PHYS 422 Lecture 20 Coupled Oscillations and Normal Modes III

PHYS 422 Lecture 20 Coupled Oscillations and Normal Modes III

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l d O ill ti d Coupled Oscillations and ormal odes III Normal Modes III
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Many Coupled Oscillators Real physical systems contain many particles and can often be described as a coupled set of oscillators Consider the following system fixed fixed l initial tension let’s restrict ourselves to small transverse 0 N+1 2 1 N N-1 3 T displacements of the masses α 2 T T 1 cos l l α = l α 1 for α 1 small 2 1 1 cos 1 2 ≈− 0 2 1 3 2 1 1 2 ll ⎛⎞ ⇒≈ + ⎜⎟ ⎝⎠ 2 1 2 l l Δ= any increased tension due to Δ l is negligible
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Many Coupled Oscillators T t’s look at particle 2 l’ α 1 α 2 T let s look at particle 2 the x component of the force is 22 2 1 2 1 cos cos () TT T α αα +− ± 02 13 12 2 very small term ignore id l di l t ( << consider only y displacements (y << l ) y component of the force on the p th particle is n sin T T + the sines are given by 1 sin p pp FT =+ 1 y y y + so the force is 11 1 sin sin p p yy ll −+ = = T ( ) ( ) p p p F y y =− +
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Many Coupled Oscillators T plying Newton II l’ α 1 α 2 T applying Newton II () 2 11 2 2 0 p pp p dy TT yy y dt ml ml +− + −+ = 02 13 2 0 T ml ω≡ we can write similar equations for each particle from 1 to N (y 0 = 0, y N+1 = 0) ecial cases: special cases: N=1 m 2 d y 2 T T T 2 1 01 2 20 y dt ω + = 0 2 ml ω= ll
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PHYS 422 Lecture 20 Coupled Oscillations and Normal Modes III

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