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_lec36_ppt_ - Lecture 36 P112 Announcements Final Exam...

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Lecture 36 P112 Apr 20, 2007
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Announcements ± Final Exam
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Agenda for today ± Energy in SHM ± Taylor Expansions ± Simple Pendulum
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Energy in SHM- ideal spring ± The total energy ( K + U ) of a system undergoing SHM will always be constant, since there are only conservative forces present, hence K+U energy is conserved. -A A 0 s U U K E
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Total mechanical Energy in SHM of an ideal spring ± Total mechanical energy is conserved: ± Inserting v(t)=- ± A sin( ± t + ² ) and x(t)= A cos ( ± t + ² ) : E tot = 1 2 mv 2 + 1 2 kx 2 1 2 mv 2 + 1 2 kx 2 = 1 2 kA 2 sin 2 ( ± t + ² ) + 1 2 kA 2 cos 2 ( ± t + ² ) = 1 2 kA 2 E tot = 1 2 kA 2
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Question ± Using energy considerations, derive a formula that relates the oscillation’s maximum speed, v MAX , to its amplitude A.
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Taylor Expansion ± TE is an approximation of a function as a sum of terms calculated from the values of its derivatives ± General form: ± Example: f ( n ) ( a ) n ! n = 0 ± ² ( x ³ a ) n sin( x ) ± x ² x 3 3! + x 5 5!
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