Lecture9-SHM - So far … •  fluids: Density, pressure, buoyancy •  Equa9on of con9nuity& Bernoulli’s equa9on today

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Unformatted text preview: So far … •  fluids: Density, pressure, buoyancy •  Equa9on of con9nuity & Bernoulli’s equa9on today •  simple harmonic mo9on •  defini9ons of period (T), frequency (f), angular frequency (ω) Note: office hours TODAY shiKed to 4:30 – 5:30 PM in Hennings Rm. 334 Oscilla9ons •  Demo: Mass swinging at the end of a spring. •  Why is this interes9ng? •  Model system for everything that oscillates and vibrates, for example: –  Molecules –  Loudspeakers –  Buildings and Bridges video: Tacoma Narrows Bridge position plot x(t ) = A cos(ωt + φ ) Question What is the amplitude of the shown harmonic motion ? A)  -4.0 m B)  2.0 m C)  4.0 m D)  8.0 m E)  12.0 m Question What is the period of the shown harmonic motion ? A)  1.0 s B)  2.5 s C)  4.0 s D)  5.0 s E)  6.0 s Mass oscilla9ng at the end of a spring •  Observa9on: Mo9on repeats, periodic mo9on •  Do you know other examples of periodic mo9on?  rota9ng wheel •  What do these mo9ons have in common, or in other words: What makes a mo9on periodic? •  Where does this equa.on come from? x(t ) = A cos(ωt + φ ) Simple Harmonic Mo9on (SHM) and Circular Mo9on x(t ) = A cos(ωt + φ ) We can use rota9onal variables to describe uniform circular mo9on: period T, frequency f, and angular speed (frequency) ω •  1 revolu9on = 2 π rad •  The period T is the 9me it takes for one revolu9on. •  Frequency f = number of revolu9ons per unit 9me φ ω = dφ dt worksheet Observe the shadow on the wall and draw a mo9on diagram for the ball: Label each posi9on with a 9ck from 0 to 16. φ ω = dφ dt position plot x(t ) = A cos(ωt + φ ) write an equation for harmonic motion x = A cos ! x (t ) = A cos(! t ) consider INITIAL angle (or phase) x (t ) = A cos(! t + "0 ) position & velocity plots x(t ) = A cos(ωt + φ ) d x(t ) d v(t) = = ( A cos(ωt + φ ) ) dt dt v(t) = − (ωA) sin(ωt + φ ) v max ≡ ωA When the displacement is maximum (± A) the velocity is zero. When the velocity is maximum (± vmax ) the displacement is zero worksheet Consider the yellow dot on a rotating bicycle wheel. At t = 0 the yellow dot has the position indicated in the figure. y x Bicycle Wheel y Consider the yellow dot on a rotating bicycle wheel. At t = 0 the yellow dot has the position indicated in the figure. x a)  Which of the graphs corresponds to x position versus time? A B C D E. none of these Bicycle Wheel y Consider the yellow dot on a rotating bicycle wheel. At t = 0 the yellow dot has the position indicated in the figure. x a)  Which of the graphs corresponds to angular position versus time? A B C D E. none of these Bicycle Wheel y Consider the yellow dot on a rotating bicycle wheel. At t = 0 the yellow dot has the position indicated in the figure. x a)  Which of the graphs corresponds to y velocity versus time? A B C D E. none of these position & velocity & acceleration plots x(t ) = A cos(ωt + φ ) d x(t ) d v(t) = = ( A cos(ωt + φ ) ) dt dt v(t) = − (ωA) sin(ωt + φ ) d v(t ) d a(t) = = (− ωA sin(ωt + φ ) ) dt dt a(t ) = ! (! 2 A) cos(! t + ! ) = !" 2 [ x(t )] SHM: sign of acceleration is always opposite to the sign of the displacement. ...
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This note was uploaded on 01/15/2012 for the course PHYS 101 taught by Professor Bates during the Winter '08 term at The University of British Columbia.

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