phy2048-ch9 - Chapter 9 Center of mass and linear momentum...

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1 Chapter 9 – Center of mass and linear momentum I. The center of mass - System of particles / - Solid body II. Newton’s Second law for a system of particles III. Linear Momentum - System of particles / - Conservation IV. Collision and impulse - Single collision / - Series of collisions V. Momentum and kinetic energy in collisions VI. Inelastic collisions in 1D -Completely inelastic collision/ Velocity of COM VII. Elastic collisions in 1D VIII. Collisions in 2D IX. Systems with varying mass X. External forces and internal energy changes I. Center of mass The center of mass of a body or a system of bodies is a point that moves as though all the mass were concentrated there and all external forces were applied there. - System of particles: M x m x m m m x m x m x com 2 2 1 1 2 1 2 2 1 1 + = + + = General: - The center of mass lies somewhere between the two particles. - Choice of the reference origin is arbitrary b Shift of the coordinate system but center of mass is still at the same relative distance from each particle. M = total mass of the system
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2 I. Center of mass - System of particles: Origin of reference system coincides with m 1 d m m m x com 2 1 2 + = 3D: = = = = = = n i i i com n i i i com n i i i com z m M z y m M y x m M x 1 1 1 1 1 1 = = n i i i com r m M r 1 1 a a - Solid bodies: Continuous distribution of matter. Particles = dm (differential mass elements). 3D: M = mass of the object Assumption: dV dm V M ρ = = The center of mass of an object with a point, line or plane of symmetry lies on that point, line or plane. = = = dV z V z dV y V y dV x V x com com com 1 1 1 Uniform objects b uniform density The center of mass of an object does not need to lie within the object. Examples: doughnut, horseshoe = = = dm z M z dm y M y dm x M x com com com 1 1 1 Volume density Linear density: λ = M / L b dm = λ dx Surface density: σ = M / A b dm = σ dA
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3 Problem solving tactics: (1) Use object’s symmetry. (2) If possible, divide object in several parts. Treat each of these parts as a particle located at its own center of mass. (3) Chose your axes wisely. Use one particle of the system as origin of your reference system or let the symmetry lines be your axis. II. Newton’s second law for a system of particles It moves as a particle whose mass is equal to the total mass of the system. Motion of the center of mass: com net a M F a a = - F net is the net of all external forces that act on the system. Internal forces (from one part of the system to another are not included). -
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phy2048-ch9 - Chapter 9 Center of mass and linear momentum...

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