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z_chapter9momentum_part1of3

# z_chapter9momentum_part1of3 - z-axis ⊗ cm r r dm r r 6...

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1 Systems of Particles Center of Mass (CM) : We wish to show that a body or system of particles move as if 1. all of its mass were concentrated at one geometric point in space called the CM, and 2. the net external Force was applied there. How many particles are in this system? mg

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2 m 1 m 2 Consider a system of two point-particles. 2 1 2 2 1 1 m m x m x m x cm + This gives us the average position of the particles, weighted by their mass. ( It’s a weighted average). If m 1 = m 2 = m Then ft m m ft m ft m x cm 3 ) 4 ( ) 2 ( = + = , just as expected. x = 0 x = 2 ft x = 4 ft x-axis
3 So, for n-particles (discrete masses) = = = + + + + + + + + = n i i n i i i n n n cm m x m m m m m x m x m x m x m x 1 1 3 2 1 3 3 2 2 1 1 ... ... and = = = + + + + + + + + = n i i n i i i n n n cm m y m m m m m y m y m y m y m y 1 1 3 2 1 3 3 2 2 1 1 ... ...

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4 cm r r = the position of the CM of the system. j m y m i m x m r n i i n i i i n i i n i i i cm ˆ ˆ 1 1 1 1 = = = = + = r ( 29 + = = = n i i i i n i i cm j y i x m m r 1 1 ˆ ˆ 1 r cm r r 1 r r 2 r r n r r y-axis x-axis (x n , y n ) (x cm , y cm ) i r r
5 = = = n i i n i i i cm m r m r 1 1 r r for discrete masses ( point-particles) For a continuous distribution: = dm dm r r cm r r x-axis y-axis

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Unformatted text preview: z-axis ⊗ cm r r dm r r 6 Ex: (1-dimension) Find the position of the center of mass of a thin homogeneous rod of length L and mass M. (ex: a meter stick) ∫ ∫ = dm dm x x cm the linear mass density λ is: length mass = λ x m = x m = so dm = λ dx ∫ ∫ = dx dx x x cm (limits?) x-axis dm x M, L 7 We can also find an objects CM by hanging it from a string. We can show that the CM of a body moves according to Newton’s 2 nd Law: 8 To do this, consider a system of n-particles (connected by tiny, massless, rigid rods) on which a quick force F r acts. How would this system of particles move? What if n was large? Is Σ F = ma ? Which m? Which a? x-axis y-axis ⊗ CM m n m 2 m 1 F r cm r r x-axis y-axis ⊗ CM ruler F r ~ 10 23 particles...
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z_chapter9momentum_part1of3 - z-axis ⊗ cm r r dm r r 6...

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