PHY 113 A General Physics I 11 AM – 12:15 PM TR Olin 101 Plan for Lecture 15:
PHY 113 C General Physics I 11 AM – 12:15 PM TR Olin 101 Plan for Lecture 16:
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Transcript of PHY 113 C General Physics I 11 AM – 12:15 PM TR Olin 101 Plan for Lecture 16:
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PHY 113 C Fall 2013 -- Lecture 16 110/22/2013
PHY 113 C General Physics I11 AM – 12:15 PM TR Olin 101
Plan for Lecture 16:Chapter 16 – Physics of wave motion
1. Review of SHM2. Examples of wave motion3. What determines the wave velocity4. Properties of periodic waves
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PHY 113 C Fall 2013 -- Lecture 16 210/22/2013
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PHY 113 C Fall 2013 -- Lecture 16 310/22/2013
Some comments on Simple Harmonic Motion
k
mktAtx
xmk
dtxddtxdmmakxF
2
2
2
2
2
ωthatprovided
)φω(cos)( :solution general
:equation aldifferenti
Suppose that you know: (in standard units)
x(t)=2 cos(2pt+3p)
What is k ? k = m(2p)2
m=1 kg
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PHY 113 C Fall 2013 -- Lecture 16 410/22/2013
Some comments on Simple Harmonic Motion
k
mktAtx
xmk
dtxddtxdmmakxF
2
2
2
2
2
ωthatprovided
)φω(cos)( :solution general
:equation aldifferenti
Suppose that you know: (in standard units)
x(t)=2 cos(2pt+3p)
What is the frequency of oscillations ? w=2pf=2p f=1 Hz
m=1 kg
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PHY 113 C Fall 2013 -- Lecture 16 510/22/2013
Some comments on Simple Harmonic Motion
k
mktAtx
xmk
dtxddtxdmmakxF
2
2
2
2
2
ωthatprovided
)φω(cos)( :solution general
:equation aldifferenti
Suppose that you know: (in standard units)
x(t)=2 cos(2pt+3p)
What is amplitude of the displacement ? xmax=2m
m=1 kg
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PHY 113 C Fall 2013 -- Lecture 16 610/22/2013
Some comments on Simple Harmonic Motion
k
mktAtx
xmk
dtxddtxdmmakxF
2
2
2
2
2
ωthatprovided
)φω(cos)( :solution general
:equation aldifferenti
Suppose that you know: (in standard units)
x(t)=2 cos(2pt+3p)
What is the maximum velocity ? v(t)=-2(2p) cos(2pt+3p) vmax=4p
m=1 kg
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PHY 113 C Fall 2013 -- Lecture 16 710/22/2013
Some comments on Simple Harmonic Motion
k
mktAtx
xmk
dtxddtxdmmakxF
2
2
2
2
2
ωthatprovided
)φω(cos)( :solution general
:equation aldifferenti
Suppose that you know: (in standard units)
x(t)=2 cos(2pt+3p)
What is the maximum acceleration ? a(t)=-2(2p)2 cos(2pt+3p) amax=8p2
m=1 kg
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PHY 113 C Fall 2013 -- Lecture 16 810/22/2013
Some comments on Simple Harmonic Motion
k
mktAtx
xmk
dtxddtxdmmakxF
2
2
2
2
2
ωthatprovided
)φω(cos)( :solution general
:equation aldifferenti
Suppose that you know: (in standard units)
x(t)=2 cos(2pt+3p)
What is the displacement at t=0.3 s ? x(0.3)=2 cos(2p(0.3)+3p) =2(0.309)=0.618 m
m=1 kg
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PHY 113 C Fall 2013 -- Lecture 16 910/22/2013
Some comments on driven Simple Harmonic Motion
k
)
)
)
mk
tmFtAtx
tFxmk
dtxd
dtxdmmatFkxF
2
220
02
2
2
2
0
ω where
sin/)φω(cos)( :solution general
sin :equation aldifferenti
sin
w
F(t)=F0sin(t)
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PHY 113 C Fall 2013 -- Lecture 16 1010/22/2013
Webassign question (Assignment 14)
Damping is negligible for a 0.175-kg object hanging from a light, 6.30-N/m spring. A sinusoidal force with an amplitude of 1.70 N drives the system. At what frequency will the force make the object vibrate with an amplitude of 0.430 m?
)mktmFtAtx
2
220 ω sin/)φω(cos)(
w
) 00 amplitude sin)( XtXtx
0
022
00
//
mXF
mk
mkmFX
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PHY 113 C Fall 2013 -- Lecture 16 1110/22/2013
The wave equation
Wave variableWhat does the wave equation mean?Examples Mathematical solutions of wave equation and
descriptions of waves
2
22
2
2
xyc
ty
),( txyposition time
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PHY 113 C Fall 2013 -- Lecture 16 1210/22/2013
Source: http://www.eng.vt.edu/fluids/msc/gallery/gall.htm
Example: Water waves
Needs more sophistocated analysis:
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PHY 113 C Fall 2013 -- Lecture 16 1310/22/2013
Mechanical waves occur in continuous media. They are described by a value (y) which changes in both time (t) and position (x) and are characterized by a wave velocity c: y=f(x-ct) or y=f(x+ct)
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PHY 113 C Fall 2013 -- Lecture 16 1410/22/2013
Waves on a string:
Typical values for c:
3x108 m/s light waves
~1000 m/s wave on a string
343 m/s sound in air
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PHY 113 C Fall 2013 -- Lecture 16 1510/22/2013
Transverse wave:
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PHY 113 C Fall 2013 -- Lecture 16 1610/22/2013
Longitudinal wave:
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PHY 113 C Fall 2013 -- Lecture 16 1710/22/2013
General traveling wave –
t = 0
t > 0
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PHY 113 C Fall 2013 -- Lecture 16 1810/22/2013
iclicker question:
t=0t=1 s t=2s
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PHY 113 C Fall 2013 -- Lecture 16 1910/22/2013
Basic physics behind wave motion --
example: transverse wave on a string with tension T and mass per unit length m
y
ABT
dtydx
xm
TTdtydm
xy
xyμ
μ
θsinθsin
2
2
AB2
2
qBy
x
BBB x
y θtanθsin
2
2
0 xy
xy1
xy
xLimABx
2
2
2
2
μ xyT
ty
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PHY 113 C Fall 2013 -- Lecture 16 2010/22/2013
The wave equation:
Solutions: y(x,t) = f (x ct)
2
22
2
2
xyc
ty
string) a(for
μ where Tc
function of any shape
) )
) ) )
) ) ) 22
22
2
2
2
2
2
22
2
2
2
2
Let :Note
cuuf
tu
uuf
tuf
uuf
xu
uuf
xuf
xu
uuf
xuf
ctxu
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PHY 113 C Fall 2013 -- Lecture 16 2110/22/2013
iclicker question Is it significant to write the wave equation with the special symbols?
A. YesB. No
2
22
2
2
xyc
ty
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PHY 113 C Fall 2013 -- Lecture 16 2210/22/2013
Examples of solutions to the wave equation:
Moving “pulse”:
Periodic wave:
)20),( ctxeytxy
) )φsin),( 0 ctxkytxy
“wave vector”not spring constant!!!
ωπ2π2λπ2
fT
kc
k
cTT
txytxy
λ φλ
π2sin),( 0
phase factor
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PHY 113 C Fall 2013 -- Lecture 16 2310/22/2013
cTT
txytxy
λ φλ
π2sin),( 0
Periodic traveling waves:
Amplitude
wave length (m)
period (s); T = 1/f
phase (radians)
velocity (m/s)
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PHY 113 C Fall 2013 -- Lecture 16 2410/22/2013
Snapshot of periodic wave at t=t0
Time plot of periodic wave at x=x0
l
1/f
f l = c
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PHY 113 C Fall 2013 -- Lecture 16 2510/22/2013
φ
λπ2sin),( φ
λπ2sin),( 00 T
txytxyTtxytxy leftright
Ttxytxtxy leftright
π2cosφλπ2sin2),(y),( 0
“Standing” wave:
Combinations of waves (“superposition”)
) ) BABABA 21
21 cossin2sinsin
: thatNote
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PHY 113 C Fall 2013 -- Lecture 16 2610/22/2013
Summary of wave properties:
constant) al(fundament /102.9979
:fields magnetic and electric coupled todue Light wave :Example
: density and pressureair with in waveSound :Example
:h mass/lengt and ion with tensstring aon Wave:Example
occurs. avein which w mediumand/or process on the depends speed Wave
8 smc
pcp
TcT
c
mm
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PHY 113 C Fall 2013 -- Lecture 16 2710/22/2013
Example from webassign:
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PHY 113 C Fall 2013 -- Lecture 16 2810/22/2013
cTT
txytxy
λ φλ
π2sin),( 0
Periodic traveling waves:
Amplitude
wave length (m)
period (s); T = 1/f
phase (radians)
velocity (m/s)
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PHY 113 C Fall 2013 -- Lecture 16 2910/22/2013
Example from webassign:
0
0
0
2 :speed e transversmaximum
φλ
π2cos2),(
λ φλ
π2sin),(
yT
Ttxy
Tttxy
cTT
txytxy
p
p