CHAPTER 4 CONTINUED ACCRETION PART I
Applying these mathematical considerations to the 3P rotating case
The 3P particle with rotating and linear motion (at the
speed of light) from chapter 3 is the key. From this particle "accretion" can continue toward a Nucleon. If, with similar processes as described in chapters 2 and 3 a
fourth particle is added, making three in a row and one perpendicular
at the midpoints, this 4P case would then be balanced and all the
rotational motion would, under the right circumstances, become linear
motion, accelerating all four particles to a unison linear speed.
This speed however would now be > c. It
could maybe be accelerated and decelerated much as miniature nucleon.
For 3P-1P collisions the number of cases to consider is necessarily greater than those considered in 2P-1P collisions. They are:
DIRECT HITS The linear motion of a single particle (D) is directed at the linear motion of a 3P triplet (ABC).
The long arm of A is rotating away from its linear motion (toward back).
1 D hits off the long arm of A rotating toward back.
2 D hits off the short arm of A rotating toward front.
3 D hits off C.
4 D hits off B.
The long arm of A is rotating toward its linear motion (toward front).
5 D hits off long arm of A rotating toward front.
6 D hits off the short arm of A rotating toward back.
7 D hits off C.
8 D hits off B.
OVERTAKING HITS (D overtakes ABC)
The long arm of A is rotating toward back
9 D hits off the long arm of A rotating toward back.
10 D hits off the short arm of A rotating toward the front.
11 D hits off C.
12 D hits off B.
The long arm of A is rotating toward the front.
13 D hits off the long arm of A rotating toward the front.
14 D hits off the short arm of A rotating toward the back.
15 D hits off C.
16 D hits off B.
Case by Case from Chart 4.1
1. A) When D hits A, all of D's linear and A's linear are stopped. Therefore BC linear is also stopped , its motion going rotation ABC. If D's motion is fast enough it torques and follows and rotates ABC, elsewise it fly's past as ABC rotate.
B) If B ends up rotating and reaches the direct hit side, all motion BC goes into A and all A is stopped and rotates on D, XXXas A re-hits BC then as in Case #2B.
C) Else as in Case #2C
Excepting in B and C above the end result is BC motionless in space and A moving off fast, and D reverting to its original speed when torque is released, or if rotating when A moves away it continues to rotate around its virtual point, stationary in space.
2. A) All motion of D stops and D torques. All motion of A stops and goes into torque. All motion of BC is stopped and goes into A, speeding A's torque. So BC are left motionless, and AD are torquing to each other. Depending on the angle of D on A;
B) If A will in torqueing re-hits BC, it try's to rotate off of C orbiting B, but can't as same scenario will repeat. Here we resort to principle 15.1.7 & 18.6 and postulate that this forces A to accelerate BC linear. So ABC go off with equal linear speeds (without rotation) and D is freed up for a unison linear motion and torque, then rotatation on A.
a) From here if it reaches midpoint on the direct hit side it all excess motion (over the unison linear motion of following) goes into A, and A rotates on D. BC goes off linear. A will either 1) rotate all around D and re-hit BC, causing both torque and rotation on two different bodies so it rebounds away. D, and BC are left with linear speeds and A goes off with its previous linear and rotational now combined linear. Or 2) the backswing will re-hit BC causing A to now torque and rotate on BC, all the way around BC and most likely re-hit D again, whence it cannot torque on D and rotate and BC (torque or rotate) simultaneously so it rebounds and everything as in ending for 2Ba1.
C) Else A in torqueing will not immediately re-hit BC AD torque and rotate each other, wherein A should then in backswing or rotation all around D re-hit BC and as above reasoning under 2B.
3. A) D hits C, both lose their linear motion, and C loses its orbital motion. However when C torques it re-hits A and B and accelerating them both but their vectors compound, so B runs into A accelerating it, with all establishing a equilibrium of motion, moving away from CD. So A-B go off with a speed > c.
B) As AB move off B hits D. A continues off with its linear motion. B and D torque. D is torquing to B and C at the same time, so it rebounds as per principle 18. This rebounding results in a linear motion traveling perpendicular to the two points of contact DC and DB. When B torque is released it reverts to linear motion (principle 18.3) and follows (its speed being greater in most cases, but perhaps not always) D re-hits it, treat like overtaking hit chapter 2. C is left stationary in space.
4. A) When D hits B. B and D's linear motion stops. D and B torque. When B torques it re-hits C, accelerating C linear, which vectorizes with C's current linear motion. A goes off at c. C goes off > c. Because of this resultant in C it does not re-hit D.
B) So C goes off linear at a certain new angle and B goes linear at c perpendicular to its previous motion. D fly's bye if its speed is too slow. Else it torques and follows B and they end up as a doublet.
5. A) All of D and A's motions are stopped and they torque to each other. BC go off at c with their orbital rotation continuing around the virtual contact point ABC.
B) AD proceed as an direct hit chapter 2, with finial speed > c because of A's rotational velocity added to things.
6. A) D hits A and all A's and D's linear motion is stopped. BC move off at c with their orbital rotation continuing around the virtual contact point ABC.
B) A continues to rotate around its virtual contact point ABC, while also torquing. So it rotates thus as D torques and follows (per principle 18.2), D with a "virtual" rotation, which is really a continuously changing linear motion. Thereafter D finishes its torque and proceed to rotate. When D reaches its midpoint A's virtual rotation get changed to "real" re-rotation on D as it is accelerated by D. This happens rather D accelerated from the front or back relative to A's rotation. It simply switches its virtual rotation to off of D because it’s a new and real rotation point. So from the backside it actually flips rotation to a reverse direction. These are added to the rotation from the torque motion forming a resultant rotational speed.
Principle 18.4 A rod in virtual rotation, when given a real rotation point will always re-form a rotation off that point, regardless where that point is on the rod.
Thereafter a doublet is formed and it moves off with whatever the total motion is.
7. A) D hits C. All of D and C's linear is stopped and goes to torque. C re-hits B accelerating B by principle 9. That motion combines with its original linear motion for a resultant that moves B in a trajectory were it should not hit D. Also all of A's linear goes to rotation off C, orbiting B to any extent it is still in contact.
1) if still in contact B then slides along A as it curves away with this combined linear and orbital motion. There is some chance it now will re-hit D NOT DONE.
2) if not in contact for further orbital motion it still travels off with a combined motion.
B) Either way B is left with a unique velocity of combined linear and rotational, contained within certain parameters.
Likewise C also with what ever speed it did not impart B, and picking up rotational from A also. In fact they should travel off as a duet.
A is left rotating slowly in space, and D is left torqueing and following C!
C) This causes C's orbital motion to stop and stop D's linear follow. C's orbital goes to linear, combing with the linear it has, and C travels off, perhaps accelerating B slightly, but then parting ways with it. D re-follows C and they end up as a doublet with odd speed.
D) This leaves B as a possible light particle also.
8. A) D hits B. All of A's rotation and linear motion are stopped going to torque. C and B's rotation drain into A and are stopped likewise. B's linear is stopped likewise. All A's motion goes into B's torque. When B torques it re-hits C, B and C go off linear in unison by principle 9.
B) D if too slow passes by, otherwise it torques and follows BC. A is motionless in space.
C) After torque D rotates on BC (while having a unison motion component). when it reaches its midpoint it may so it while rounding a P, or when falling across each "flat".
1)If on rounding it would give its motion to the particle rotated on, which when re-rotating on D re-hots its neighboring P and they go off with a new linear which combines with the unison motion already present. D continues with its formally unison motion it had had.
2) If on the flat the midpoint will not as it where fall on a rod but between them. Not sure, but will guess that per principle 17.5 rotation will occur of one of B or C and the other will attempt to be orbited, but instead will cause that P to re-rotate on D, draining all of D's excess motion. When, say, B is at midpoint D will need to torque and rotate of both the other P's, so it rebounds.
9. A) Treat as overtaking hit chapter 2. All rotation ABC is stopped. A and D torque. If in torquing A re-hits BC it is torquing both them and D so rebounds with speed of its linear motion plus speed of all the rotational motion that was in ABC. BC continue on "together" at their linear speed of c.
B) D follows behind A most likely. It depends on its original speed, whether it will overtake A or not, if it does it will forms a doublet by same process as overtaking hits chapter 2.
10. A) D hits on A, treat as overtaking hit chapter 2. That is:
1. D's motion swings to unison with A per principles 15.4 - 15.6.
2. D torques and follows also per principles 15.2 and 18.2.
3. A) D rotates per principle 13 and 15.6. Because It loses its above mechanism for following the rotation of A. D instead will have a virtual rotation as A pulls away, but this virtual rotation should be enough to still cause D to hit A, but off center, so D again would torque, and also form a resultant with the unison motion, breaking that "pure" unison motion to one at an angle again to A's unison motion, but keeping the point of contact AD the same. This all happens in the first instance, and repeats for every other "instance". Therefore D rotates A while following its rotation around by this new mechanism. But D has to reach its midpoint before the length of A is in line with its linear motion, else D will impend it.
1) If it does so reach its midpoint before then, and before the situation in B below, then D will impart speed to A causing it to rotation on D, but then immediately re-rotates BC again, accelerating that rotation to the point it loses contact pressure from D. Thereafter A would slide beyond and/or separate from A as BCA rotates. However A may also rotate around and re-hit D with its long arm, thereafter it would rotate on D, orbiting BC as in C) below until DABC are together in alignment.
2) if D impends ABC as in 3A) last sentence, then A and D torque and rotate each other. BC fly off, but should re-hit D, and all becomes a rather chaotic situation, hard to follow, but the end results if probably break-up of all of most of the particle group.
B) Once rotation begins and follows as outlined in 3A here, if, and likely, D rotates to direct hit side it is actually impeding the rotation of A. When this happens D is at top of A. All unison motion is diminished to zero. A rotates on D. All rotation ABC goes into A's rotation on D, as it BC has indirectly been impeded by D also though A. Depending on position D and A may reach mid-points and go off as a doublet with speed > c. Or more likely A rotates all around D, each torqueing and rotating on each others, and then either A o D re-hits BC and rebounds as per principle 18, leaving the other P in space with virtual rotation and it should then probably hit BC And torque to it and form a triplet. Else it just rotates in space and BC is left completely motionless.
C) But if by chance D is on the overtaking side relative to A's rotation when it reached midpoint, it accelerates A to re-rotate back around D. But it can’t as that motion runs into BC. This causes again rotation off of C, but with more motion, moving the end of the arm A away from D so that D's release linear has a amount that re-hits A above midpoint, and off balance some, causing the process of torque and follow, and rotation downward until the Particle D is parallel with BC. Then CBAD synchronizes, even though the feedback off B to C to A if necessary, until DCB are aligned. That is DABC can re-rotate off of D without rebound because rotation accelerates B at midpoint which rotates back though C and equalizes the system, until all three particles BCD are in line and touching, but unbalanced all on one side from the midpoint to midpoint contact of B to A, for the ensemble ABCD which moves off at c (original velocity of 3P) and with a rotational velocity also.
11. D hits C. All rotation ABC is stopped. Here things get kind of stuffed up, like in my cabin. All rotation goes into C (easiest outlet) and C and D torque, but C re-torques to B and A, then accelerates both A and B by principle 9 because it has a midpoint alignment to both, though with B a whole line of contact. Also, as A and B are perpendicular to C, this causes a dual motion (compounded motion) to occur. However, when C pushes A it causes A to rotate on C back to B and then to C and as C can't rotate so the whole process “starts over”.
So by principle 5, the only outlet for all rotational motion to go is into accelerating B. Here B has that linear acceleration and its original linear motion, which come together as a compounded resultant. From there B accelerates A, by principle 9, such that the result is A moving off >c, and B sliding down it and away with a diminished compounded motion. C is left moving at C and D overtaking it from the get go as per overtaking hit chapter 2, resulting in a doublet forming.
12. D hits B. All rotation ABC is stopped. Then thing happen almost as in as in #11 but in reverse. Here all rotational motion ends up in B. B torques to D but re-torques to C and A both, so rebounds (first toward D actually, then back between AC). Here the motion ends up again split in this case between A and C. Both A and C get accelerated immediately until their speeds are such that B traveling in a diagonal without pressing C and A. Note: first draw the vector of B rebound together with its original linear motion, then split it between C and A, but any acceleration of C must come from the compounded motion component that is in that direction. Likewise in #11 above).
This will leave A going off > c and BC sliding past it in unison motion > c. D, depending on its velocity may or may not keep up.
If it does it would do the torque and follow thing after B, then rotates on B. BC would eventually slide all off A and when D reaches its midpoint, if then D at midpoint press on C or B they would rotate on D but immediately rerotate on B or C accelerating BC off with compounded motion, usually away from D, but in some cases back into D causing the process to repeat until the compounded motion becomes BC sliding off of D.
13. D hits A. D establishes unison motion sliding across A. Excess over that goes to torque on A and following A’s rotation. Now a lot may depend on where in A’s rotation the first impact is, and where in that rotation each subsequent event occurs. But I will follow this scenario. When D begins to rotate, its testing and unison motion merge to the plane of that testing motion. From there D may continue to torque, follow and rotate due to the factors mentioned in previous cases. Each rotation point changes the unison motion if it is getting closer to the linear motion of A, but here there is no merging to that point because D is not pointed in that plane. However, as A rotates it will pass a point where it is perpendicular to its linear motion. With D following through this point, the vectorization of D’s unison motion will be in a plane equal to A’s linear motion and be locked in after that in D on that plane but not line of A’s motion.
Consider this, if in the unison motion of D, one first has the Unison motion at impact, that is still in the linear direction of D’s motion before impact. Then after torque and follow, that unison linear motion can vecotize with the testing motion that occurs before rotation. But it occurs to me these two vectors may resolve across an arc that intersects the linear motion of ABC. This would depend in large upon the angle in which D impacts A, and the rotation of D on A so the "testing" for each point of rotation is in the direction on the other side of the plane of ABC linear motion from the original overtaking motion vector of D. But in the case where it does, the linear motion of ABC should act as a limit to the resolving of the two vectors as per previous ideas and principles. In that case once the resultant is equal to the linear motion ABC that gets locked into D, allowing it to rotate freely on A by the torque and follow from the consistently open window of contact presented by A’s rotation, as previously expounded on in other cases. But only up until the point that D rotates to the front of A’s rotation. Then all rotation ABC is stopped, A tries to re-rotate on D, but immediately re-rotates on C again.
But there is a continuity here, in that (explained below) here A can rotate off D pushing both BC forward and then immediately upward around D. See D’s first rotation block of A occurs, of course, just after it has rotated to the “top” of A. If it were to block A on the top A would be able to rotate up past BC and around D. But it is after the top, though no point can be ascribed to it, it still is a continuity such that the point of rotation up and also the point re-hitting BC are such that they are adjacent in style, though not in measure, and the rotation of ABC can turn up and around D. 3-1-18 bit more complicated still So because his is in "instant type pattern" A can push "under" BC causing BC to rotate but in a direction away from D . All torques and rotates with following, both D and A and BC. D may reach its midpoint sometime and then BC continue to rotates around A as A rotates (scoots) around D as itself is rotated by A! Until at some point BC are in parallel to D then rotation occurs locked into that plane and brings BC into contact with D so that they all comes together. Once that is all so the sides around the midpoint contact A-B are balanced and all motion reverts to linear.
So ABC re-rotate off D both by principle 17.5, and because of the feedback from B to C to A. When A rotates off of C it goes through B to A and back to C. This locks them together as it were. Even when A rotates off of D this mechanism holds then together even while being rotated off D, a continuous interplay allowing the rotation of D to override the rotation off of C.
SO THERE ARE OTHER SENERAIOS here that have not been done, and I am a little confused why they seem to be more possibilities based on position here but not in the other cases, but cannot do now.
14. D hits on A, all rotation ABC is stopped, D forms unison motion the rest goes to torque on A.
A torques away from BC. A and D torque each other, and
then rotate each other, D or A re-hits BC. Rather on overtaking or direct hit side and rather A or D that particle cannot torque both to BC and the other particle so it rebounds. So without going into more details, the system breaks up. As in BC going off, perhaps with some spin left as when A torques it may give free space for a little spin to remain, and a doublet, though most likely BC and two individual particle, one shooting off, the other rotating and linear, which may in fact re-hit BC also. In which case it should torque, rotate and follow (any spin) of BC until it reaches its midpoint. Then any spin (rotation) of BC should be stopped and D (or A if such) then torques to BC and rotates. When D reaches its midpoint it accelerates BC to rotates on it, rather it is direct or overtaking hit doesn't matter because B or C immediately re-rotate into its partner particle accelerating both again and D moves off separately with its leftover unison motion. Though there maybe a nuisance here regarding figuring on the compounded motion, see Appendix A, Section 4, but I think the results should be the same.
15. A hits on C, all rotation ABC is stopped. C and D torque, with all rotational velocity ABC going into C first. As in case #11 C in torqueing re-hits A and B accelerating them each linearly. C and B form a new compounded motion, it being the resultant of B and C's original linear motion together with equal parts of its linear motion derived from the rotational motion. So they form a pair moving off together with a new compounded linear motion. Any that went to A causes A to rotate on C through B back to C as case #11, But being stopped again by D, the easiest outlet is all the motion to go into accelerating C and B more. So A is left moving at c, as BC move off > c. D follows BC if it has the velocity to do so.
D then torques to BC and rotates. When D reaches midpoint it accelerates BC to rotates on it, rather it is direct or overtaking hit doesn't matter because B or C immediately re-rotate into its partner particle accelerating both again and D moves off separately with its leftover unison motion. Though there maybe a nuisance here regarding figuring on the compounded motion, see Appendix A, Section 4, but I think the results should be the same.
16. D hits on B. D forms a unison motion and torques to B as it follows it,. All rotation ABC continues. It is somewhat sililar to case #12, but here D should never achieve a perfect unison motion with ABC.
ABC will slide along or past D. If D starts to rotate and reached its midpoint because ABC slide across it, D accelerates B to rotate it, but it immediately re-hits C and A accelerating them as compounded (with unions motions) or combined motions in different trajectories. Note A will not rotates as in case 15 as its contact with B is at midpoint.
If D is rotating toward front, its unison diminishes to zero, so ABC would slide to its midpoint as above.
Else if rotating toward the back and it reaches its midpoint still same as above.
Much to work with here as far as a system goes, only the case #10 & 13 accretes to a larger particle which I will follow up on in Chapter 4.5