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Lecture 5

# Lecture 5 - Lecture 5 SHM Pendulum Damping and Resonance Ch...

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April 15, 2008 Physics 40B Lecture 5 1 Review: Simple Harmonic Motion (SHM) of Mass-Spring System Hooke' s Law F s = ! kx U s = 1 2 kx 2 " = k m d 2 x dt 2 + 2 x = 0 (differential equation) Solution: x = A cos( t + # ) \$ T = 2 % = 2 m k and f = 1 T = 1 2 k m v = dx dt v max = A a = dv dt = !" 2 x a max = 2 A Initial Conditions: x = x o , v = v o at t = 0 tan ! = " v o x o A = x o 2 + v o \$ % & 2 Lecture 5 SHM: Pendulum, Damping and Resonance Ch. 13:5-8 Same equations for vertical oscillations with mass & spring. Use k = mg/ Δ L for spring constant where L is initial displacement in quilibrium positon due to mass.

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April 15, 2008 Physics 40B Lecture 5 2 Energy of Mass/Spring in SHM E = K + U = 1 2 mv 2 + 1 2 kx 2 E = 1 2 m ! A sin( t + " ) ( ) 2 + 1 2 k A cos( t + ) ( ) 2 Substituting 2 = k m E = 1 2 kA 2 is constant K = 1 2 m A sin( t + ) ( ) 2 = E sin 2 t + ( ) and U = E cos 2 t + ( ) K = E # U = 2 # 1 2 2 = 1 2 mv 2 \$ v = ± k m 2 # x 2 ( ) Review: Energy of Mass/Spring System
April 15, 2008 Physics 40B Lecture 5 3 Simple Pendulum UCR Physics Foucault Pendulum: L~25 ft~8 m T = 2 π (8/9.8) 1/2 =5.7 s " = I #\$" = % mg sin( & ) L Use small angle approx.

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Lecture 5 - Lecture 5 SHM Pendulum Damping and Resonance Ch...

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