus, \[\vec{F}_{\mathrm{A}} = - \vec{F}_{\mathrm{A}^{\prime}}, \\ \vec{F}_{\mathrm{A}} \neq \text{const.} \text{ and } \vec{F}_{\mathrm{A}^{\prime}} \neq \text{const.} \Rightarrow \frac{\mathrm{d} \vec{\omega}}{\mathrm{d} t} \neq \vec{0}, \\ \sum \vec{\tau} = \vec{\tau}_\mathrm{A} + \vec{\tau}_\mathrm{A^{\prime}} = \left(\vec{r} \times \vec{F}_\mathrm{A} \right) + \left(-\vec{r} \times \vec{F}_\mathrm{A^{\prime}} \right), \\ \sum \vec{\tau} = 2 \vec{r} \times \vec{F}_\mathrm{A}, \\ \vec{r}: \text{translation screw radius}, \\ \vec{F}_\mathrm{Coriolis} = \vec{0} \text{ and } \vec{F}_\mathrm{Centrifugal} = \vec{0}, \\ \vec{F}_\mathrm{I} = \sum \vec{F}_\mathrm{Inertial} = -\vec{F}_\mathrm{Euler} = - m_{\mathrm{T}} \frac{\mathrm{d} \vec{\omega}}{\mathrm{d}t} \times \vec{r}. \] An alternative way to derive the inertial force that causes the acceleration of mass \(m_{\mathrm{T}}\) is through the conservation of angular momentum. Hence, \[\sum\vec{\tau}_\mathrm{ext} = \vec{0} \text{ and } \vec{F}_{\mathrm{A}} = - \vec{F}_{\mathrm{A}^{\prime}}, \\ \sum\vec{\tau}_\mathrm{int} = \left(\vec{\tau}_\mathrm{A} + \vec{\tau}_\mathrm{A^{\prime}}\right) + \vec{\tau}_\mathrm{Inertial} =\vec{0}, \\ \left(2 \vec{r} \times \vec{F}_\mathrm{A} \right) + \vec{\tau}_\mathrm{Inertial} =\vec{0}, \\ \overbrace{\left(2 \vec{r} \times \vec{F}_\mathrm{A}\right) }^{\curvearrowleft} + \overbrace{\vec{\tau}_\mathrm{Inertial}}^{\curvearrowright} = \vec{0}. \] At this point, developing a general expression for the inertial force requires the introduction of the dimensionless factor \(n_\mathrm{r}\) (ideal mechanical advantage) along with a definition of the inertial torque. Thus, \[\mathrm{d} \vec{\omega} = \vec{0} \Rightarrow n_\mathrm{T} = \frac{\vec{\omega} \times \vec{r}}{\vec{u}_\mathrm{T}} = \frac{2 \pi \vec{r}}{\vec{l}_\mathrm{T}}, \\ \mathrm{d} \vec{\omega} \neq \vec{0} \Rightarrow n_\mathrm{r} = \frac{\mathrm{d} \vec{\omega} \times \vec{r}}{\mathrm{d}\vec{u}_\mathrm{T}} = \frac{2\pi \mathrm{d}\vec{r}}{\mathrm{d}\vec{l}_\mathrm{T}}, \\ \overbrace{n_\mathrm{r}\left(2 \vec{r} \times \vec{F}_\mathrm{A}\right) }^{\curvearrowleft} + \overbrace{n_\mathrm{r} \vec{\tau}_\mathrm{Inertial}}^{\curvearrowright} = \vec{0}, \\ \vec{\tau}_\mathrm{I} = n_\mathrm{r} \vec{\tau}_\mathrm{Inertial}, \\ \overbrace{n_\mathrm{r}\left(2 \vec{r} \times \vec{F}_\mathrm{A}\right) }^{\curvearrowleft} + \overbrace{\vec{\tau}_\mathrm{I}}^{\curvearrowright} = \vec{0}. \] Dividing by the position-vector magnitude \(2|\vec{r}|\) yields \[\frac{n_\mathrm{r}\left(2 \vec{r} \times \vec{F}_\mathrm{A}\right)}{2|\vec{r}|} + \frac{\vec{\tau}_\mathrm{I}}{2|\vec{r}|} = \vec{0}, \\ n_\mathrm{r} \frac{\left(2 \vec{r} \times \vec{F}_\mathrm{A}\right)}{2|\vec{r}|} = n_\mathrm{r}\cdot m_{\mathrm{T}} \frac{\mathrm{d} \vec{u_{\mathrm{T}}}}{\mathrm{d}t}, \\ \vec{F}_\mathrm{I} = \frac{\vec{\tau}_\mathrm{I}}{2|\vec{r}|} \Rightarrow \vec{F}_\mathrm{I} = - n_\mathrm{r} \cdot m_{\mathrm{T}} \frac{\mathrm{d} \vec{u_{\mathrm{T}}}}{\mathrm{d}t} = -m_{\mathrm{T}} \frac{\mathrm{d} \vec{\omega}}{\mathrm{d}t} \times \vec{r}.\] The above equation reveals the inertial force does not possess reaction; therefore, it enables the acceleration of mass \(m_{\mathrm{T}}\) that leads to the redeployment of the center of mass and acceleration of the system as a whole. Thus, \[\frac{\mathrm{d} \vec{p}}{\mathrm{d}t} = \vec{F}_\mathrm{I} = - m_{\mathrm{T}} \cdot n_\mathrm{r} \frac{\mathrm{d} \vec{u_{\mathrm{T}}}}{\mathrm{d}t} = -m_{\mathrm{T}} \frac{\mathrm{d} \vec{\omega}}{\mathrm{d}t} \times \vec{r}, \label{eq33} \\ m\left(\vec{u}^{\prime} - \vec{u} \right) = - m_\mathrm{T} \left(\vec{u}^{\prime}_{\mathrm{cm}} - \vec{u}_{\mathrm{cm}}\right) , \\ \mathrm{d} \vec{u_{\mathrm{T}}} \neq 0 \Rightarrow \vec{u}_\mathrm{rel} = \vec{u}^{\prime}_{\mathrm{cm}} - \vec{u}_{\mathrm{cm}} \neq \vec{0}, \\ \vec{u}^{\prime} \neq \vec{u} \Rightarrow \vec{u}^{\prime}_{\mathrm{cm}} \neq \vec{u}_{\mathrm{cm}} \Rightarrow \vec{a} \neq \vec{0}, \\ \vec{u}_\mathrm{rel}: \text{system's center of mass relative velocity.} \]

IF you carefully read the 18 points I shared on that thread and keep your focus on that Fig and think clearly without being influenced about how you can solve regular classical mechanics problems then there is a high probability to start seeing the light.