Chapter_9 - Chapter 9 Center of Mass and Linear Momentum In...

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Chapter 9 Center of Mass and Linear Momentum In this chapter we will introduce the following new concepts: -Center of mass ( com ) for a system of particles -The velocity and acceleration of the center of mass - Linear momentum for a single particle and a system of particles We will derive the equation of motion (Newton’s 2 nd law) for the center of mass, and discuss the principle of conservation of linear momentum Finally we will use the conservation of linear momentum to study collisions in one and two dimensions and derive the equation of motion for rockets
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1 2 1 1 2 1 1 2 2 2 1 2 2 Consider a system of two particles of masses and at positions and . We define the position of the center of mass (com) as follow ( ) / 2 s: + + = + The Center of Mass : com x x i m x m x x x m m m m x x 1 2 1 2 = f m m if m m = 1 1 2 2 3 3 1 1 2 3 2 3 1 ... We can generalize the above definition for a system of particles as follows: Here is the total mass ... We can generalize the def 1 ... = + + + + = = + + + = + + + + + n n n n com i i i n m x m x m x m x x m x m m m m M M n M m m m m 1 inition for the center of mass of a system of particles in three dimensional space. We assume that the -th particle has a mass and its position vector is 1 = = n com i i i i i i m r r m r M r r r (9-2)
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1 1 The position vector for the center of mass is given by the equation: ˆ ˆ ˆ The position vector can be written as: The components of are given by the n com i i i com com com com com r m r M r x i y j z k r = = = + + r r r r 1 1 1 1 1 1 equations: n n n com i i com i i com i i i i i x m x y m y z m z M M M = = = = = = The use of center of mass simplifies description of motions of multiple particles because it has the following property: The com of a system of particles moves as if all the mass is concetrated at th This statement will be proved later. For example, A baseball bat is flipped into the air and moves under the influence of the gravitati e com, and the sum of all external forces is applied at the com. on force. The com (black dot) follows a parabolic path as discussed in Ch4 (projectile motion). All the other points of the bat follow more complicated paths. (9-3)
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A solid body can be considered as a system with continuous distribution of mass The sums used for calculating com of discrete systems now become integrals: The Center of Mass for Solid Bodies i i i m x The integrals above are rather complicated. If we assume constant density 1 1 1 1 1 1 , a nd = = = = = = = = com com com com com xdm x xdm y ydm z zdm M M M x x dV xdV dm dV M V y ydV V V V ρ For objects with symetry elements (symmetry point, symmetry line, symmetry plane) it is necessary to evaluate the integrals. The center of mass lies on the symm 1 etry element ( = not geome com z zdV V ). For example the com of a uniform sphere is at the center, the com of a uniform
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Chapter_9 - Chapter 9 Center of Mass and Linear Momentum In...

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