Midterm_1_EQ

Midterm_1_EQ - L x ae cos( w n t + f n ) n = np L T m f n =...

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Physics 1B MIDTERM 1 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 2 = k M T = 2 p w 0 f = 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 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 0 2 = k M ν = b M x ( t ) = x 0 e - gt cos w d t γ = 2 d 2 = w 0 2 - n 2 4 t damp = 1 g (i) overdamping 0 < n 2 no oscillations (ii) critical damping 0 = n 2 no oscillations; fastest decay (iii) underdamping 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 1
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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 ae è ç ö ø ÷ K is maximized when max = w 0 2 - n 2 2 Q (quality) factor Q = w 0 w » w 0 n w = width at K max 2 higher Q = sharp peak Waves – STANDING z ( x , t ) = A sin( kx )cos( wt ) kl = 2 p = kv v = fl T P = 2 p w L = nl 2 k = np L f n = nf 1 = n 2 L T m z ( x , t ) = z n sin np
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Unformatted text preview: L x ae cos( w n t + f n ) n = np L T m f n = n 2 L T m T P = 1 f Wave Equation 2 z t 2 = T 2 z x 2 2 z t 2 = v 2 2 z x 2 TRAVELING Equation for a moving point: x = x + vt Wave speed v = T m = kv T P = l v = 2 p w k = 2 p l f = v l z ( x , t ) = z sin[ k ( x vt )] = z sin[ kx wt )] +v : moving right-v : moving left In general z ( x , t ) = z sin kx cos wt = 1 2 z sin( kx + wt ) + sin( kx-wt ) { } = F ( kx + wt ) + G ( kx + wt ) 2...
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This note was uploaded on 10/15/2009 for the course PHYS 318007200 taught by Professor Cowley during the Spring '08 term at UCLA.

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Midterm_1_EQ - L x ae cos( w n t + f n ) n = np L T m f n =...

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