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

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