1 PHY41M1 CLASSICAL MECHANICS M Chirwa, Chemical & Physical Sciences Department, Faculty of Natural Sciences, Walter Sisulu University, Nelson Mandela Drive, P/Bag X1, Mthatha 5117, South Africa [email protected], [email protected](12 March 2017 – 29 June 2017) 1. Mechanics of a System of Particles 1.1. Equations of motion In a system of particles, for the ithparticle with mass imand velocity tiiiddrrvr&rr=≡at position irr, the equation of motion is ijiiitFFpprrr&r+==dd(1.1) where ip&ris the time rate of change of the momentum iiimrp&rr=, and iFris the external force on the ithparticle, while ijFris the internal force on the ithparticle due to another (jth) particle in the same system. Note that 0Frr=iias a particle cannot exert a force onto itself. Furthermore, to satisfy Newton’s third law of motion, the equality of action-reaction forces, we must have the relation: 0FFrrr=+jiij(1.2) Then for the entire system of particles, the equation of motion is found by summing over all the particles in the system ()FFFFFFFrpPrrrrrrrr&r&r∑∑∑∑∑∑∑=≡++≡+===jiiijiijiijiijiiiiiiimt,,2221dd(1.3) The second equivalence is after applying equation (1.2). 1.2. Moments as composite physical quantities1.2.1. Moment of a scalar quantity The moment of a scalar physical quantity biof a single particle located at position irris defined as the vector iibrrThen for a system of such particles characterized by the overall physical quantity ∑=iibB(1.4)
2 the corresponding overall moment of Bis ∑=iiibBrRrr(1.5) where the position vector Rris known as the centreof the quantity Bof the system. For a system where the scalar quantity Bis continuously distributed, the above summations become integrals: ∫=bBd (1.6) and ∫=bBdrRrr(1.7) If the elemental quantity is the mass =vmddddρσλal(1.8) with line λ, surface σand volume ρdensities, then the total mass is ∫=mMd(1.9) while the overall mass momentis ∫=mMdrRrr(1.10) where vector Rris the position vector of the centre of mass (com) relative to the origin. Fig.1.1:A system’s ithobject has mass imand the positions irrand ir′rrelative to an origin O and the system’scomrespectively. O comimirrRrirr′
3 Substituting (1.10) into (1.3) yields FRPr&&r&r==M(1.11) Thus the centre of mass (com) moves as if the total external force were acting on the entire mass concentrated at the com. Hence internal forces in a system have no effect on the motion of the com. The total linear momentum is MmtiiiRrP&rrr==∑dd(1.12) That is, it is equal to the velocity R&rof the comtimes the total mass M.