PHY41M1 CLASSICAL MECHANICS.20 March 2018.pdf - PHY41M1 CLASSICAL MECHANICS M Chirwa Chemical Physical Sciences Department Faculty of Natural Sciences

# PHY41M1 CLASSICAL MECHANICS.20 March 2018.pdf - PHY41M1...

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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 i th particle with mass i m and velocity t i i i d d r r v r & r r = at position i r r , the equation of motion is ij i i i t F F p p r r r & r + = = d d (1.1) where i p & r is the time rate of change of the momentum i i i m r p & r r = , and i F r is the external force on the i th particle, while ij F r is the internal force on the i th particle due to another ( j th ) particle in the same system. Note that 0 F r r = ii as 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: 0 F F r r r = + ji ij (1.2) Then for the entire system of particles, the equation of motion is found by summing over all the particles in the system ( ) F F F F F F F r p P r r r r r r r r & r & r = + + + = = = j i i i ji ij i i j i ij i i i i i i i m t , , 2 2 2 1 d d (1.3) The second equivalence is after applying equation (1.2). 1.2. Moments as composite physical quantities 1.2.1. Moment of a scalar quantity The moment of a scalar physical quantity b i of a single particle located at position i r r is defined as the vector i i b r r Then for a system of such particles characterized by the overall physical quantity = i i b B (1.4)
2 the corresponding overall moment of B is = i i i b B r R r r (1.5) where the position vector R r is known as the centre of the quantity B of the system. For a system where the scalar quantity B is continuously distributed, the above summations become integrals: = b B d (1.6) and = b B d r R r r (1.7) If the elemental quantity is the mass = v m d d d d ρ σ λ a l (1.8) with line λ , surface σ and volume ρ densities, then the total mass is = m M d (1.9) while the overall mass moment is = m M d r R r r (1.10) where vector R r is the position vector of the c entre o f m ass ( com ) relative to the origin. Fig.1.1: A system’s i th object has mass i m and the positions i r r and i r r relative to an origin O and the system’s com respectively. O com i m i r r R r i r r
3 Substituting (1.10) into (1.3) yields F R P r & & 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 M m t i i i R r P & r r r = = d d (1.12) That is, it is equal to the velocity R & r of the com times the total mass M .
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