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Physics 2101 Section 6
November 8th: finish Ch.16
Lecture Notes: http://www.phys.lsu.edu/classes/fall2012/phys2101-6/
Announcement: • Exam # 3 (November 13th) Lockett 10 (6 – 7 pm) Nicholson 109, 119 (extra
time 5:30 – 7:30 pm) Covers Chs. 11.7-15
Transverse Traveling Wave
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y(x,0) = ymax sin kx( )
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k =2πλ
Spatially Periodic ( repeats ) : kλ = 2π
Wave number Wavelength
Transverse: Displacement of particle is perpendicular to the direction of wave propagation
Longitudinal: Displacement (vibration) of particles
is along same direction as motion of wave
Traveling Waves - they travel from one point to another Standing Waves - they look like they’re standing still
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vwave =dxdt
=ωk
=λT
= λf
Wave Speed
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vwave =τµ
= λf For transverse wave in physical medium
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k =2πλ
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ω =2πT
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phase : kx ±ωt
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kx +ωt⇒
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kx −ωt⇒ Wave traveling in + x direction
Wave traveling in - x direction
Interference Waves
Problem 16-‐33: Interference of Waves Two sinusoidal waves with the same amplitude ym=9.00 mm and the same wavelength λ travel together along a string that is stretched along the x axis. Their resultant wave is shown twice in the figure, as the valley A travels in the negative direction by a distance d=56.0 cm in Δt=8.0 ms. The tick marks along the x axis are separated by Δx=10 cm, and the height H is 8.0mm. Assume the first wave is given by Find (a) y’m, (b) k, (c) ω, (d) φ2, and the sign in front of ω.
y1(x,t) = ym sin(kx ±ωt)
The two waves have the same λ and kThey are in the same string so the velocity is the same
v =ωk
: so ω is same in both waves
y '(x,t) = 2ym cosφ2
2$
%&'
()sin(kx + ±1ωt ±2 ωt
2+φ2
2)
y 'm =H2= 2ym cos
φ22
cosφ22=
H4ymv ≡ ±1ω ±1 ω
2k=dΔt
need positive sign on both
ω =dkΔt
Standing Waves
Wavelength: Define:
Standing Waves
frequency
wave velocity
Standing Waves
Example
Solution: use
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T1 +T2 = 2T = Mg
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T1 = T2 =12Mg
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ν2 =T2µ2
=Mg2µ2
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ν1 =T1µ1
=Mg2µ1
Example -‐ con>nued
Solution:
use
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T1 = M1g
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T2 = M2g
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ν2 =T2µ2
=M2gµ2
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ν1 =T1µ1
=M1gµ1
Problem 16-‐58
b) If the mass of the block is m, what is the corresponding n?
Problem 16-‐49: Standing waves/resonances
A nylon guitar string has a linear density of µ=7.20 g/m and is under a tension of τ=150 N. The fixed supports are a distance D=90. cm apart. It oscillates with the pattern shown in the figure. Calculate the (a) speed, (b) wavelength, and (c) frequency of wave.
(a) Speed of wave
v =τµ=
150N7.20x10−3kg / m
= 144.3ms
(b) Wavelengt λ: look at figure
D =3λ2
: λ= 23D = 0.60m
(c) frequency f= vλ
f =144.3m / s
0.6m= 241Hz
Problem 16-‐59
Solu>on for Problem 16-‐59
From
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m = ρV = ρAL
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ν2 =τµ2
=τρ2A
Problem 16-‐11
From
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ν 2 =τ 2µ2
=τ 2µ
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ν1 =τ1µ1
=τ1µ
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L1 = L =n1λ12
=n1v12 f
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L2 = L =n2λ22
=n2v22 f
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n1v1 = n2v2
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n1n2
=v2v1
=τ 2τ1
frequency
wave speed
Chapter 18: Temperature, Heat, and Thermodynamics
Definitions
“System”- particular object or set of objects
“Environment” - everything else in the universe
What is “State” (or condition) of system?
- macroscopic description - in terms of detectable quantities:
volume, pressure, mass, temperature (“State Variables”)
Study of thermal energy --> temperature
Temperature & Thermometers
Linear scale : need 2 points to define
Fahrenheit [° F] body temp and ~1/3 of body temp ~100 ° F ~33 ° F Celsius [° C] “freezing point” and “boiling point” of water 0 ° C 100 ° C
Conversion factors K→ ° C ° C → ° F
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TF = 95TC + 32
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TC =TK − 273.15 (1 ΔK = 1 ΔC)
Kelvin [K] Absolute zero and triple point of water 0 K 273.16 K
18-‐3: Zeroth Law of Thermodynamics
Defines THERMAL EQUILIBRIUM If two systems are in thermal equilibrium with a third, then they are in
thermal equilibrium with each other T1 = T2 = T3
No Heat flow
In this case: a) A is in thermal equilibrium with T
b) B is in thermal equilibrium with T
c) A & B are in thermal equilibrium
18-‐4: Measuring Temperature
Phase Diagram of Water
Need two points and linear scale T=absolute zero
Water triple point.
18-‐4: Measuring Temperature
Triple Point of Water: Defined as T3=273.16 K
A gas filled bulb is connected to a Hg manometer. The pressure volume can be maintained constant by raising or lowering the the Hg level in reservoir R.
The Constant-Volume Gas Thermometer
T of liquid defined at T=Cpp = p0 + ρg(−h)
T = T3pp3
!
"#$
%&= (273.16K ) p
p3
!
"#$
%&
(C=constant)
18-‐4: Measuring Temperature
A gas filled bulb is connected to a Hg manometer. The pressure volume can be maintained constant by raising or lowering the the Hg level in reservoir R.
The Constant-Volume Gas Thermometer
T of liquid defined at T=Cpp = p0 + ρg(−h)
T = T3pp3
!
"#$
%&= (273.16K ) p
p3
!
"#$
%&
Still have a problem because answer depends upon p.
T = (273.16K ) lim p→0pp3
"
#$%
&'
Keep V fixed: Figure shows Measurement for boiling water
Checkpoint 1: The figure here shows three linear temperature scales with the freezing and boiling points of water indicated.
18-‐4: Temperature Scales
(a) Rank the degrees on these scales by size, greatest first.
Checkpoint 1: The figure here shows three linear temperature scales with the freezing and boiling points of water indicated.
18-‐4: Temperature Scales
(b) Rank the following temperatures, highest first: 50oX, 50o W and 500 Y
Thermal expansion Most substances expand when heated
and contract when cooled
ZrW2O8 is a ceramic with negative thermal expansion over a wide temperature range, 0-1050 K
The change in length, ΔL ( = L - L0 ), of almost all solids is ~ directly proportional to the change in temperature, ΔT ( = T - T0 )
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ΔL =αL0ΔTL = L0 1+αΔT( )
α = coefficient of thermal expansion
What causes thermal expansion?
Thermal expansion of the Brooklyn Bridge
Problem 1: Brooklyn Bridge Expansion The steel bed of the main suspension bridge is 490 m long at + 20°C. If the extremes in temperature are - 20°C to + 40°C, how much will it contract and expand?
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α steel =12 ×10−6(°C)–1
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ΔL =α steelL0ΔT=12 ×10−6(°C)–1(490m)(60°C)= 35 cm
The solution is to use expansion joints
Thermal expansion and a Pendulum Clock
A pendulum clock made of brass is designed to keep accurate time at 20°C. If the clock operates at 0°C, does it run fast or slow? If so, how much?
Problem 2: Pendulum Clock
If the original period was 1 second
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T = 2π Lg
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L = L0 + ΔL= L0 +αbrassL0ΔT
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L0 =1s2π#
$ %
&
' ( 2
g = 24.824 cm
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L = 24.824 cm 1+ (19 ×10−6 /°C)(−20°C)( )= 24.824 cm 0.9996( )= 24.814
The new period is:
T = 2π 24.8149.8
= 0.9998 s
It runs slow (less time per tick) at 20°C at 0°C: fewer ticks = 1.7hr/yr
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# ticks = 24 *60*60 = 86400# ticks = 86400 *0.999 = 86383
Example: Bimetal Strip
Common device to measure and control temperature
F = kx = kL0 1+αΔT( )
18-‐6 Area Expansion
Expansion in 1-D
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ΔL =αL0ΔTL = L0 1+αΔT( )
A = L0 1+αΔT( )#$ %& W0 1+αΔT( )#$ %&Expansion in 2-D
ΔA = A0 1+αΔT( )2 − A0= A0 2αΔT + αΔT( )2( )≅ A0 2α( )ΔT≅ A0βΔT
β = 2α
Thermal expansion of holes
Do holes expand or contract when heated?
Does radius increase or decrease
when heated?
The hole gets larger too!
When the temperature of the piece of metal shown below is increased and the expands metal expands, what happens to the gap between the ends?
1. It becomes narrower 2. It becomes wider 3. It remains unchanged
Clicker Question
Volume expansion
Expansion in 1-D
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ΔL =αL0ΔTL = L0 1+αΔT( )
width
heig
ht
Expansion in 3-D
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V = L0 1+αΔT( )[ ] W0 1+αΔT( )[ ] H 0 1+αΔT( )[ ]
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ΔV =V0 1+αΔT( )3 −V0=V0 3αΔT + 3 αΔT( )2 + αΔT( )3( )
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≅V0 3α( )ΔT≅V0βΔT β = coefficient of volume expansion
Problem 3: Gas tank in the sun The 70-L steel gas tank of a car is filled to the top with gasoline at 20°C. The car is then left to sit in the sun, and the tank reaches a temperature of 40°C. How much gasoline do you expect to overflow from the tank? [gasoline has a coefficient of volume expansion of 950×10-6/°C ]
Volume expansion coefficients
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solids : 1− 87 ×10−6 C liquids : 210 −1100 ×10−6 C gasses : 3400 ×10−6 C
