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Unformatted text preview: Oscillations Summary Free Oscillations: Dierential Equation: Solution:d2 x dt2+ 2x = 0x(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 mbt bt0 = 0 =k m g L mgd I0 =Use Forces or Work/Energy to obtain 0 The amplitude is time-dependent ! Frequency Shift ! System oscillates at 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 = 0b 1 2( 2m0 )2tan( ) =b 2 m(0 2 )...
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- Winter '10