NEWTON'S SECOND LAW AND CURVILINEAR MOTION OF A MATERIAL PARTICLE. Yuriy Zhivotov Yuzhnoye State Design Office, Dniepropetrovsk, Ukraine

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1 NEWTON'S SECOND LAW AND CURVILINEAR MOTION OF A MATERIAL PARTICLE Yuriy Zhivotov Yuzhoye State Desig Office, Diepropetrovsk, Ukraie Summary The Newto's secod law is cosidered with referece to motio of a material particle alog a curviliear trajectory. It is show that a chage i mometum of a material particle creates a tagetial ad cetripetal force. The equatios of motio of a particle were obtaied. It is show that motio of a particle requires applicatio of a mometum force at presece of a reactio force. It is show that a work of a tagetial ad cetripetal force chages the magitude ad directio of a kietic eergy vector. Newto gave the followig defiitio of the secod law of motio with referece to the rectiliear motio of a material particle with a mass ad acceleratio d mv is i proportio to the applied m a : Chage of mometum ( ) mometum force F ad takes place i the lie of the straight lie alog which this force acts. F = ma = d ( mv ) / dt () This law provides for a chage i mometum of the material particle without a chage i the directio of its motio. However, while a free material particle moves alog a curviliear trajectory, the mometum magitude ad directio may vary []. Let us assume that the material particle moves alog a covex curviliear trajectory from left to right, the directio of the velocity chagig clockwise. Let us assume that withi a short period of time t t 0 the mometum of the material particle mv 0 has icreased ad is equal to mv, a small agle ϕ havig bee formed betwee the directio of the mometum vector mv ad the directio of the mometum vector mv 0 (Figure ). Figure Let us itroduce a additioal mometum vector mv 0 so that a magitude of the mometum vector mv0 + mv is equal to that of the mometum vector 0 mv. Let us joi the termius of the additioal vector mv 0 ad the termius of the vector mv ad we shall obtai a vector mv 02. Therefore, we ca write dow the followig vector equatio i view of the small agle ϕ = siϕ ad si 0 = 0 Proceedig to the limit, if t t, we shall obtai ( ) 0 0 mv mv0 = mv0 + mv02 (2) d( mv)/ dt = ma + ma = F + F (3) where ma = mv siϕ si 0 / dt = mvϕ/ dt = mvw, ω is the agular velocity, F is the tagetial force taget to the curviliear trajectory, F is the cetripetal force with the directio perpedicular to the force F. I this defiitio of the forces brigig about the chage i mometum of the material particle, the directio of the force F coicides with that of the vector a, ad the directio of the force F, with that of the vector a. The equatio of a accelerated motio of the free material particle alog the curviliear trajectory is as follows: F + F = ma + ma (4) For the purpose of cosideratio, we selected a covex trajectory of the material particle motio. If we assume that the covex trajectory gradually turs ito a cocave trajectory, the the chage of the velocity directio will happe aticlockwise. The positio of the istataeous radius of rotatio will chage cosetaeously. The equatios

2 (2), (3) ad (4) will ot chage. If mechaical costraits impose requiremets of kiematical or geometric ature, the the chage of mometum of the material particle should be writte as follows where ( ) / d mv dt = F + R (5) R is the costrait reactio force that requires a additioal tagetial force for ow creatio. I this case, the motio equatio for a costraied particle will be as follows F = ma + ma (6) Let us cosider some features of the equatio (4). If F = 0, the ma = 0. If F is the gravitatio force, the from the equatio (4) we shall obtai a certai equatio of motio of the satellite i circular orbit. The force F esures the material poit motio roud a circle. This force is ot a mometum force. I this case we obtai that F should be equal to ma + ma. If we additioally assume tha the acceleratio a ca be equal to zero, we shall obtai that F = mvω. It results from this that at presece of costraits a uiform motio of the material particle roud a circle requires applicatio of a tagetial force. It is also worthwhile otig that the defiitio of the forces (3) brigig about the chage i mometum of the material particle alog the curviliear trajectory does ot have a effect o the defiitio of the chage i a momet of mometum relatively to the istataeous ceter of rotatio. The chage of the momet of mometum of the material particle relatively to the istataeous ceter of rotatio has its usual view: ( ) ( ) dmomv [ ]/ dt= Mo F (7) This is explaied by the fact that the directio of vector F passes through a poit relatively to which the momet is defied, ad the momet from the force F is therefore equal to zero. The equatio (3) allows a trasitio to the otio of a kietic eergy ad a work of forces. Let us assume that i a time t the material particle moved from the poit M A to the poit M B of the trajectory. Let us break the traslatio s ito sectors s. Let us multiply the right ad left parts of the equatio (3) by the mea velocity s / t (a scalar). This operatio does ot chage the magitudes or directios of the vectors i the equatio (3). As a result, we obtai the followig vector equatio: or where 2 F s = mv ϕ s s s d( mv) = ma dt+ ma dt (8) t t t mv v /2 mv v /2= F s + F s (9) 0 0 I view of the traslatio s of the material particle from the poit M A to the poit M B withi the time t, we obtai mv ( ) v ( ) /2 mv ( ) v ( ) /2 = A M M M M ( M M ) + A ( MA MB),, B B A A A B (0) The equatio (0) shows that the chage of the magitude ad directio of kietic eergy is accomplished by the work of force for traslatio ad the work of force for chagig the kietic eergy vector directio. Here, the work of force is a scalar. CONCLUSIONS The obtaied research results allow obtaiig differetial equatios of motio ad solvig problems of traslatory motio of a material particle ad a body alog a curviliear trajectory. Refereces [] Тарг С.М.: Краткий курс теоретической механики. Москва. Высшая школа стр. /Targ S.M. Cocise Course i Classical Mechaics, Moscow, Vysshaya Shkola, 966, 46 p./ Cotiuatio o the followig page!

AUGUST 24 30, 2008 Dr Yuriy Zhivotov Mechaics 3, Krivorozhskaya Street, Diepropetrovsk 49008, UKRAINE Diepropetrovsk Depropetrovsk regio, Ukraie Adelaide, April 22, 2008 PROFESSOR ERNIE TUCK

3 AUGUST 24 30, 2008 Dr Yuriy Zhivotov Mechaics 3, Krivorozhskaya Street, Diepropetrovsk 49008, UKRAINE Diepropetrovsk Depropetrovsk regio, Ukraie Adelaide, April 22, 2008 PROFESSOR ERNIE TUCK PRESIDENT, ICTAM2008 LEVEL 4, 0 PULTENEY STREET THE UNIVERSITY OF ADELAIDE SA 5005 AUSTRALIA Dear Dr Zhivotov, I am writig o behalf of the Cogress Committee of lutam with referece to your paper with ICTAM etitled NEWTON'S SECOND LAW AND CURVILINEAR MOTION OF A MATERIAL PARTICLE submitted to the 22 d Iteratioal Cogress of Theoretical ad Applied Mechaics (ICTAM2008). Ufortuately the umber of papers submitted greatly exceeded the umber that could be accommodated i the program ad I regret to iform you that we could ot accept your paper for presetatio. I hope you will, evertheless, be able to participate i the Cogress, whose success depeds o those who atted the lectures ad participate i discussios as much as o those who make the formal presetatios. Complete details about the Cogress, icludig registratio, may be foud o the web site The orgaizers joi me i expressig their hope that you will participate i this importat mechaics evet. A formal ivitatio to atted ICTAM2008 follows. I look forward to seeig you i Adelaide. With best wishes, Yours sicerely Timothy J. Pedley Secretary, Cogress Committee of IUTAM Zhivotov s aswer! (O 2/05/2008, at 9:20 PM) ICTAM 2008, Presidet Professor Erie Tuck, The Uiversity of Adelaide. eotuck@iterode.o.et Dear Professor Erie Tuck. I have prepared a paper «NEWTON'S SECOND LAW AND CURVILINEAR TRANSLATIONAL MOTION OF THE BODY» (idetificatio umber 03) for participatio i the XXII Iteratioal Cogress of Theoretical ad Applied Mechaics, to be held i Adelaide, Australia betwee August I a paper it is show, that: I a paper it is show, that:

4 - the moder formulatio of the Newto's secod law is erroeous: the formulatio of the law substitutes idea of Lagrage to express ay force through force of iertia. The ote: The moder formulatio of the secod law substitutes the formulatio of the Newto's third law; - chage of quatity of movemet of a body ca occur both o magitude, ad o a directio. Thus forces should make work for chage of quatity of movemet of a body o magitude ad a directio; - chage of quatity of movemet demads the appedix to a body of two mutually perpedicular forces: drivig force ad cetripetal force. Cetripetal force very much frequetly is abset i egieerig practice. I this case drivig force should compesate absece of cetripetal force. I.e. i mechaisms cetripetal forces are created due to work of drivig forces; - eve uiform movemet of a material poit cocerig the ceter of rotatio demads the appedix of drivig force which makes work o chage of a directio of its movemet. Hece, uiform rotatio of a material poit o a circle demads expeses of eergy; - the existig mechaics is ot suitable for the descriptio of curviliear movemet of a body ad the decisio of this problem is resulted. The give paper is the basic documet, which advaces ew ideas i theoretical mechaics. These ideas will result i radical processig mechaics. Orgaizers of the cogress have set the followig message: I am writig o behalf of the Cogress Committee of lutam with referece to your paper with ICTAM etitled NEWTON'S SECOND LAW AND CURVILINEAR MOTION OF A MATERIAL PARTICLE submitted to the 22d Iteratioal Cogress of Theoretical ad Applied Mechaics (ICTAM2008). Ufortuately the umber of papers submitted greatly exceeded the umber that could be accommodated i the program ad I regret to iform you that we could ot accept your paper for presetatio. However orgaizers of the cogress have ot set commets of reviewers o a paper. Absece the commet does ot allow to reveal a mistake of reviewers ad to exclude artificial obstacles for promotio of ew ideas i mechaics. I this coectio, I ask you to direct to my address of commets of reviewers. Best regards, Yu. Zhivotov The aswer to the Zhivotov's letter! (O AM 8:08:27) Dear Professor Zhivotov, Thak you for your . Ufortuately the umber of papers submitted greatly exceeded the umber that could be accommodated i the program ad we regret to iform you that we caot accept your paper for presetatio.

5 We hope you will, evertheless, be able to participate i the Cogress, whose success depeds o those who atted the lectures ad participate i discussios as much as o those who make the formal presetatios. To participate i this importat mechaics evet please go to our registratio page. Best regards Jim Deier Associate Professor Jim Deier Secretary Geeral ICTAM2008 School of Mathematical Scieces The Uiversity of Adelaide South Australia 5005 Australia Phoe: (+68) Zhivotov s aswer! (Date: ) ICTAM 2008, Presidet Professor Erie Tuck, The Uiversity of Adelaide. eotuck@iterode.o.et Dear Professor Erie Tuck. I i the previous letter (O AM 8:08:27) asked to direct resposes of reviewers which have served as the reaso for a deviatio of my paper: NEWTON'S SECOND LAW AND CURVILINEAR MOTION OF A MATERIAL PARTICLE to my address. However from Secretary Geeral ICTAM2008 I have received the iadequate aswer Thak you for your . Ufortuately the umber of papers submitted greatly exceeded the umber that could be accommodated i the program ad we regret to iform you that we caot accept your paper for presetatio. I ask you oce agai will direct resposes of reviewers to my address! Best regards, Yu. Zhivotov Last aswer to the Zhivotov's letter! ( :02:0) Dear Professor Zhivotov, Thak you agai for your . All papers for the cogress were reviewed by the Iteratioal Papers Committee appoited by IUTAM. As local orgaisers, we are ot ivolved i the review process or are we privy to their deliberatios. For these reasos I am uable to provide you with ay feedback from the Iteratioal Papers Committee. Best regards Jim Deier Cotiuatio o the followig page!

6 New aswer of Zhivotov! (Date: July 9, 2008) Presidet IUTAM Professor L.B. Freud Copy ICTAM 2008, Presidet Professor Erie Tuck Dear Presidet IUTAM, Professor L.B. Freud. I have addressed to orgaizers of cogress ICTAM 2008 with the request to give resposes of reviewers o a paper «NEWTON'S SECOND LAW AND CURVILINEAR TRANSLATIONAL MOTION OF THE BODY» (idetificatio umber 03). Prof. Jim Deier has iformed, that he has o resposes of reviewers ad orgaizers of the cogress have o feedback with Iteratioal Papers Committee. I hope, that the maagemet of the IUTAM has a feedback with Iteratioal Papers Committee. Therefore I ask you to assist i receptio of resposes of reviewers o my paper. Best regards Yu. Zhivotov P. S. I iform also, that the same history has happeed to a paper TORQUE EQUATION FOR RIGID ROTOR WITH ELASTIC SHAFT (ID 0856), which has bee set o the cogress ICTAM04. The give paper has bee published i magazie ad placed for the review Similar actios Iteratioal Papers Committee do ot promote promotio of mechaics.

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