Classical Mechanics
                   LECTURE21:
        SYSTEMSOFPARTICLES
      ANDMOMENTOFINERTIA
                      Prof. N. Harnew
                    University of Oxford
                          HT2017
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  OUTLINE:21. SYSTEMSOFPARTICLESANDMOMENT
                               OFINERTIA
       21.1 NII for system of particles - translation motion
          21.1.1 Kinetic energy and the CM
       21.2 NII for system of particles - rotational motion
          21.2.1 Angular momentum and the CM
       21.3 Introduction to Moment of Inertia
          21.3.1 Extend the example : J not parallel to ω
          21.3.2 Moment of inertia : mass not distributed in a plane
          21.3.3 Generalize for rigid bodies
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                   21.1 NII for system of particles - translation motion
                 Reminder from MT lectures:
                       ◮ Force on particle i:                                           m d2(r ) = F ext +F int
                                                                                             i  dt2          i                i                   i
                                N                  2                           N                                           N
                             X d                                             X                                          X                                       P
                       ◮               mi               (r ) =                         Fext +                                     Fint                    = NFext
                                              dt2            i                             i                                           i                               i        i
                             |i             {z                }              |i {z               }                      |i {z }
                                    all masses                           external forces                      internal forces = zero
                                    P mr
           ◮ r               = N i i
                    CM                    i        M
                                                   P
                 where M =                               N m
                                                         i         i
                                                       P mr˙
           ◮ v                =r˙                = N i i
                     CM                CM                     i       M
                                               P
                 → P = Nmir˙ =Mv
                               CM                     i             i                    CM
                                                       P m¨r                          Fext
           ◮ a                =¨r               = N i i = i
                    CM                 CM                    i        M                  M
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                   21.1.1 Kinetic energy and the CM
                                           P
         ◮ Lab kinetic energy :     T = 1    N mi v2 ; v = v′ +v
                                         2   i      i      i    i    CM
            where v′ is velocity of particle i in the CM
                     i
         ◮ T = 1 P m v′2 + 1 P m v2          +Pmv′·v
                 2   i  i  i    2   i  i  CM      i  i  i   CM
                 P                  Pmv′
         ◮ But       mv′·v       = i i i ·Mv
                    i  i i   CM        M          CM
                                   |   {z  }
                                      =0
       ◮   T =T′+1Mv2
                     2    CM
         Sameexpression as was
         derived in MT
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