PY2101 Classical Mechanics Dr. Síle Nic Chormaic, Room 215 D Kane Bldg

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1 PY2101 Classcal Mechancs Dr. Síle Nc Chormac, Room 215 D Kane Bldg s.ncchormac@ucc.e Lectures stll some ssues to resolve. Slots shared between PY2101 and PY2104. Hope to have t fnalsed by tomorrow. Mondays 12-1pm ORB 303 Tutoral or Lecture Mondays 1-2pm G7 Lecture Thursdays 11-12pm G7 Lecture Frday 10-11am G7 Lecture? Uncertan due to tmetable clash Frday 12-1pm G7 Lecture Frday 2-3pm G7 Lecture? Uncertan Tutorals to be held every 2 weeks. PY2101 tutorals commence 19 th October and PY2104 commence 12 th October durng the Monday 12-1pm slot. Recommended text book: Classcal Mechancs by Kbble and Berkshre. Buy a copy or get from lbrary!

2 Tomorrow s lectures (+ next week) 10-11am or 2-3pm whch does the group want? It wll be a PY2101 slot. 12-1pm wll be a PY2104 slot. Next week, all four slots wll be Classcal Mechancs PY2101 startng on Monday at 1-2pm slot n G7. No tutorals next week.

3 Assessment Total Marks 100 End of Year Wrtten Examnaton 80 Marks (1.5 hours) Contnuous Assessment 20 Marks (Weekly problem sets no late submssons accepted) Module Objectve To advance the student s knowledge of classcal mechancs

4 Classcal Mechancs Chapter 1 Laws of classcal mechancs formulated by Galleo and Newton. Vald n many stuatons but nvald n two regmes: Phenomena on atomc and nucle scales requre quantum mechancs Phenemona close to or at the speed of lght (c = 3 x 10 8 ms -1 ) requre specal relatvty (relatvstc mechancs) Classcal mechancs s excellent over a range of phenomena larger than atomc scales (everyday objects, galactc dstances, ) for speeds not close to c.

5 Example of Newton s Laws Applcable n the followng: Apple fallng from tree, d ~ 3m, v ~ 8m/s = % c Earth s moton around sun, d ~ 1.5 x m, v ~ 30 km/s = 0.01% c Sun s moton around galactc centre, d ~ 2.6 x10 20 m, v ~ 220km = 7.3% c Phenomena cover 20 orders of magntude n dstance and 5 orders of mag. n speed huge range descrbed by classcal mechancs. Some aspects of physcs may be unversal e.g. conservaton of energy and momentum always apples.

6 Assumptons of classcal mechancs 1. There s such a thng as a unversal tme system (two observers who have synchronsed ther clocks wll always agree about the tme of the event). Volated at fast speeds of relatvstc mechancs! 2. The geometry of space s Eucldean (any 2 ponts can be joned by a straght lne, etc. ) 3. We can, n prncple, measure all postons and veloctes Volated on wth small arbtrarly scales hgh of quantum accuracy. mechancs!

7 The Relatvty Prncple There s no absolute poston poston only has meanng relatve to a specfed pont (e.g. the centre of the earth). Velocty s also relatve. Ths s not true for acceleraton. Relatvty Prncple: Gven 2 bodes movng wth some constant relatve velocty, t s mpossble to determne expermentally whch of the two bodes s at rest (f ether) and whch s (are) movng. Example: Sttng on a bus hard to tell f car next to you s movng or f bus s movng. If two unaccelerated observers perform a measurement they get same result. For accelerated observers ths s not true.

8 Frame of Reference Must defne a reference frame Lmerck s 100 km east and 50 km north s meanngless f we don t defne the pont from whch we are measurng. An event occurred at some tme 12 mnutes and 13 seconds also s meanngless f we don t defne the orgn of tme. Ths s what we mean by a frame of reference. Example: Cartesan coordnate system, poston defned by x, y, z and tme t. In another reference frame use x, y, z and t.

9 From the relatvty prncple, frames of reference used by dfferent unaccelerated observers are completely equvalent.e. laws of physcs stay the same regardless of reference frame chosen. Unaccelerated reference frames are called nertal frames. In an nertal frame an observer would deduce the correct laws of physcs frame nether rotatng nor acceleratng. Fxed reference frames can be defned relatve to dstant galaxes for example Internatonal Celestal Reference Frame s based on poston of 212 extragalactc sources dstrbuted over entre sky.

10 Can also defne poston usng a vector r relatve to the coordnate orgn O: r = x + y j + z k,, j and k are unt vectors along x, y and z axs. Use a hat to denote other unt vectors e.g. vector n the drecton of vector r. rˆ s a unt Intally, treat object of nterest as a pont partcle, located at centre of mass of the object

11 Consder solated system of N bodes, = 1, 2,, N. (assume all other bodes far enough away that ther nfluence s neglgble). Poston, velocty and acceleraton of body are gven by: r r& () t ; v () t = r& () t a ( t) = v& ( t) () t = ; dr () t etc. dt Momentum of the body s gven by: Newton s second law: p & = = ma F () t m v ( t) m s body mass. p =,, total force actng on body. F N = j= 1F, F s force on body by body j. j j Two-body forces (only nvolve two bodes).

12 Two body forces must satsfy Newton s thrd law: F = j F j Because of relatvty prncple, F j can only depend on relatve poston and relatve velocty of two bodes: r j = r r ; j r r j v j = v v r r j j If we know force between two bodes as a functon of relatve poston and relatve velocty, use Newton s second law and eqns. relatng acceleraton, velocty and poston to predct future moton of the two bodes. For n bodes, must do ths for whole system ncludng forces between each par of bodes.

13 Central, Conservatve Forces 1. Forces that are drected along the lne connectng the two bodes (central). 2. Forces that depend only on relatve postons of two bodes (conservatve). Man feature of conservatve forces s that total energy of the system s conserved. F = f j r j ( ) rˆ j f s scalar functon of dstance between two bodes r j. Newton s Law of Gravtaton: Coulomb force between two charged bodes: f f Gm m kg 2 r j ( r ) = ; G = Nm j j 12 1 ( r ) = = ; ε = j kq q r q q j 2 0 Fm 2 j 4πε 0r j j attractve or repulsve attractve (negatve)

14 We wll concentrate on central, conservatve forces. But not all forces must be central and conservatve. Need to use quantum mechancs for forces between partcles on atomc scale Frctonal forces appear non-conservatve on large scale Forces between 2 charges n relatve moton are nether central nor conservatve requres ntroducton of electrc feld concept.

15 Mass and Force Should only ntroduce new quanttes f they can be measured so that theory can be tested. Newton s laws use velocty, acceleraton, mass and force. Can use dstance and tme to measure v and a. How can we measure m and F? Inertal mass Newton s 2 nd law Gravtatonal mass Newton s law of gravtaton Need to measure each separately (e.g. cannot use scales to measure nertal mass snce ts operaton s based on gravtatonal force due to Earth. Actually measure gravtatonal mass!)

16 Inertal Mass We could measure nertal mass of 2 bodes by subjectng each to equal force and comparng acceleraton need to ensure both forces are equal. Consder two bodes nteractng wth each other and solated from other bodes and forces. Inertal masses must obey: F 12 = F 21 m 1 a 1 = - m 2 a 2 Can measure rato of nertal masses by measurng acceleratons. Eg. f two bodes collde, mass rato can be determned by measurng veloctes before and after collson snce total momentum must be conserved. After ntegraton: m 1 a 1 + m 2 a 2 = 0 m 1 v 1 + m 2 v 2 = constant Conservaton of momentum

17 Fundamental Axom For any solated par of bodes, the acceleratons always satsfy the relaton a 1 = k 21 a 2, where k 21 s a scalar that does not depend on the postons, veloctes or nternal states of the two bodes Need to choose some body as standard and assgn a mass (e.g. 1 kg). We can then defne mass of any other body relatve to t by k 21 = m 2 /m 1. We can also assume that: In a system wth more than two bodes, the acceleraton of any one of the bodes s equal to the vector sum of the acceleratons t feels due to each of the other bodes ndvdually m 1 a 1 = F 12 + F 13 m 2 a 2 = F 21 + F 23 = - F 12 + F 23 m 3 a 3 = F 31 + F 32 = - F 13 F 23 Now add three equatons: m 1 a 1 + m 2 a 2 + m 3 a 3 = 0 If bodes 2 and 3 are bound together, a 2 = a 3 m 1 a 1 = - (m 2 + m 3 )a 2