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Corresponding Relation Between SHM and UCMThe simple harmonic motion is the side view of circular motion.Draw x-t Diagram Using Circle of ReferenceExampleUsing the phasorExampleExample: π=+0v >ExampleExampleExampleExample:A wooden block floats in water. We press it until its upper surface2Example ——Vertical SHM:Suppose we hang a spring with force constant k and suspend from it a body with mass m. Oscillation will now be When the body hangs at rest, in equilibriumTake x=0 to be the equilibrium position, andThe body ’s motion is still SHMwith the angular frequency:kmω=ExampleWhen the body is at the position x, the total 0=)0l mg ∆−=0Example cont ’d§6 Damped OscillationsThe dissipative force causes the decrease in amplitude ——damping, the corresponding motion is called damped oscillation.Restoring force:Resistance force:Newton ’s second law:The solution:s F kx=−R bv=−F kx bv ma=−−=∑220d x b dx k x dt m dt m ++=(/2)cos()cos()b m t x Ae t A t ωφωφ−′=+=+22(/2)20, 22b m t k b b A Ae m m m ωω−⎛⎞⎛⎞′==−=−⎜⎟⎜⎟⎝⎠⎝⎠τ=2m /b is called damping time constant, or mean lift time.If the system no longer oscillates,and is called critically dampedDamped Oscillations Cont ’d§7 Forced OscillationsA forced oscillator is damped oscillator driven by an external force that varies periodically.A sinusoidally varying driving force:Newton’s Second Law:The solution:0()sin F t F t ω=202sin dxd xF t b kx m dt dt ω−−=202sin F d x b dx k x tdt m dt m m ω++=cos()x A t ωφ=+()022220/F mA b m ωωω=⎛⎞−+⎜⎟⎝⎠The forced oscillator in its “steady state”isoscillated with the frequency of driven force.The goblet breaks as it vibrates in the resonance In 1940, the Tacoma Narrows Bridge collapsed four months and six days after it was opened for traffic, due to gusty ResonanceUsing Circle of Reference12212cos()A A φφ+−1122122sin sin cos cos A A φφφφ++Chapter 12 Oscillatory Motion are in phase, the resultant amplitude take are out of phase, the resultant amplitude Chapter 12 Oscillatory MotionExample。
