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Summary Oscillations Free Oscillations: Dierential Equation: Solution: d2 x dt2 + 2x = 0 x(t) = A cos(t + ) Common Oscillators: Mass on Spring: Simple Pendulum: Physical Pendulum: Uncommon Oscillators: Damped Oscillations: Solution: x(t) = A0 e 2m cos(t + ) A(t) = A0 e 2m = 2 0 ( 2b )2 m bt bt 0 = 0 = k m g L mgd I 0 = Use Forces or Work/Energy to obtain 0 The amplitude is time-dependent ! Frequency Shift ! System at oscillates Classication: if is real, the system is underdamped. If it is imaginary, the system is overdamped. If it is zero, the system is critically damped. Energy (Lightly Damped): E = 1 kA(t)2 2 Driven Oscillations: Solution: x(t) = F0 2 m2 (0 2 )2 +(b )2 (or similar) cos( t + ) is the angular frequency of the driver. Phase dierence between driver and oscillation: resonant = 0 b 1 2( 2m0 )2 tan( ) = b 2 m(0 2 ) ... View Full Document