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PHYSICS 231 Lecture 34: Oscillations & Waves . Period T 6 3 2 Frequency f 1/6 1/3 ½ (m/k) 6/(2) 3/(2) 2/(2) (2)/6 (2)/3 (2)/2 . Remco Zegers Question hours: Thursday 12:00-13:00 & 17:15-18:15 Helproom. Harmonic oscillations vs circular motion. v 0. t=0. t=1. - PowerPoint PPT Presentation
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PHY 2311
PHYSICS 231Lecture 34: Oscillations & Waves
Remco ZegersQuestion hours: Thursday 12:00-13:00 & 17:15-
18:15Helproom
Period T 6 3 2Frequency f 1/6 1/3 ½(m/k) 6/(2) 3/(2) 2/(2) (2)/6 (2)/3 (2)/2
km
fT
221
PHY 2312
Harmonic oscillations vs circular motiont=0 t=1 t=2
t=3 t=4
v0=r=A
v0
=t=t
A
v0
vx
PHY 2313
time (s)
A
-A
-kA/m
kA/m
velocity v
a
x
A(k/m)
-A(k/m)
xharmonic(t)=Acos(t)
vharmonic(t)=-Asin(t)
aharmonic(t)=-2Acos(t)
=2f=2/T=(k/m)
PHY 2314
Another simple harmonic oscillation: the pendulum
Restoring force: F=-mgsinThe force pushes the mass mback to the central position.
sin if is small (<150) radians!!!
F=-mg also =s/Lso: F=-(mg/L)s
PHY 2315
pendulum vs spring
parameter spring pendulum
restoring force F
F=-kx F=-(mg/L)s
period T T=2(m/k) T=2(L/g)*
frequency f f=(k/m)/(2)
f=(g/L)/(2)
angular frequency
=(k/m) =(g/L)
* gL
LmgmT 2/
2
PHY 2316
example: a pendulum clockThe machinery in a pendulum clock is keptin motion by the swinging pendulum.Does the clock run faster, at the same speed,or slower if:a) The mass is hung higherb) The mass is replaced by a heavier massc) The clock is brought to the moond) The clock is put in an upward accelerating
elevator?L m moon elevato
rfaster same slower g
LT 2
PHY 2317
example: the height of the lecture room
demo
22
2
25.04
2
TgTL
gLT
PHY 2318
damped oscillationsIn real life, almost all oscillations eventually stop due to frictional forces. The oscillation is damped. We can alsodamp the oscillation on purpose.
PHY 2319
Types of damping
No dampingsine curve
Under dampingsine curve with decreasingamplitudeCritical dampingOnly one oscillations
Over dampingNever goes through zero
PHY 23110
Waves
The wave carries the disturbance, but not the water
Each point makes a simple harmonic vertical oscillation
position x
position y
PHY 23111
Types of waves
Transversal: movement is perpendicular to the wave motion
waveoscillation
Longitudinal: movement is in the direction of the wave motion
oscillation
PHY 23112
A single pulse
velocity v
time to time t1
x0 x1
v=(x1-x0)/(t1-t0)
PHY 23113
describing a traveling wave
While the wave has traveled onewavelength, each point on the ropehas made one period of oscillation.
v=x/t=/T= f
: wavelengthdistance betweentwo maxima.
PHY 23114
example2m A traveling wave is seen
to have a horizontal distanceof 2m between a maximumand the nearest minimum andvertical height of 2m. If itmoves with 1m/s, what is its:a) amplitudeb) periodc) frequency
2m
a) amplitude: difference between maximum (or minimum) and the equilibrium position in the vertical direction (transversal!) A=2m/2=1mb) v=1m/s, =2*2m=4m T=/v=4/1=4sc) f=1/T=0.25 Hz
PHY 23115
sea wavesAn anchored fishing boat is going up and down with thewaves. It reaches a maximum height every 5 secondsand a person on the boat sees that while reaching a maximum, the previous waves has moves about 40 m awayfrom the boat. What is the speed of the traveling waves?
Period: 5 seconds (time between reaching two maxima)Wavelength: 40 m
v= /T=40/5=8 m/s
PHY 23116
Speed of waves on a string
LM
Fv
/
F tension in the string mass of the string per unit length (meter)
example: violin
L M
screwtension T
v= /T= f=(F/)
so f=(1/)(F/) for fixed wavelength the frequency willgo up (higher tone) if the tension is increased.
PHY 23117
exampleA wave is traveling through thewire with v=24 m/s when thesuspended mass M is 3.0 kg.a) What is the mass per unit length?b) What is v if M=2.0 kg?
a) Tension F=mg=3*9.8=29.4 N v=(F/) so =F/v2=0.05 kg/m b) v=(F/)=(2*9.8/0.05)=19.8 m/s
PHY 23118
bonus ;-)
The block P carries out a simple harmonic motion with f=1.5HzBlock B rests on it and the surface has a coefficient ofstatic friction s=0.60. For what amplitude of the motion doesblock B slip?
The block starts to slip if Ffriction<Fmovementsn-maP=0smg=maP so sg=aP ap= -2Acos(t) so maximally 2A=2fAsg=2fA A= sg/2f=0.62 m
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