PHYSICS 149: Lecture 22
• Chapter 11: Waves
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Lecture 22
11.1 Waves and Energy Transport
11.2 Transverse and Longitudinal Waves
11.3 Speed of Transverse Waves on a String
11.4 Periodic Waves
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Midterm Exam 2
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Wednesday, November 17, 6:30 PM – 7:30 PM
Place: PHY 114
Chapters 5 - 8
The exam is closed book.
The exam is a multiple-choice test.
There will be ~15 multiple-choice problems.
– Each problem is worth 10 points.
• Note that total possible score for the course is 1,000 points (see the
course syllabus)
• The difficulty level is about the same as the level of textbook
problems.
• You may make a single crib sheet
– you may write on both sides of an 8.5” × 11.0” sheet
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ILQ 1
A bolt requires 110 N⋅m of torque to be
unscrewed. If the maximum force you can apply
is 198 N, what is the shortest wrench you can
use to unscrew the bolt?
A)
B)
C)
D)
28 cm
56 cm
1.8 m
0.556 cm
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ILQ 2
A)
B)
C)
D)
E)
A uniform bar of mass m is supported by a pivot
at its top, about which the bar can swing like a
pendulum. If a force F is applied
perpendicularly to the lower end of the bar as in
the diagram, how big must F be in order to hold
the bar in equilibrium at an angle from the
vertical?
2mg
mg sinθ
2mg sinθ
(mg/2) sinθ
(mg/2) cosθ
Lecture 22
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Linear and Angular
Lecture 22
Linear
Angular
Displacement
x
θ
Velocity
v
ω
Acceleration
a
α
Inertia
m
I
Kinetic Energy
½ m v2
½ I ω2
Newton’s 2nd
Law
F = ma
τ = Iα
Momentum
p = mv
L = Iω
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What is a Wave
• A wave is a disturbance that travels away from its
source and carries energy.
• A wave can transmit energy from one point to
another without transporting any matter between
the two points.
• Examples:
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Lecture 22
Sound waves (air moves back & forth)
Stadium waves (people move up & down)
Water waves (water moves up & down)
Seismic waves (earth moving up & down)
Light waves (what moves ??)
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Waves
• A wave is a disturbance that travels away from its
source.
– Mechanical waves
• Waves traveling through a material medium.
• Examples include water waves, sound waves, and the seismic
waves caused by earthquakes.
• Particles in the medium oscillate (or vibrate) around their
equilibrium position but do not travel.
– Electromagnetic waves
• Waves in which the disturbance consists of oscillating
electromagnetic fields.
• Example: visible light, radio waves, infrared waves, and
ultraviolet waves.
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Why are Waves Important
• How do we transport energy from one place to
another through matter or a medium?
• Electromagnetic waves transport energy
(electromagnetic energy in the form of light) from
the Sun to the Earth.
• In wave motion energy is transported, but matter
is not.
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Energy Transport
• A wave can transmit energy from one point to
another without transporting any matter between
the two points.
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Intensity
• Average power per unit area carried by the wave
past a surface which is perpendicular to the
direction of propagation of the wave.
• Intensity decreases with distance
• Unit: W/m2 (recall 1 W = 1 J/s)
Note: surface area of a sphere = 4πr
• I ∝ A2
2
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Example: Intensity
• The intensity of sunlight that reaches the Earth’s
atmosphere is 1400 W/m2. What is the intensity of the
sunlight that reaches Saturn? Saturn is 9.5 times as far
from the Sun as Earth.
Average power of the Sun:
P = IE⋅(4πrE2)
Intensity of the sunlight at the Saturn: Sun
IS = P / (4πrS2)
= IE⋅(4πrE2) / (4πrS2)
= IE⋅(rE/rS ) 2
= (1400 W/m2) ⋅ (1 / 9.5)2
= 16 W/m2
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rE
rS
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ILQ
If the distance to a point source of sound is
tripled, by what factor does the intensity of the
sound change?
A)
B)
C)
D)
Ifar = 9 Inear
Ifar = 3 Inear
Ifar = (1/3) ⋅ Inear
Ifar = (1/9) ⋅ Inear
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Types of Waves
• Transverse: The medium oscillates perpendicular to the
direction the wave is moving.
– Water (more or less)
– Slinky
energy transport
• Longitudinal: The medium oscillates in the same direction
as the wave is moving.
– Sound
– Slinky
energy transport
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ILQ
•
In a transverse wave, the individual particles of
the medium
a) move in ellipses.
b) move in circles.
c) move perpendicularly to the direction of the
wave's travel.
d) move parallel to the direction of the wave's
travel.
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ILQ
Consider a wave on a string moving to the right,
as shown. What is the direction of the velocity
of a particle at the point labeled A?
a)
b)
c)
d)
Right
Zero
Left
Down
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Simple Harmonic Motion
• Vibrations
– Vocal cords when singing/speaking
– String/rubber band
• Simple Harmonic Motion
– Restoring force proportional to displacement
– Springs F = -kx
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Hooke’s Law relaxed position
The force exerted by a
spring is proportional
to the distance the
spring is stretched or
compressed from its
relaxed position.
– FX = -k x
x
is the displacement
from the relaxed
position and k is the
constant of
proportionality.
FX = 0
x
x=0
FX = - kx < 0
x
x>0
x=0
FX = -kx > 0
x<0
x=0
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Energy in SHM
• A mass is attached to a spring and set to
motion. The maximum displacement is x=A
ΣWnc = ΔK + ΔU
0 = ΔK + ΔU or Energy U+K is constant!
Energy = ½ k x2 + ½ m v2
PES
– At maximum displacement x=A, v = 0
Energy = ½ k A2 + 0
– At zero displacement x = 0
Energy = 0 + ½ mvm2
0
Since Total Energy is same
½ k A2 = ½ m vm2
m
vm = sqrt(k/m) A
x=0
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x
x
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Springs and Simple Harmonic Motion
X=0
X=A; v=0; a=-amax
X=0; v=-vmax; a=0
X=-A; v=0; a=amax
X=0; v=vmax; a=0
X=A; v=0; a=-amax
X=-A
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X=A
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Simple Harmonic Motion
What does moving in a circle have to do with moving back &
forth in a straight line?
x = R cos θ = R cos (ωt)
since θ = ω t
x
x
1
1
2
R
3
R
8
θ
2
8
y
7
4
6
0
-R
5
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3
θ
6
4
5
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Simple Harmonic Motion
x(t) = [A]cos(ωt)
x(t) = [A]sin(ωt)
v(t) = -[Aω]sin(ωt)
a(t) = -[Aω2]cos(ωt)
OR
v(t) = [Aω]cos(ωt)
a(t) = -[Aω2]sin(ωt)
xmax = A
Period = T (seconds per cycle)
vmax = Aω
Frequency = f = 1/T (cycles per second)
amax = Aω2
Angular frequency =
ω = 2πf = 2π/T
For spring: ω2 = k/m
since F = ma = -kx
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Period of a Spring
• Simple Harmonic Oscillator
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ω=2πf =2π/T
x(t) = [A] cos(ωt)
v(t) = -[Aω] sin(ωt)
a(t) = -[Aω2] cos(ωt)
• For a Spring F = -kx
– amax = (k/m) A
– Aω2 = (k/m) A
ω = sqrt(k/m)
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Pendulum Motion
• For small angles
– T = mg
– Tx = -mg (x/L) Note: F proportional to x!
– Σ Fx = m ax
-mg (x/L) = m ax
ax = -(g/L) x
– Recall for SHO a = -ω2 x
L
ω = sqrt(g/L)
T = 2 π sqrt(L/g)
T
Period does not depend on A, or m!
x
m
mg
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Simple Harmonic Motion
• Occurs when having linear restoring force F= -kx
– x(t) = [A] cos(ωt)
– v(t) = -[Aω] sin(ωt)
– a(t) = -[Aω2] cos(ωt)
• Springs
– F = -kx
– U = ½ k x2
– ω = sqrt(k/m)
• Pendulum (small oscillations)
– ω = sqrt(L/g)
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Waves on a String
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Velocity of Waves
A spring and slinky are attached and stretched. Compare the
speed of the wave pulse in the slinky with the speed of the wave
pulse in the spring.
A) vslinky > vspring B) vslinky = vspring C) vslinky < vspring
Slinky is stretches more, so it has a smaller mass/length μ.
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Harmonic Waves
y(x,t) = A sin(ωt –kx)
A = amplitude
ω = angular frequency
k = wave number = 2π/λ
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Amplitude and Wavelength
y(x,t) = A cos(ωt – kx)
Wavelength: The distance λ between identical points on the wave.
Amplitude: The maximum displacement A of a point on the wave.
Angular Frequency ω: ω = 2 π f
Wave Number k: k = 2 π / λ
Recall: f = v / λ
y
Wavelength
λ
Amplitude A
A
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Period and Velocity
l
l
Period: The time T for a point on the wave to undergo one
complete oscillation.
Speed: The wave moves one wavelength λ in one period T
so its speed is v = λ / T.
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