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R01_a (1) - REVIEW Oscillations Waves Electric Force Field...

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REVIEW: Oscillations, Waves, Electric Force, Field, and Flux Oscillations: Hooke’s Law: kx F x = (linear restoring force) Simple Harmonic Motion: o Displacement obeys differential equation x dt x d 2 2 2 ω = o For mass and spring, m k = 2 ω Solution of differential equation is: ( ) ( ) φ ω + = t A t x cos o Important: argument ( ) φ ω + t must be in radians o Constants A , ω , and φ must be found from details of motion
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Relations between Position, Velocity, and Acceleration in SHM ( ) ( ) A x t A t x + = φ ω cos ( ) ( ) A ω v t x t A dt dx t v x x + = = ), vs of (slope sin φ ω ω ( ) ( ) 2 2 2 2 ), vs of (slope cos A ω a t v t A dt x d t a x x x + = = φ ω ω
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Period: time in which phase increases by π 2 o ω π 2 = T and π ω 2 1 = = T f Energy for mass m on spring k so that m k / 2 = ω o ( ) φ ω + = t A x cos and ( ) φ ω ω + = t A v sin o Kinetic Energy: ( ) φ ω + = = t kA mv K 2 2 2 sin 2 1 2 1 o Potential Energy: ( ) φ ω + = = t kA kx U 2 2 2 cos 2 1 2 1 o Mechanical Energy: ( ) ( ) ( ) 2 2 2 2 2 1 cos sin 2 1 kA t t kA U K E = + + + = + = φ ω φ ω
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Learned how to get ( ) A and , , φ ω from description of specific motion o Angular frequency ( ) ω depends on properties of oscillator i.e. m k / = ω for spring/mass system o Get Amplitude and phase constant from x and v at specific time (i.e. initial displacement and velocity) 2 2 2 ω i i v x A + = i i x v ω φ = tan
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Simple Pendulum o for small θ , θ θ L g dt d = 2 2 o Looks like SHM with L g = 2 ω Note, does not depend on m ! o Period is g L T π ω π 2 2 = = Physical Pendulum o for small θ , θ θ I mgd dt d = 2 2 o Looks like SHM with I mgd = 2 ω
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Underdamped Oscillation: o ( ) φ ω + = t Ae x m bt cos 2 / o 2 2 0 2 = m b ω ω Resonance: o Periodic applied force ( ) t F t F ω sin 0 = o Damping force v b R r r = o Resulting oscillation is at driving frequency ( ) ( ) φ ω ω + = t A x cos o Frequency dependent amplitude is ( ) ( ) ( ) 2 2 2 0 2 0 2 / / m b m F A + = ω ω ω
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WAVES: Wavefunction: o ( ) ( ) vt x f t x y = , for wave going to right with speed v o ( ) ( ) vt x f t x y + = , for wave going to left with speed v Sinusoidal wave: two ways to represent.
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