This preview shows page 1. Sign up to view the full content.
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. ...
View
Full
Document
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.
 Winter '08
 BATES
 Physics, Buoyancy

Click to edit the document details