NEW ROTATIONAL DYNAMICS

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1 NEW OTATIONAL DYNAMICS haate of statis Inetia toque piniple and the foe oent the GuagSan Yu ( Habin Mao Dynais Institute , P.. China ) E-ail:sxzyu5@hotail.o ( ) Abstat: Textual point of view, geneate in a seies of otational dynais expeient. Initial eseah, is wish find a ethod oveoe the oentu onsevation. But futhe study, again deteted inside the lassial ehanis, the eo of the piniple of foe oent. Then a seies of, oentous the eo of inside lassial theoy, disove oe out. The oentu onsevation law is wong; the newton thid law is wong; the enegy onsevation law is also an supass. Afte edess these eo, the new theoy naely pefoe bing. This will involve the lassial physis and ehanis the foundation fation, textbooks of physis foundation pat should poeed the gand odifiation. Key Wods: igid body; Inetia toque; Centoid oent; Centoid a; Statis; Stati foe; Dynais; Consevation law PACS:45.0.Dd, àf, àj, Dd 0 Intodution Textual aguentation is to bases on the seveal siple physis expeient, these expeients pass two videos the douent to poeed to deonstate. These two videos is:.expeient testify oentu is not onsevation;.the expeient of physis of ehanis of the Inetia toque. Also still have elevant the atile in go along to explain and disusses [,]. Figue expeient testify oentu is not onsevation Figue The expeient of physis of ehanis of the Inetia toque Figue and Figue is these two videos the pitues espetively. Figue the expeient the show, is do onening oentu is not onsevation one ealiest expeient. On this foundation the

2 passage deepen the eseah, just so it beoe this textual a standpoint, and suession opleted suh as the Figue the expeient. Textual point of view piaily be fo figue and figue the expeient geneates. Theefoe textual aguentation is haved the expeient the evidene and sustaining, don't is siple easoning o hypothesis o onjetues. The notion the Inetia toque Inetia toque is objet inetial ass and the a of foe the podut. Figue onening inetia toque Why an save labou it the leve? The oent of foe ey do to foe the hange? Do it nope to the objet buthen the ass ouene hange? The Inetia toque, naely, suh as figue the show, the patile (the is also its ass) is a to O the slewing adius, so it the Inetia toque is: I (.0.) The Inetia toque is siila with oent of foe, to the sae of oigin, it is a onstant. So by the oigin the othe deivation veto, fo exaple in gaph, naely ust poeed the hange. I If q well then q (.0.) Theefoe hee, naely, inish ultiple of q. It is an invese popotion to the veto quantity hange ultiple. So, Inetia toque is that podut to that objet ass and foe a. It of tue eaning, in fat be the plae end in foe a, that to two side tangent line the dietion, by the inetial ass of the objet. Theefoe, an alulated fo the Inetia toque, at dietion of tangent line the ass of objet. I Suh as the oespondene in Inetia toque:. I Its dietion of tangent line the ass, naely: (.0.)

3 But the oespondene in the of the foula (.0.), the dietion of tangent line the ass: I q (.0.4) Theefoe the foula (.0.) and the foula (.0.4) is not the siila. The ux is disiinative at both, the of ass of the foula (.0.), is the ass of the atual patile in objet; But the of ass of the foula (.0.4), it is on the diffeent foe a, to the the ass the invese popotion apping. It is not atual patile. Theefoe in an objet, abitaily the Inetia toque of the foe a of the veto quantity the tangential ass, ae ay be tue ass, o is not tue ass the apping ass. M { I } { } M ρ I ( ρ ) f : M M f ( ) ρ (.0.5) Inside the foula the is tue ass of the patile of the Inetia toque, the is the apping ass of patile of diffeent foe a of the apping of the Inetia toque. The apping ass and tue ass, the quality is diffeent. Tue ass ebodient the objet tue exist, so it is in objet, having the atual influene to the objet the inetia. But apping ass, it oneself is not ey the esse. It eely be, when the objet is ation by the oent of foe, pesent in a kind of inetial buthen. Theefoe in the ation of foe it deteined, fo this a oent is to the foe will be how big holdout in the inetia. Beause the ass the apping on the oent of foe, the atual is while thee is extenal foe the ation, the pefoane is an inetial buthen, naely. So it is Inetia toque a ingedient of the ass of buthen. The ass of buthen of the Inetia toque, is to abitaily the a of foe of the objet, it suation to the tue ass and the ass of apping. The Inetia toque is to the inetial ak of the objet otation.. The expeient of Inetia toque Figue is a dynais expeient the video photo [], this expeient to beoe suppoted in theoy of the Inetia toque. In this iage, oute ing is a big pitue, idst is a s pitue. Two pitue shoot use sae equipent, the ovelay in togethe is fo utuy opae, with display the expeient the esult. Two the expeient is sae devie to use. Naely a stik fo the evolution a is an the agility otation, has the etain ass; and a sping it an eleased in oentay by eation thust. While expeienting eleased suddenly the sping, geneate an ipulse foe, push the evolution a to apid iuvolved. Two expeient, by exat adjust, do the sping push two evolution a, the distanes is sae. Theeby, the sping thusts to two evolution a, also is sae. Its diffeent, is two evolution a it one is in / a the push, and one is in a the push. The esult of the expeient, fo the iage an fisthand aquisition. Two evolutions a evolved with sae angula aeleation and the angula veloity. This is vesus Inetia toque theoy fisthand and dependable sustain.

4 θ F a β t θ F a β t (..) (..) Inside the foula, the θ is ental angle tuned by evolutions a, the β is an angula aeleation. Theefoe two expeient, the ipulse foe of the sping is the sae. But its buthen ass diffes is doubled, the invese atio of lineaity aeleation diffes is doubled, only the angula aeleation is sae. State explain the expeient and Inetia toque foula the (.0.) and (.0.) et is paelis, testifyed the Inetia toque theoy is exatitude. But oe ipotant, was this expeient to negated, in lassial theoy the " oent of inetia" and " law of otation of igid body", et. The otational dynais of the physis, need do the ipotane odifiation.. The total Inetia toque of the otational igid body The igid body of the fixed axis otation, have affiatoy Inetia toque. Naely, the foula (.0.) and foula (.0.), et. But odinay igid body, usuy be onstituted by a lot of patiles. Cetainly aong the eah a patile, egadless it is how any agnitude vetos, also that is etain have a the Inetia toque. Suh as foula (.0.) et. But at igid body the Inetia toque of eah a patile, it is a onstant. Be naely When the veto agnitude hange. the ass of its apping, will is the invese atio the hange. So its Inetiatoque is not beause the veto the hange. Theefoe, beause this eason, do that patile in inside in a igid body the Inetia toque, to dietly plus, an get the total Inetia toque of that igid body. Hee whethe ae dietly the Inetia toque of the patile, o is the Inetia toque of the apping ass, will ath its total Inetia toque in any veto. So when a igid body of fixed axis otation, be onstituted by soe patiles. That it eah patile, have a ass the i, and the foe a i of to the shaft. So the total Inetia toque of this igid body is: I n n i i (..). The entoid oent of the slewing igid body The total Inetia toque of the slewing igid body is: I i i (..) But the entoid oent of the slewing igid body, is say the igid body total ass M and foe a the podut, is equal to the total Inetia toque I. I M i i Naely: (..) The foula I epesents the entoid oent naely. We have obviously: 4

5 ( i i ) M (..) Naely total Inetia toque I by total ass M divide, inoe foe a, is unique foe a to that igid body entoid oent I a oespondene. Beause the is exlusive, theefoe will it entoid a its definition. By the entoid a a end, do the ile o uves the line, it is epesents the otation ente of ass line of the igid body. The total Inetia toque of the igid body, is a onstant. Theeby but entoid a, any othe foe a i. I Mt t i i It's : (..4) ( ) i i M t t (..5) In foula the t is a foe a but, M t is elatively in total Inetia toque I and foe a t, the total buthen ass of equivalent of the foe. The total buthen ass is an efeene, oesponding igid body one foe a, the ass of its the patiles of the iufeene; and igid body patiles of othe pat, apping to the suation of ass of this foe a. M + t i i (..6) Inside the foula i it's on iufeene that tue patile the ass, the is othe egion of igid body the apping ass of patiles. Fo foula (..5) then, on the slewing igid body, if the foe a diffe, the total buthen ass of the end of foe a, fo the invese popotion hange of the foe a hange. Then foe a if hange Q ultiple, the ass of total buthen to hange / Q ultiple. Foe a if inease, the ass of total buthen is inish. Vie vesa. Theefoe in igid body diffeent foe a end, the ass of total buthen is diffeent. Theefoe, be the t at the >, o < it of both sides hange. Fo exaple at t > and tends to infinity, the M t tends go zeo. Wheeas when the t < and tends to infinitesial, the M t tends go infinity. Bak a kind, is the foe a tends to 0, the equal foe to though the shaft, theefoe egadless the foe is how big, the shaft also an't sew. The foula (..5) enuniation, in abitaily slewing igid body, take the aleatoi foe a, the oespondene in the end of its foe a, ae one onfi the ass of otation buthen. Fo exaple the foula (..6) is the ass of total buthen naely. In otation of igid body, it's the ass of buthen of otation and by the ass of eality of objet be equivalent. it's via easue of ehanis to easued..4 The piniple of Inetia toque Abitaily the igid body of otation of fix shaft, is one deided paaete I fo the Inetiatoque. I n n i i (.4.) 5

6 Theefoe, it's in igid body that patile the ass and foe a to the podut the suation. It is a onstant to this igid body. Fo it to divide abitaily the of the foe a, naely gained the igid body of in foe s a the tangent dietion buthen ass I i M (.4.) In foula, the M been by foe a, that igid body tangent the dietion buthen ass. Fo the level of the dynais, the M in the foe a tangent dietion, ath in the Newton seond law. Naely: θ d θ F a M M M β t dt (.4.) The Inetia toque in otational dynais is an objet by onening easue of the inetiy. It been ey, and fo attibute of dynais of the objet, by definite onstaint effet paaete. The attibute of statis of the foe oent The oent of foe is the onept of a kind of statis, it in the dynais use will ause istake. Fo exaple soe siple ahiney, and the ahiney sale that weigh fo exaple. Suh as the figue 4 show, is the piniple of a ahiney sale o leve. In figue on the fulu O of the leve L, its in both sides of leve L to the length and the weight of objet, ake it etain the equipoise. The piniple of the oent of foe, be appliable to the Stati foe in the statis only. When the foe in its point of ation, engende pessue, but it in the efeene fae is Stillness, it is d l Fq i ( a) i ( ) Stati foe. Theefoe, naely: dt (.0.) Beause, Stati foe the F q an podue the pessue, but have no the displaeent, so it is a foe of iaginay nube [7,8]. Figue 4 Moent of foe in the statis But, if the otion of the foe, ath the ondition of the statis equilibiu [9], naely it is the otion but has no aeleation. Then it still is the Stati foe. Naely: 6

7 d l Fq S ( i a) S ( i ) S dt (.0.) So, the Stati foe in by the podut of its otion distane, equal the Stati foe to does a wok. This by in dynais the foe to does wok, is alike. But its the otion has no aeleation. The Statifoe does the wok in ehanis the is not few, fo exaple, be the foe push the objet, oveoing the fition foe to otion, is a Stati foe to ake the wok. The ation of the Stati foe, by is ath the oent of foe piniple. Fo instane, in equal to zeo of veto suation of oplete extenal foe oent, objet is in state of the equilibiu, it also in state of the stillness, theeby hee the foe is Stati foe. So: τ τ + τ F F 0 (.0.) τ But if + τ F + F 0 τ That F τ F (.0.4) State the foe oent τ and τ dietion is evese. But when foe oent dietion evese, the foe(foe oent) esists utuy, theeupon its ation is that quiesene, naely enuniation is a Stati foe status ation. Theefoe states, the piniple of foe oent in fat is a kind statis piniple. It is appliable to the Stati foe only. Theeupon pass the hange of the foe a, an vay the big o s of the foe, in fat only have the Stati foe. Stati foe that onfo the statis, have in the physis ehanis a good any. As long as is in quiesene o equilibius, an geneate to pessue the foe, is the Stati foe. Fo instane gavity, fition and eletoagnetis the foe, and the wate pessue et, these foes ae the Stati foe. All an is via vay the foe a, by the bulk of the hange foe. But it is in stillness, o in equilibiu state (naely unifo otion status) to the ealization. So, foe oent piniple at, evese dietion foe oent to utual withstand, o in foe oent by equilibiu state un fo Stati foe, that just be appliable. The inetia of the objet, naely inetial ass, in dynais, is tantaount to a kind spae estition foe. It obviously too is a kind Stati foe. Theefoe inetia and ass of objet, onfo in the piniple of foe oent. Hee its atual pefoane, is an Inetia toque. Naely inetia and ass, hange beause of the hanges of the foe a. When the ation of the foe, ake the oveent status ouene hange of the objet. In dynais ation, does the piniple of foe oent be appliable? Answebak is, in the statis be appliable, not be neessaily in dynais appliable. Beause the ondition of the ehanis, aleady ou the essenes and vey big the hange. Theeupon, igid body in the lassial ehanis the otation law, the fato is wong. The haateistis of dynais of Inetia toque piniple Beause the piniple of Inetia toque, any otation objet had the new dynais the 7

8 haateistis.. The linea oentu of the otation igid body and evey kind of angula oentu The total Inetia toque of the slewing igid body is this: I n n i i If the angula veloity of the slewing igid body is a, then its otation linea oenta is that: (..) dθ d θ Pl ( M ) ( M ) ( ii ) ( i i ) dt dt (..) We awaeness, the M is equal to the total Inetia toque a I, is a onstant. Then fo a slewing igid body, at its the angula veloity the is asetains, then its the otation linea oenta is a P l, it is also a etainty quantity. Naely suh as above foula (..). On a slewing igid body, its Inetia toque is a onstant. Theeupon, take egadless how big of foe a an, it vesus otation the ass a M the buthen, by invese atio hange. The fo this eason Inetia toque hold is onstant. So at angula veloity a is etain, its otation linea oenta is also a hold hangeless. egadless is in this slewing igid body that, the ando foe a (naely ando adius ), its otation linea oenta is sae. Pl ( M ) (ρ ) Naely: ρ (..) The angula oentu of the as to otation igid body, it took plae a kind of stange hange. Conening the definition of the angula oentu, it is the objet enile the iula otion the line oentu, with its otation adius by the podut. But beause Inetia toque piniple, a otation igid body abitaily the otation line oentu( inlude the ass of the tue patile and the ass of the othe adius patile of the apping) of the adius, is sae. When the angula veloity is etain, it is a onstant. Theefoe, inside of foe lassial ehanis, total angula oentu of igid body, is in igid body that patile the angula oentu the suation. Fo exaple the expession below: i n L l + l + + l l n i i (..4) i n n n i i i L (..5) It is to don't be eaningful. Beause, on diffe the adius that patile the angula oentu at this tie, the inapability is with auay to onvet utuy. Fo exaple the angula oentu of the patile on the adius, if onvet to the adius : Suppose α (..6) ( ) α α So α α (..7) 8

9 The value of the angula oentu beause the ultiple wee ultiplied the squae, theefoe aleady taken plae the hange. It with atual angula oentu in objet, aleady diffeent. Theefoe be the otation igid body inside abitaily the angula oentu of the patile, if onvet to the diffeent adius, the angula oentu of its a apping has taken plae the hange. So as foula(..5) the angula oentu the su, been nonsense in fat. But the angula oentu of the tue patile in igid body, also is not signifiane without opletely. Beause when by objet o patile on the otation igid body, the ouene adial ove. Oiginal tue angula oentu in that objet o patile, be as to it's at tavel heeafte, the total angula oentu in igid body hanges, having got the deisive ation. So to the total tue angula oentu of the igid body, an eod fo: { } {,,, } ( ),( ),,( ) L l l l n n n (..8) It is an finite olletion. But as the foula(..5), to patile the angula oentu the su, been nonsense. With this opposite, anothe a kind show ethod should be: ( ) L x n n (..9) It is igid body patile the angula oentu veto quantity the odule total su x, and angula oentu the unit the. Beause a otation igid body fo having patiula evolving speed, its abitaily the otation line oentu of the foe a is sae. Theefoe it had the onept of anothe angula oentu. Naely its otation line oentu fo having in patiula, with the podut of its vaious adius. Loplex Pl ( M ) (ρ ) ρ (..0) Inside the foula the is an hange, but the otation line oentu P l is unhanged. Theefoe its oespondene in diffeent adius, thee will be a seies of angula oentu L oplex. Suh angula oentu L oplex, it as the oposition angula oentu of the igid body. Eah a igid body oposition angula oentu L oplex, be onstituted by a seies of diffeent value. And with adius diet popotion.. New igid body otation law and foes at diffe a of foe does the wok When extenal foe is ated in slewing igid body abitaily foe a, so it would angula aeleation hange how? In lassial ehanis, this iustane was been the otation law of the igid body by the foulation [4], naely: du F I dt β d dt (..) The I in the foula is to the point the oent of inetia, diffeent fo textual the Inetia toque. Aoding to the otation law of the igid body, the angula aeleation of the igid body, with its esultant extenal foe oent is diet popotion. Is shown as foula (..) naely. But this is wong. Aoding fo textual to the Inetia toque piniple, in asetained to total Inetia toque of a igid body, oespondene in its diffeent foe a, its the ass of buthen the M is with the invese atio 9

10 hange. Suh as the foula (.4.). I i M Aoding the iula otion of the patile, the line quantity and the angle quantity the elation of onvesion [4]. d θ U dt d a β And dt (..) The foe akes patile eation line aeleation and angula aeleation is it: du F a dt d F a β And dt (..) Theefoe, the patile the line veloity and line aeleation and angula veloity and angula aeleation, with foe a is to insepaable. So fo a igid body of otation, when its angula veloity and angula aeleation is asetained, by dissiila the foe a the oespondene, its line veloity and line aeleation will is dissiila. Naely at hee its line veloity and line aeleation, will in the foe a fo diet popotion to hange. Aoding to the foula (.4.), a slewing igid body, it abitaily the a of foe the ass of buthen the M, ae to foe a invese atio hange. Naely its Inetia toque is a onstant. I M onstant (..4) But an also get fo the foula (..), when the M and the is etain, ake sue the foe of the size, also the angula aeleation it is engende ake sue. d F a M dt (..5) This is vey the geeze, beause this show fo a igid body, it on the extenal foe oent the ation, by angula aeleation fo poduing, is with its the extenal foe of tangent line dietion the size to diet popotion. But have nothing to do with the size its foe a. This oplete subvesion lassial ehanis inside, the otation law of the igid body. This is in the Inetia toque piniple, new the igid body otation law. The (..5) is the foula of new igid body otation law. Theefoe, egadless in any foe a (etainly it be unequal to the zeo o infinity), to ake a igid body eation deteinate a angula aeleation, the size of a foe fo needing is unifo. Cetainly, the ating foe akes slewing igid body engende angula aeleation. Although in dissiila the foe a, the size of the foe is sae. But oespondene in sae ipulse, the foe a wok fo ake also is dissiila obviously. d dp M dt M d Fo exaple: dp Fdt and dt (..6) A Fdl this is the ipulse and angle ipulse. And that: (..7) this is the foe a wok fo ake. But L (is l )in the foula should fo: 0

11 L Udt d l dl du dt And dt dt (..8) With the foula (..5) opposite, the show when the foe a is big, the buthen ass M is s, but linea veloity should be big. Linea veloity U deision foe F the otion quantity L the size, both in eality is a diet popotion elation(..8). Theefoe although is as big as the foe, but when foe a is big, the foe F the otion quantity L also is big. So, the foe will also ake still geate a wok. So, the foe akes igid body otation, in the dissiila foe a, the size of the foe is sae. But the foe ake a wok, when the foe a is big oe, the wok be also oe big.. Many uliple slewings kineti enegy of the slewing igid body One slewing igid body, when its otation the angula veloity is etain, its otation linea oentu, at abitaily the foe a is sae. P ( M )(ρ ) u ρ l (..) It explains, when the foe a at hange, the igid body buthen ass of this foe a in oespondene, by invese popotion hange with foe a. ( M )(ρ ) ( M )(ρ U ) u ρ ρ (..) So, a slewing igid body, at its diffeent foe a, its buthen ass and linea veloity u, also hange with the invese atio. When the foe a is big oe, the linea veloity u also big oe, but buthen ass is then s oe. Vie vesa. Thee is a iustane at this tie, been no ow to neglet. Aoding to the definition of the kineti enegy of the objet, the kineti enegy of the objet is it: E u E And (..) So than foula (..) and (..), the slewing igid body the buthen ass and foe a and linea veloity u, hange with the invese popotion. But in opute of kineti enegy, the ass is a linea funtion, the foe a and linea veloity u but a quadati funtion. Theeupon, at ondition sae, naely sae slewing igid body and sae otation angula veloity, it in vaiant foe a, slewing kineti enegy is divese. < ρ ρ (..4) Theefoe, fo the hange agnifiation of the foe a by to seond powe. So, on a slewing igid body, when its the otation adius be big oe, its otation linea veloity seond powe also be big oe, theeupon its slewing kineti enegy is also big oe. So, any a slewing igid body, its slewing kineti enegy, don't is a single value. It also have the ulti

12 ply slewing kineti enegy. Its slewing kineti enegy, at its otation adius fo s aive the big that diffe plae, will display to is oe and oe big. E E ρ E ρ ρ (..5) That a slewing igid body has the ulti ply slewing kineti enegy, this a iustane henged the enegy onsevation law. Beause, at need pass the ollision, ake slewing igid body stop tuned. Then in otation adius s and otation adius big the plae, will eit fewness and any two kind heat enegy. An objet, its the kineti enegy has fewness and any, diffe iustane. This be with the enegy onsevation law, the enegy eation an't too an't disappea, been ay fo a fo onvesion is anothe a fo is abivalent. Beause when the enegy of the objet, thee is ultiple value, be so it onvet anothe the fo of enegy, etainly be an the fewness and the any. So in this tie of enegy onsevation law, is also a obviously none exatness. 4 igid body by uliple foe oent ation A igid body possibility is in sae tie, by uh foe oent ation. These foe oent ay be foe oent of the powe, also ay be the foe oent of the Stati foe; ay be diffe the diension a foe a, still ay be the dietion eah othe ontay. This is the opliated iustane. So suffe the opliated status of any foe oent ation in the igid body, should how ount? 4. igid body uh the ount of foe oent Should ount fist, igid body suffe foe oent of dietion of onefold that veto su. τ τ + τ + F + F + Fi i i + n (4..) + τ + τ + τ 4 + F + F4 4 + Fi i i + n (4..) Suh as the foula (4..) and (4..), with inus sign and positive sign espetively epesents the left hand tuning and ight hand tuning of the foe oent. Then ount, hinde the Stati foe foe oent of the igid body otation(fo instane the foe oent of the fition foe). Beause this Stati foe foe oent, also ay have the dietion, so the ount also fo inus sign and positive sign, distinguish its dietion. σ σ + σ + Fq + Fq + Fqi i i + n (4..) + σ + σ + σ 4 + Fq + Fq 44 + Fqi i i + n (4..4)

13 In foula the σ is delegate the Stati foe foe oent. The Stati foe foe oent ay have the dietion, and also the possibility do not have the dietion. If do not have the dietion, so inus sign and the Stati foe foe oent of the positive sign, will be the sae that. Beause the oent of foe of Stati foe ay have the dietion, also ay have no the dietion. So the igid body otation the left hand tuning and ight hand tuning the foe oent, should distinguish the oputation. Naely: Fi i + Fi i + Fqi i i + n i + n i + n ( τ ) ( τ ) ( σ ) Fi i + Fi i + Fqi i i + n i + n i + n ( τ ) ( τ ) ( σ ) (4..5) (4..6) Then the left hand tuning oent of foe subtats the ight hand tuning oent of foe and the oent of foe of Stati foe; o is the ight hand tuning oent of foe subtats the left hand tuning oent of foe and the oent of foe of Stati foe. Gained the left hand tuning powe oent of foe and ight hand tuning powe oent of foe of espetively. Aoding to the above ondition and alulate foula, then igid body by any foe oent to the otation is ay onfi, its the angula aeleation is a left hand tuning o ight hand tuning. O be quiesent, o be the otation of nothing angula aeleation. 4. igid body by uh the foe oent fo the ation esult Beause only thee is the oent of foe of the powe, then an push the igid body otation. So left hand tuning and the ight hand tuning the total oent of foe in powe, naely oe to a deision the otation dietion of the igid body possibility. at this tie an tepoay to the oent of foe of Statifoe be no onside. + + Fi i + Fi i i + n i + n ( τ ) ( τ ) (4..) In the left hand tuning and two total oent of foes of the ight hand tuning, the absolute value big subtats the absolute value s, then egad big sybol in absolute value as the sybol. Coe to a deision naely to the igid body an atual eation ating oent of foe, whih is a dietion in left hand tuning o ight hand tuning. Fi i + Fi i ± τ f i + n i + n (4..) In foula the τ f be left hand tuning and ight hand tuning the oent of foe to utuy subtat, but obtain the tue oent of foe. It atual plus o inus sybol, naely the ation dietion that epesent it. Then use it subtat the oent of foe of Stati foe that be as the otation esistane: ( ) ( ) ± τ σ τ + + τ ± σ f (4..) Naely total oent of foe and total Stati foe oent of foe utuy ation total outoe. In the igid body siila to fition oent of foe and Stati foe oent et, fo an ation at obstuts

14 to otation. Theefoe while total Stati foe oent of foe opae total ation oent of foe be big, the igid body will keep quiesene and no otation. When ation oent of foe the equal to Statifoe oent of foe, the igid body will oveoe the Stati foe oent of foe dag esistane foe ation, but etain otation state. It is an even veloity otation at this tie. ( ) 0 τ < σ τ f σ 0 Naely: and (4..4) f ( ) When the total oent of foe exeed the oent of foe of Stati foe, its exeed Stati foe oent of foe pat, beoe the ipulse to the igid body naely, ake igid body eation angula τ aeleation. At this tie: f τ d + τ s τ 0 But s σ (4..5) In foula the τ d is exeed Stati foe oent of foe pat, the τ s is equal to the Stati foe oent of foe pat. d τ τ d + s M + ( ) dt (4..6) So, the τ d esults in the angula aeleation of the igid body. The τ s suppots the igid body oveoes Stati foe the dag esistane foe, but even veloity the otation. 4. Foe and the size of the foe a and the sequene of the oputation The left hand tun oent of foe of the igid body utuy subtat with ight hand tun oent of foe, big that pat in oent of foe, also ay be ulti ply the oent of foe. Naely suh as foula (4..) o (4..) show. Beause the foe akes igid body eation angula aeleation, hiefly fo the foe the size deision, have nothing to do with the size of the foe a. But fo positive and negative the dietion ountewok in the oent of foe, the foe and foe a size is elevant. So engende at this tie, foe and the foe a that size, whih is ipotant of atuy? Whih fist ation? Suh question. Theefoe should onside fist, when positive and negative the dietion oent of foe utuy ountewok, the deision igid body go whih is the dietion of otation. Beause the ation of the foe a is vey big at this tie. When foe a is big, an use the s foe, esisting the bigge foe. τ > + τ Suppose: (4..) That: b i i qi i i + n i + n a ( τ ) + ( + τ ) + ( + σ ) F + F + + F + F + + F + n x n x n n ( F ) (4..) The size peutation of the foe a of the foula inside oent of foe, is sae with subsipt nube size peutation the dietion. So when of subsipt of the nube oe big, of the value is also oe big. Suppose in foula the vinulu a the veto quantity su of oent of foe, equal to vinulu b oent of foe and the oent of foe of Stati foe the veto quantity su. 4

15 That: ( τ ) ( τ ) ( σ ) F + F + + F τ d + n x + n x (4..) Naely left hand tun and ight hand tun oent of foe and the oent of foe of Stati foe the vetos su, equal to inside vinulu the oent of foe the veto quantity su. Theefoe in above oputation, is fist by the big the a of foe in the oent of foe, anel out eah othe with the oent of foe of Stati foe and left hand tun and ight hand tun. This is big beause of the foe a, the foe then is ay s, theefoe ath the hoie of the best the foe. Want to alulate at this tie, the foe akes igid body eation angula aeleation. Aoding to the piniple of Inetia toque, any igid body ontain etain the paaete of Inetia toque. I i i But ake igid body eation angula aeleation, aoding to new igid body otation law, the angula aeleation of the igid body, with its otation that tangent dietion the foe the size diet popotion. But have nothing to do with the foe a size of the foe. Hypothesis at this tie the vetos su of the oent of foe is that foula (4..), so its foe veto quantity su is: F F + F + F + F + + F s + n x (4..4) In the foula, via oent of foe ultiply the sae subsipt eipoal, oing to expuntion foe a. This poess is ust. Beause uh oent of foe, ust ai at the patiula oent of foe, afte doing away with its foe a get the patiula foe of a oespondene. Aoding to the new otation law: d d F a M Fs M dt dt F s d M dt (4..5) Theefoe, use the vetos su of these foe F s at this tie divide with the Inetia toque of the igid body, then get the angula aeleation of the igid body. So to the oputation of any foe oent of the igid body, it is ahieve. In foula (4..4), eah the oent of foe fo that thow off the oespondene the a of foe, to get the atual foe. This is vey ipotant. When the igid body only have positive and negative dietions the oent of foe by the ation, the ay also equie to alulated like this. τ > + τ ( τ ) + ( + τ ) Faa + Fb b Fx a Fo exaple: So (4..6) d Fxa Fx M a Thow off the foe a a : a dt (4..7) 5

16 Two the veto quantity su of oent of foe, get new oent of foe, its the a of foe with big the veto quantity the odule the foe a of oent of foe sae. Naely a in the foula. Thow away that paaete, then get the ating foe, and beoe the angula ipulse to the igid body. d Px Fxdt M a dt dt (4..8) Above of the disuss indiate, when any foe oent is ation at the sae tie in a igid body, fo ehanis and physis egulation the deision. Fist these oent of foes, will autoatiy with the oent of foe in big the a of foe, oe the opae and the antagonize. Toing deide igid body is a stillness o a otation, and its otation will whih is dietions. Aftewad fo the se the a of foe that oent of foes of that pat the the su of the foe, oe to beoe the ipulse to the igid body. Make igid body eation angula aeleation. Theefoe, the otation of the igid body whethe and to one dietion eation angula aeleation, was two diffe poess. Have got the pateny the hange in font and bak. 5 The angula oentu onsevation of the otation objet and that is not onsevation When the objet do the iula otion is the angula oentu onsevation, below will study angula oentu onsevation o that is not onsevations of the slewing igid body. 5. The angula oentu onsevation of the patile iula otion Angula oentu in the lassial ehanis is onsevation, is beause the patile o the objets the iula otion, when the otation adius hange, the entipetal foe akes the veloity between the patile o the objet, ouene and the hange of the adius invese popotion. Suh as the figue 5 and figue 6 the show. Figue 5 The fo big hange to s Figue 6 The fo s hange to big 6

17 Is two kinds iustanes of the adius fo big diinish and fo s beoe the big. Fo the figue inside to look, the entipetal foe ake the objet podue a oveent quantisty to the ente of a ile dietion, it to plus with the hange pevious the objet linea veloity the veto quantity, then obtain a new line veloity of the objet. Inside the figue of the veloity veto quantity u u and slewing adius, and entipetal kineti veto quantity in the objet, onstituted soe ight tiangle, and these tiangles ae siila tiangles. Theeupon these veto onfo the oposition of the veto quantity and the elation of the deoposition. But veto quantity u u and adius onstitute two ight tiangles, is siila tiangles. Also explain it will have elation as follows. Naely: u u (5..) So at hange font and bak, the veloity veto quantity and slewing adius is hange with invese popotion. Theeupon in above poess, without hange the angula oentu of the objet. l u u (5..) Foula (5..) and (5..) vesus figue 5 and figue 6 usable siilaly. Coe see is not diffiult, figue 5 and figue 6 the show, when the adius ouene hange of the iula otion in objet o patile, its the oveent linea veloity will hange with invese popotion. This kinds iustane is fo the piniple deision of deoposition(figue 6) and esultant(figue 5) of the veto quantity, theeupon is a kind oveent onening atte and the natual selet of the piniple of the foe. It is inevitable, any objet o patile at do iula otion the tie, follow this egulation. Naely its slewing adius and veloity veto is hange with invese popotion. When the hange of the adius of the iula otion of the objet, its in blink the ealization, then an adopt diffeential the algoith at this tie. d du l ± dt u dt ± d u du dt dt ( ) ( ) ( ) (5..) u du ( ± d ) u At this tie obviously: (5..4) So the hange in is in ealy o late, the objet still keep the angula oentu is invaiability, naely angula oentu onsevation. The iula otion between objet o patile the possess angula oentu onsevation, the best the pateny instane in this kinds iustane, naely inside univese evey kind of elestial bodies unning. Moeove, this kind of iustane is appliable to only, objet o patile suffe the entipetal foe ating iustane only. If diffe the objet is in iula otion, the inteation of the siila ollision in ouene. Then beause of new objet inteation law [5] : Diffe the objet of the ass when the inteation, the inteation foe is diffeent. So the ollision is in ealy o late, the angula oentu of these objet and the angula oentu the su, will is hange the ouene. Theefoe at this tie, the angula oentu between objet o patile, will be what is not onsevation. 5. The angula oentu onsevation of the igid body 7

18 Beause the piniple of Inetia toque of the igid body, in igid body the angula oentu hange is a opliated poess. Suh as the figue 7 show: Figue 7 the igid body of angula oentu onsevations Suppose that igid body be onstituted by patile 4. They the otation of with the angula veloity of iled the oigin o. Aong the the otation adius of the patile is the, the patile 4 the otation adius is the. The total angula oentu of the syste is of at this tie: L (5..) If the patile go to otion of the ente of a ile dietion, aive its otation adius equal to. Beause saying in font, the objet the angula oentu onsevation. The patile be inlined to the new angula veloity in engende, naely: ( l ) l (5..) Beause the onstaint of the igid body, the patile 4 should always have the sae angula veloity. Theefoe at this tie, fo the the angula veloity fo inlining to hange, will ake patile and 4 to an ation the foe. F F + F (5..) Inside the foula the F is patile do suffeed, tendeny the foe that obstuts its angula veloity hange. The F is the patile 4 do suffeed, tendeny the foe of the its angula veloity in enlageent. Two dietions of foe is ontay, F dietion and oiginal angula veloity in igid body the is sae, the F then is ontay with it. Beause two objets at utuy, the size agnitude of the foe and two objets ass geoeti popotion [5]. So: F F (5..4) F t ( ) F t (5..5) (5..6) 8

19 The is when a patile do ove to the otation adius, fo to no an enlaging that angula veloity. The is patile 4 an angula veloity fo enlaging. Theefoe: + (5..7) In figue 7 is iustane that >, if is a < that iustane, the in the foula(5..5) is a patile to tavel the otation adius, that an angula veloity fo no an inish. But in the foula(5..6) then patile 4 an angula veloity fo inish. The foula(5..7) also will beoe at this tie: + (5..8) Theefoe, be to have a patile o a objet on the otation igid body, take plae adial the ove. It will ake the angula veloity ouene hange of the igid body. And, this is as if angula oentu onsevation the hange. Naely the objet if tavel towad ente of a ile dietion, the angula veloity enlage; wheeas the angula veloity inish. 5. In otation igid body the ass apping and oposition ass The Inetia toque of the otation igid body, its oespondene in the ass of buthen of any foe a, is a the quantity that is etain. Usuy it is that a quantity of oposition, in it inside have tue patile the ass, also have fo othe the foe a the patile, the ass apping of apping. Also inlude the tue ass, also inlude the ass apping, this a iustane is the ass oposition. { } B { } C x,,,,, ( ) A x,,,, { } (5..) A C B C f : A B g : B C h : A C g f : A C (5..) A ' {,, } {, } ' A A {, } ' A B {, } ' A C (5..) The above foula the and x is tue ass, the is the ass apping, the is the ass oposition. Expess the ass oposition, an epesent the tue and apping the ass, ontol the otation of the igid body, and to alulate the igid body otation oveent. Aong the olletion A, B, C epesentative otation igid body not the adial ove the pat, but olletion A'is on the otation igid body, taking plae the adial ove the pat of the objet. 5.4 Inside otation igid body fo the objet to adial ove the angula veloity hange and the ass the elation When the objet o patile adial ove on the otation igid body, indue angula veloity hange of the siila angula oentu onsevation. Its angula veloity hanges with diffe patial of the igid body, is haved soe elation in the ass. Be got by foula (5..4) (5..5) (5..6): 9

20 F F t + + F F t + + ( ) 4 4 (5.4.) ( )( ) ( ) and (5.4.) So, ove the objet fo without an hanging an angula veloity, with the angula veloity of igid body the est of ouene hange of two angula veloities it atio; equal to the squae of igid body the est of asses, with ove the objet ass squae these two ass squae it the atio. Theefoe, when the objet ouene adial on the otation igid body the ove, its angula veloity ouene hange. The size of its angula veloity hange, with the ass of the igid body and the ass of the ove objet, the like foula (5.4.) to onfi the elation. Cetainly, the above oputation is a igid body to be liited by only suh as the figue 7 show, only have patile 4 fou patiles a vey siple syste fo onstituting. Unde the ajoity iustane, the stutue of the igid body is ipossible to be so siple. Theefoe an is divided into the igid body at this tie, Taking plae the adial ove the olletion A'the pat, and that has no aises the adial ove that the pat the olletion C. Is shown as foula (5..) and (5..), The olletion A ' epesentation the objet of ove, theeinto tenate eleent the podut be used as an eleent, by it is an unit eleent olletion. The olletion C epesentative the otation igid body, it is a finite olletion, aong the the eleent and to opposite in eah othe, thei podut is this igid body( do not inlude ove on this igid body the objet) the Inetia toque naely. The foula(5.4.) will beoe at this tie: (5.4.) Hee beause the token of the ass of the objet of adial oveent to at this tie, and that use to the igid body the adius the oposition ass, epesent 4 thee patile the ass. At this tie is yet the igid body adius physial tangential dietion load ass. Theeupon it possibility inlude the tue patial ass, also ay inlude this adius to othe pat igid body the ass apping. The expession of the eobination ass is: I C In foula the I C is Inetia toque by olletion C. (5.4.4) The eobination ass of the igid body an epesent the tuth and ass of the apping, oe to slewing that onstain the igid body, and ealization the igid body slewing oveent ekon. So an by eobination at this tie the ass to epesent 4 thee asses of patial, go along ekon suh as foula(5.4.), thus gained foula(5.4.). The piniple is alike, expessing suh ass of the igid body the elation, deiding the elation of its angula veloity hange. Theeupon by slewing on igid body to the ass of the objet of adial oveent of ouene, and igid body othe pat is in that ove objet in it aive the adius the buthen ass, then get the and of atio. Pass again the foula (5..7) and (5..8) the tansfo, an get the physial angula veloity hange value of the igid body again: 0

21 is an angula veloity hange that igid body ou fat. (5.4.5) (5.4.6) 5.5 The opute of the slewing igid body angula veloity hange Aoding to above of analysis (5..), was to know the ass of the objet in ove in slewing igid body, and it the foe and aft the adius and, an naely obtain it beause the angula oenta onsevation, but the value of the angula veloity that ouene hange: A l (5.5.) The atual angula veloity hange that afte whole igid body will ou, with and, and and the elation, be eant by foula (5.4.5) and (5.4.6). Be got by foula (5.4.): (5.5.) Get this foula joined (5.4.5) and (5.4.6): Tansfo expession: + + (5.5.) (5.5.4) (5.5.5) (5.5.6) Till: + (5.5.7) And + (5.5.8) It is that with and and and and the algebas elation. As long as ontoled above

22 and and and eah paaete, the substitution foula pogess alulates, naely getting the value of the hange of the angula veloity that igid body will take plae then. Theefoe, foula (5.5.7) and (5.5.8), naely is the foula to opute the igid body angula veloity hange. The anteio is iustane that >, the posteio is iustane <. Below: d + (5.5.9) d + and (5.5.0) This is the objet that adial ove to least that diffeential oputation foula. When the objet poseution in the igid body a line of onseutive adial ove, that hange of the igid body angula veloity, it is applied the definite integal to alulate: n f (, ) d + (5.5.) n f (, ) d + and (5.5.) The is the adius to tavel the fist in objet, n is its aive finy the adius. The in the foula although is also a vaiable, but it is that dependent vaiable of adius. So in oputation, eely to with vaiable elevant paaete pogess diffeential. In addition aoding to foula (5.5.7) and (5.5.8): ( ) f, (5.5.) Theefoe via to the ontinuous oputation of the definite integal (5.5.) and (5.5.), get the last auate esult naely. n Beg the appoxiate value to the integal of the foula(5.5.) and (5.5.), get: I C f (, ) d I C n I C aaaa a n n (5.5.4) n I C f (, ) d I C n I C aaaa a n n (5.5.5) If the objet on the otation igid body that adial ove, is a feedo and onstaint not fo to in the inteval. Afte siply aived, just again at with igid body utuy, ake olletion C with olletion A'by togethe onnet again. Theefoe its angula veloity will onstaint beause of eah othe it to the assiilate. At this tie the oputation of the angula veloity hange onening this igid body,

23 be by foula (5.5.7) and (5.5.8), oe to atualize in biefness. Theefoe by foula (5.5.7) to (5.5.), an atualize the oplete oputation that the angula veloity of the otation igid body hange. 5.6 The hange of the otation igid body angula oentu and hangeless? When that objet olletion A', on olletion C that otation igid body to adial ove, that objet olletion A'and olletion C that otation igid body, they total angula oentu do an hange? This point will deide to be below this iustanes, this otation's igid body whethe angula oentu onsevation? Abitaily the total Inetia toque of the igid body is: I n n i i In textual subjet suppose this I is a total Inetia toque in olletion C the igid body. Abitaily the total angula oentu of the igid body is: ( ) L x n n In textual subjet too suppose this L is olletion C the fist total angula oentu in the igid body, aong the the is naely the. On that olletion C otation igid body of behind adial ove of olletion A'objet, between the olletion A'objet and olletion C otation igid body, will engende the inteation foe. F d dt (5.6.) F d dt (5.6.) The F is a foe to olletion A'the objet suffe, the F is olletion C the otation igid body, in its the ass oposition is and adius is suffe a foe. In fat F fo an angula ipulse on olletion A'objet: F t Theefoe it ake olletion A'objet ouene angula oentu hange. Sae eason, F likewise vesus olletion C the otation igid body engende an angula ipulse: F t (5.6.) (5.6.4) It also akes olletion C otation igid body ouene angula oentu hange. Disove is not diffiult the foe F is a dietion with foe F ontay, theefoe thei angula ipulse is also a dietion ontay. So olletion A'the objet and olletion C the igid body, the angula oentu hange of the ouene, also the dietion is ontay. Both the odule of the veto subtats utuy, also will beoe to olletion A'the objet and olletion C the igid body, total angula oentu hange. On a igid body fo with angula veloity otation, on the angula veloity of the patile, is a also. Theefoe, the angula veloity of the igid body is with its angula oentu diet popotion. Theefoe a igid body, its the patile the and invaiability, but the hange ultiple of the angula

24 veloity, naely with the hange ultiple of its angula oentu is saeness. In anteio analyse, fist angula veloity in igid body is, the objet olletion A ' behind ove, the angula veloity that olletion C igid body will hange the. So at this tie olletion C igid body angula oentu hange ate is: L hange L (5.6.5) In foula the L is olletion C ealiest angula oentu, the L hange is olletion C utative angula oentu, is utative angula veloity. But towad that olletion A' the objet of ove, if it hange plus the angula veloity, its angula oentu invaiability. But beause it with inteation that olletion C the igid body, it have the of the angula veloity did not an hange. So at this tie olletion A'the angula oentu annot hange the atio is: l hange l A + + This is that iustane >, but that iustane < it is: l hange l A (5.6.6) (5.6.7) In foula the l A is fist angula oentu of the objet of olletion A ', the l hange is utative angula oentu behind objet ove. It is not diffiult to opehension is that so ed, the olletion A'the objet angula oentu ould not hange that atio, in eality is the hange ate of its angula oentu, but is inus and dietion is anti. The olletion A ' objet and the olletion C igid body total the angula oentu, behind the objet olletion A' ove, whethe ouene hange? Obviously as long as see to olletion A' the objet and olletion C the igid body the angula oentu hange, whethe exatly is size equivaleny and dietion is ontay. Beyond doubt, two hanges of angula oentu, the dietion ontay is affiative. But whethe is a size equal, that need to the poof. Is shown as foula below: L l + + L l A A (5.6.8) (5.6.9) If these two foula tenable o is identity, then olletion A' the objet and olletion C the igid body total angula oentu invaiability. Poeed the analysis below, tansfo equality as ( abidge the < iustane): L l A + + ( ) Be got by foula (5..7) and (5..8)( egadless is iustane that > o < ): (5.6.0) 4

25 L l A (5.6.) L l A (5.6.) Aoding the foula (5.4.), the equality hanges into again: L l A Aoding to an idea an hanges fo again: n n (5.6.) (5.6.4) The equal sign the left side the nueato patial that oeffiient naely so an edution of fation. Futhe tansfo: n n ( ) Be got by foula (5.4.4): n n I I C n n C ( ) ( ) (5.6.5) (5.6.6) (5.6.7) n n ( n n ) (5.6.8) I C L o (5.6.9) On these gounds, the onlusion is pateny. The foula (5.6.8) and (5.6.9) is not the idential equation, while satisfy foula (5.6.8) and (5.6.9) they just tenable. At this tie in the equal sign both sides, the obviously possessive quantity is onstant. The ight side of the fo instane equal sign, inevitable is a positive the ational nube. When the ondition asetain, is only its the value. Theeupon if want the equality tenable, the epesentative of the left side of the equal sign the olletion A'the ove the objet the ass, also usting be single affiatoy the value, is equal to the equal sign the value of the ight side. Foula (5.6.8) and (5.6.9), display vey inteesting, a kind the attibute of the slewing igid body. It expesses, when the slewing igid body, have the objet by the adial ove, the said igid body an not neessaily keeps angula oenta onsevation. At this tie in that igid body, the ove objet of 5

26 olletion A ' and the slewing igid body of olletion C, ust be to satisfy the foula (5.6.8) and (5.6.9), is just unde the iustanes, an satisfy the angula oenta onsevation. Not so its angula oenta is not onsevation. Theefoe this foula siila is a law to that igid body angula oentu whethe onsevation, ount fo uh. If the equal sign both sides neas to the equivaleny, then it an look like to satisfy the angula oenta onsevation. But if equal sign the left side lage than to in equal sign the ight side, naely olletion A'ove the objet the ass lage than etain value, then olletion A'ove objet angula oenta hange, fo the se than the olletion C slewing igid body angula oenta hange. Contaily olletion A'ove the objet angula oenta hange, lage than to olletion C slewing igid body angula oenta hange. Will is ondue the total angula oenta ouene hange of slewing syste, theeupon angula oenta is not onsevation. Theeupon, be thee is objet on the slewing igid body along the adial ove, its total angula oenta an keep the invaiable ondition is vey igou. So is unde the this kind of iustane of the pluality, the slewing igid body is angula oenta is not onsevation. 5.7 The slewing igid body is unde ost iustanes is not onsevation the angula oenta Theeupon when the slewing igid body, have the objet to the adial ove, its angula oenta is usuy is not onsevation. But beause Newton thid law, have beened by testify be a wong [5], and fo new objet inteation law, two objet inteation opeations, the diension of foe and the ass of two objet is geoeti popotion [5]. Theefoe the objet lineaity oentu onsevation law also is wong. And oentu onsevation law sine is wong, so in a atte syste, have no the opeation of the extenal foe even, fo inside of syste the objet of inteation, also an wok syste oenta a hange. This kind the iustane is too be appliable, in slewing the atte syste the slewing lineaity oenta. Theefoe lineaity oenta of slewing objet, it is not onsevation also. So, in a slewing igid body syste, has no the opeation of the extenal foe oent even, be oved by objet in the syste adial, o the objet by opeation in the slewing positive and negative oientation, an ondue the angula oenta in igid body to hange. The angula oenta of the fo this eason slewing igid body is unde ost iustanes is not onsevation. The slewing of the onening objet, besides satisfying the foula (5.6.8) and (5.6.9), still eseve the angula oenta onsevation, is an objet o patile enile the iula otion, but its in addition to only having the adial oveent( naely hange slewing adius), without any dietion of a tangent foe( is an intenal foe o an extenal foe egadless) opeation. As long as satisfied the this kind ondition, then it is an angula oenta onsevation in hold. Fo instane elestial body in the osos, enile the slewing of the fixed sta, naely usuy has the this kinds the angula oenta onsevation. 6 Leve and oent of foe piniple the new henge Aoding to the Inetia toque piniple, the foe a of the oent of foe the hange toing the 6