Classical Mechanics ( Particles and Biparticles )

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1 Classcal Mechancs ( Partcles and Bpartcles ) Alejandro A. Torassa Creatve Commons Attrbuton 3.0 Lcense (0) Buenos Ares, Argentna Abstract Ths paper consders the exstence of bpartcles and presents a general equaton of moton, whch can be appled n any non-rotatng reference frame (nertal or non-nertal) wthout the necessty of ntroducng fcttous forces. Unversal Reference Frame The unversal reference frame S s a reference frame n whch the acceleraton å of any partcle s gven by the followng equaton: å = F m where F s the net force actng on the partcle, and m s the mass of the partcle. The unversal reference frame S s an nertal reference frame. Therefore, t can be stated that the unversal reference frame S s also a non-rotatng reference frame.

2 General Equaton of Moton The general equaton of moton for two partcles A and B, s as follows: m a m b (r a r b ) = m a m b ( r a r b ) where m a and m b are the masses of partcles A and B, r a and r b are the postons of partcles A and B relatve to a non-rotatng reference frame S, r a and r b are the postons of partcles A and B relatve to the unversal reference frame S. f m a m b = m ab,(r a r b ) = r ab and( r a r b ) = r ab, then the above equaton reduces to: m ab r ab = m ab r ab The general equaton of moton for a system of N partcles, s as follows: m m j (r r j ) = m m j ( r r j ) where m and m j are the masses of the -th and j-th partcles, r and r j are the postons of the -th and j-th partcles relatve to a non-rotatng reference frame S, r and r j are the postons of the -th and j-th partcles relatve to the unversal reference frame S. f m m j = m j, (r r j ) = r j and ( r r j ) = r j, then the above equaton reduces to: m j r j = m j r j A system of partcles forms a system of bpartcles. For example, the system of partcles A, B, C and D forms the system of bpartcles AB, AC, AD, BC, BD and CD.

3 Partcles and Bpartcles From the general equaton of moton for two partcles A and B (underlned blue equaton) the followng equatons are obtaned: BPARTCLE PARTCLE K N E M A T C S a ab = åab a ab = å ab a a = å a a a = åa v ab = v ab v ab = v ab v a = v a v a = v a r ab = r ab r ab = r ab r a = r a r a = r a K N E M A T C S D Y N A M C S m abr ab = m ab r ab m ab r ab = m ab r ab m a r a = m a r a m ar a = m a r a m abv ab = m ab v ab m ab v ab = m ab v ab m a v a = m a v a m av a = m a v a m aba ab = m abå ab m ab a ab = m ab å ab m a a a = m a å a m aa a = m aå a D Y N A M C S BPARTCLE PARTCLE 3

4 The blue equatons are vald n any non-rotatng reference frame, snce (r ab = r ab ), (v ab = v ab ) and (a ab = å ab ) The red equatons are vald n any nertal reference frame, snce (a a = å a ) The knematc equatons are obtaned from the dynamc equatons f we consder that all partcles have the same mass. Therefore, the knematc equatons are a specal case of the dynamc equatons. The dynamcs of partcles s obtaned from the dynamcs of bpartcles f we only consder bpartcles that have the same partcle. For example: f we consder a system of bpartcles AB, AC and BC, we have: AB + AC + BC = AB + AC + BC Consderng only the bpartcles that have partcle C, t follows: AC + BC = AC + BC Applyng the general equaton of moton, we obtan: m a m c (r a r c ) + m b m c (r b r c ) = m a m c ( r a r c ) + m b m c ( r b r c ) Dfferentatng twce wth respect to tme, yelds: m a m c (a a a c ) + m b m c (a b a c ) = m a m c (å a å c ) + m b m c (å b å c ) Dvdng by m c, usng a reference frame C fxed to partcle C ( a c = 0 relatve to reference frame C ) and assumng that reference frame C s nertal (a c = å c ), we obtan: m a a a + m b a b = m a å a + m b å b Substtutng å = F/m and rearrangng, fnally yelds: F a + F b = m a a a + m b a b 4

5 Equaton of Moton From the general equaton of moton t follows that the acceleraton a a of a partcle A relatve to a reference frame S (non-rotatng) fxed to a partcle S, s gven by the followng equaton: a a = F a m a F s m s where F a s the net force actng on partcle A, m a s the mass of partcle A, F s s the net force actng on partcle S, and m s s the mass of partcle S. n contradcton wth Newton s frst and second laws, from the above equaton t follows that partcle A can have non-zero acceleraton even f there s no force actng on partcle A, and also that partcle A can have zero acceleraton (state of rest or of unform lnear moton) even f there s an unbalanced force actng on partcle A. On the other hand, from the above equaton t also follows that Newton s frst and second laws are vald n the reference frame S only f the net force actng on partcle S equals zero. Therefore, the reference frame S s an nertal reference frame only f the net force actng on partcle S equals zero. Bblography A. Ensten, Relatvty: The Specal and General Theory. E. Mach, The Scence of Mechancs. R. Resnck and D. Hallday, Physcs. J. Kane and M. Sternhem, Physcs. H. Goldsten, Classcal Mechancs. L. Landau and E. Lfshtz, Mechancs. 5

6 Appendx Transformatons The unversal reference frame S s an nertal reference frame. Any nertal reference frame s a non-rotatng reference frame. Any central reference frame S cm (reference frame fxed to the center of mass of a system of partcles) s a non-rotatng reference frame. A change of coordnates x, y, z, t from a reference frame S (non-rotatng) to coordnates x, y, z, t from another reference frame S (non-rotatng) whose orgn O has coordnates x o, y o, z o measured from reference frame S, can be carred out by means of the followng equatons: x = x x o y = y y o z = z z o t = t From the above equatons, the transformaton of poston, velocty and acceleraton from reference frame S to reference frame S may be carred out, and expressed n vector form as follows: r = r r o v = v v o a = a a o where r o, v o and a o are the poston, velocty and acceleraton respectvely, of reference frame S relatve to reference frame S. 6

7 Partcles Defntons Bpartcles Mass M = m M j = j> m j Vector poston R = m r /M R j = j> m j r j /M j Vector velocty V = m v /M V j = j> m j v j /M j Vector acceleraton A = m a /M A j = j> m j a j /M j Scalar poston R = m r /M R j = j> m j r j /M j Scalar velocty V = m v /M V j = j> m j v j /M j Scalar acceleraton A = m a /M A j = j> m j a j /M j Work W = m a dr W j = j> mj a j dr j ( ) ( ) W = m v W j = j> m j vj Relatons ( ) M j R j = M R R ( ) M j V j = M V V ( ) M j A j = M A A f M /M j = k, then the above equatons relatve to the central reference frame S cm reduces to: R cm j V cm j A cm j = k R cm = k V cm = k A cm 7

8 Prncples The postons, veloctes and acceleratons (vector and scalar) of a system of bpartcles are nvarant under transformatons between non-rotatng reference frames. R j = R j = Rj cm = R j R j = R j = Rj cm = R j V j = V j = Vj cm = V j V j = V j = Vj cm = V j A j = Å j = Aj cm = A j A j = Å j = Aj cm = A j From ths prncple t follows that the acceleraton a a of a partcle A relatve to a non-rotatng reference frame S fxed to a partcle S, s gven by the followng equaton: a a = F a m a F s m s where F a s the net force actng on partcle A, m a s the mass of partcle A, F s s the net force actng on partcle S, and m s s the mass of partcle S. The acceleratons (vector and scalar) of a system of partcles are nvarant under transformatons between nertal reference frames. A = Å = A A = Å = A From ths prncple t follows that the acceleraton a a of a partcle A relatve to an nertal reference frame S, s gven by the followng equaton: a a = F a m a where F a s the net force actng on partcle A, and m a s the mass of partcle A. 8

9 W j = Work and Force The work W j done by the forces actng on a system of bpartcles relatve to a non-rotatng reference frame, s gven by: ) d (r r j ) m m j ( F m F j m j The work W done by the forces actng on a system of partcles relatve to the central reference frame, s gven by: W = F dr The work W done by the forces actng on a system of partcles relatve to an nertal reference frame, s gven by: W = F dr Conservaton of Knetc Energy f the forces actng on a system of partcles do not perform work relatve to the central reference frame, then the knetc energy of the system of partcles s conserved relatve to the central reference frame. f the knetc energy of the system of partcles s conserved relatve to the central reference frame, then the knetc energy of the system of bpartcles s conserved relatve to any non-rotatng reference frame. f the forces actng on the system of partcles do not perform work relatve to an nertal reference frame, then the knetc energy and lnear momentum (magntude) of the system of partcles are conserved relatve to the nertal reference frame; even f Newton s thrd law were not vald. 9

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