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Final_EQ

# Final_EQ - Physics 1B Simple Harmonic Motion Fspring = kx...

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Physics 1B FINAL EQUATIONS Kyle Chin Simple Harmonic Motion F spring = - kx M d 2 x dt 2 = - kx x ( t ) = A cos[ w 0 t ] x ( t ) = B sin[ w 0 t ] or F total = - Mg + kd d 2 x dt 2 = - w 0 2 x x ( t ) = A cos[ w 0 t ]+ B sin[ w 0 t ] x ( t ) = x (0)cos[ w 0 t ]+ v (0) w 0 sin[ w 0 t ] A = x (0) B = v (0) w 0 ϖ 0 = k M T = 2 p w 0 f = 1 2 p k M = 1 T Conservation of Energy : 1 2 Mv 2 + 1 2 kx 2 = 1 2 kx max 2 Pendulum θ ( t ) = q (0)cos[ w 0 t ]+ ú q (0) w 0 sin[ w 0 t ] d 2 q dt 2 = - g l sin q ϖ 0 = g l T = 2 p l g f = 1 2 p g l Damped Oscillations F D = - bv M d 2 x dt 2 = - b dx dt - kx d 2 x dt 2 + n dx dt + w 0 2 x = 0 ϖ 0 2 = k M ν = b M x ( t ) = x 0 e - gt cos w d t γ = ν 2 = b 2 M ϖ d 2 = w 0 2 - n 2 4 t damp = 1 g (i) overdamping ϖ 0 < n 2 no oscillations (ii) critical damping ϖ 0 = n 2 no oscillations; fastest decay (iii) underdamping ϖ 0 > n 2 oscillations (decreasing amplitude) Driven Oscillations M d 2 x dt 2 = - b dx dt - kx + F 0 cosW t d 2 x dt 2 + n dx dt + w 0 2 x = a 0 cosW t

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2 ν = b M ϖ 0 2 = k M a 0 = F 0 M = forcing angular frequency x ( t ) = x P ( t ) + x H ( t ) where x H ( t ) dies away over time x P ( t ) = K cos[W t + d ] K = a 0 ( w 0 2 - W 2 ) 2 + n 2 W 2 δ = tan - 1 - n W w 0 2 - W 2 æ è ç ö ø ÷ K is maximized when
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Final_EQ - Physics 1B Simple Harmonic Motion Fspring = kx...

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