7.1 Measuring rotational motion

1
Chapter 7

• 7.1 Measuring rotational motion

2
Rotational Quantities

• Rotational motion motion of a body that spins
  about an axis
• Axis of rotation the line about which the
  rotation occurs
• Circular motion motion of a point on a rotating
  object

3
Rotational Quantities

• Circular Motion
• Direction is constantly changing
• Described as an angle
• All points (except points on the axis) move
  through the same angle during any time interval

4
Circular Motion

• Useful to set a reference line
• Angles are measured in radians
• s arc length
• r radius

5
Angular Motion

• 360o 2?rad
• 180o ?rad

6
Angular displacement

• Angular dispacement the angle through which a
  point line, or body is rotated in a specified
  direction and about a specified axis
• Practice
• Earth has an equatorial radius of approximately
  6380km and rotates 360o every 24 h.
• What is the angular displacement (in degrees) of
  a person standing at the equator for 1.0 h?
• Convert this angular displacement to radians
• What is the arc length traveled by this person?

7
Angular speed and acceleration

• Angular speed The rate at which a body rotates
  about an axis, usually expressed in radians per
  second
• Angular acceleration The time rate of change of
  angular speed, expressed in radians per second
  per second

8
Angular speed and acceleration

• ALL POINTS ON A ROTATING RIGID OBJECT HAVE THE
  SAME ANGULAR SPEED AND ANGULAR ACCELERATION

9
Rotational kinematic equations
10
Angular kinematics

• Practice
• A barrel is given a downhill rolling start of 1.5
  rad/s at the top of a hill. Assume a constant
  angular acceleration of 2.9 rad/s
• If the barrel takes 11.5 s to get to the bottom
  of the hill, what is the final angular speed of
  the barrel?
• What angular displacement does the barrel
  experience during the 11.5 s ride?

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Homework Assignment

• Page 269 5 - 12

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Chapter 7

• 7.2 Tangential and Centripetal Acceleration

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Tangential Speed

• Let us look at the relationship between angular
  and linear quantities.
• The instantaneous linear speed of an object
  directed along the tangent to the objects
  circular path
• Tangent the line that touches the circle at one
  and only one point.

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Tangential Speed

• In order for two points at different distances to
  have the same angular displacement, they must
  travel different distances
• The object with the larger radius must have a
  greater tangential speed

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Tangential Speed
16
Tangential Acceleration

• The instantaneous linear acceleration of an
  object directed along the tangent to the objects
  circular path

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Lets do a problem

• A yo-yo has a tangential acceleration of 0.98m/s2
  when it is released. The string is wound around a
  central shaft of radius 0.35cm. What is the
  angular acceleration of the yo-yo?

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Centripetal Acceleration

• Acceleration directed toward the center of a
  circular path
• Although an object is moving at a constant speed,
  it can still have an acceleration.
• Velocity is a vector, which has both magnitude
  and DIRECTION.
• In circular motion, velocity is constantly
  changing direction.

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Centripetal Acceleration

• vi and vf in the figure to the right differ only
  in direction, not magnitude
• When the time interval is very small, vf and vi
  will be almost parallel to each other and
  acceleration is directed towards the center

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Centripetal Acceleration
21
Tangential and centripetal accelerations

• Summary
• The tangential component of acceleration is due
  to changing speed the centripetal component of
  acceleration is due to changing direction
• Pythagorean theorem can be used to find total
  acceleration and the inverse tangent function can
  be used to find direction

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Whats coming up

• HW Pg 270, problems 21 - 26
• Monday Section 7.3
• Wednesday Review
• Friday TEST over Chapter 7

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Chapter 7

• 7.3 Causes of Circular Motion

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Causes of circular motion

• When an object is in motion, the inertia of the
  object tends to maintain the objects motion in a
  straight-line path.
• In circular motion (I.e. a weight attached to a
  string), the string counteracts this tendency by
  exerting a force
• This force is directed along the length of the
  string towards the center of the circle

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Force that maintains circular motion

• According to Newtons second law
• or

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Force that maintains circular motion

• REMEMBER The force that maintains circular
  motion acts at right angles to the motion.
• What happens to a person in a car(in terms of
  forces) when the car makes a sharp turn.

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Chapter 9

• 9.2 - Fluid pressure and temperature

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Pressure

• What happens to your ears when you ride in an
  airplane?
• What happens if a submarine goes too deep into
  the ocean?

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What is Pressure?

• Pressure is defined as the measure of how much
  force is applied over a given area
• The SI unit of pressure is the pascal (PA), which
  is equal to N/m2
• 105Pa is equal to 1 atm

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Some Pressures
Table 9-2 Some pressures Table 9-2 Some pressures
Location P(Pa)
Center of the sun 2 x 1016
Center of Earth 4 x 1011
Bottom of the Pacific Ocean 6 x 107
Atmosphere at sea level 1.01 x 105
Atmosphere at 10 km above sea level 2.8 x 104
Best vacuum in a laboratory 1 x 10-12
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Pressure applied to a fluid

• When you inflate a balloon/tire etc, pressure
  increases
• Pascals Principle
• Pressure applied to a fluid in a closed container
  is transmitted equally to every point of the
  fluid and to the walls of a container

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Lets do a problem

• In a hydraulic lift, a 620 N force is exerted on
  a 0.20 m2 piston in order to support a weight
  that is placed on a 2.0 m2 piston.
• How much pressure is exerted on the narrow
  piston?
• How much weight can the wide piston lift?

33
Pressure varies with depth in a fluid

• Water pressure increases with depth. WHY?
• At a given depth, the water must support the
  weight of the water above it
• The deeper you are, the more water there is to
  support
• A submarine can only go so deep an withstand the
  increased pressure

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The example of a submarine

• Lets take a small area on the hull of the
  submarine
• The weight of the entire column of water above
  that area exerts a force on that area

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Fluid Pressure

• Gauge Pressure
• does not take the pressure of the atmosphere into
  consideration
• Fluid Pressure as a function of depth
• Absolute pressure atmospheric pressure
  (density x free-fall acceleration x depth)

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Point to remember

• These equations are valid ONLY if the density is
  the same throughout the fluid

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The Relationship between Fluid pressure and
buoyant forces

• Buoyant forces arise from the differences in
  fluid pressure between the top and bottom of an
  immersed object

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Atmospheric Pressure

• Pressure from the air above
• The force it exerts on our body is 200 000N (40
  000 lb)
• Why are we still alive??
• Our body cavities are permeated with fluids and
  gases that are pushing outward with a pressure
  equal to that of the atmosphere -gt Our bodies
  are in equilibrium

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Atmospheric

• A mercury barometer is commonly used to measure
  atmospheric pressure

40
Kinetic Theory of Gases

• Gas contains particles that constantly collide
  with each other and surfaces
• When they collide with surfaces, they transfer
  momentum
• The rate of transfer is equal to the force
  exerted by the gas on the surface
• Force per unit time is the gas pressure

41
Lets do a Problem

• Find the atmospheric pressure at an altitude of
  1.0 x 103 m if the air density is constant.
  Assume that the air density is uniformly 1.29
  kg/m3 and P01.01 x 105 Pa

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Temperature in a gas

• Temperature is the a measure of the average
  kinetic energy of the particles in a substance
• The higher the temperature, the faster the
  particles move
• The faster the particles move, the higher the
  rate of collisions against a given surface
• This results in increased pressure

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HW Assignment

• Page 330 Practice 9C, page 331 Section Review

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Chapter 9

• 9.3 - Fluids in Motion

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Fluid Flow

• Fluid in motion can be characterized in two ways
• Laminar Every particle passes a particular point
  along the same smooth path (streamline) traveled
  by the particles that passed that point earlier
• Turbulent Abrupt changes in velocity
• Eddy currents Irregular motion of the fluid

46
Ideal Fluid

• A fluid that has no internal friction or
  viscosity and is incompressible
• Viscosity The amount of internal friction within
  a fluid
• Viscous fluids loose kinetic energy because it is
  transformed into internal energy because of
  internal friction.

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Ideal Fluid

• Characterized by Steady flow
• Velocity, density and pressure are constant at
  each point in the fluid
• Nonturbulent
• There is no such thing as a perfectly ideal
  fluid, but the concept does allow us to
  understand fluid flow better
• In this class, we will assume that fluids are
  ideal fluids unless otherwise stated

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Principles of Fluid Flow

• If a fluid is flowing through a pipe, the mass
  flowing into the pipe is equal to the mass
  flowing out of the pipe

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Pressure and Speed of Flow

• In the Pipe shown to the right, water will move
  faster through the narrow part
• There will be an acceleration
• This acceleration is due to an unbalanced force
• The water pressure will be lower, where the
  velocity is higher

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Bernoullis Principle

• The pressure in a fluid decreases as the fluids
  velocity increases

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Bernoullis Equation

• Pressure is moving through a pipe with varying
  cross-section and elevation
• Velocity changes, so kinetic energy changes
• This can be compensated for by a change in
  gravitational potential energy or pressure

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Bernoullis Equation
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Bernoullis Principle A Special Case

• In a horizontal pipe

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The Ideal Gas Law

• kB is a constant called the Boltzmanns constant
  and has been experimentally determined to be 1.38
  x 10-23 J/K

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Ideal Gas Law Contd

• If the number of particles is constant then
• Alternate Form
• mmass of each particle, MN x m Total Mass of
  the gas

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Real Gas

• An ideal gas can be described by the ideal gas
  law
• Real gases depart from ideal gas behavior at high
  pressures and low temperatures.

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Chapter 12 Vibration and Waves

• 12.1 Simple Harmonic Motion

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Simple harmonic motion

• Periodic motion Back and forth motion over the
  same path
• E.g. Mass attached to a spring

59
Simple Harmonic Motion
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Simple harmonic motion

• At the unstretched position, the spring is at
  equilibrium (x0)
• The spring force increases as the spring is
  stretched away from equilibrium
• As the mass moves towards equilibrium, force (and
  acceleration) decreases

61
Simple harmonic motion

• Momentum causes mass to overshoot equilibrium
• Elastic force increases (in the opposite
  direction)

62
Simple harmonic motion

• Defined as a vibration about an equilibrium
  position in which a restoring force is
  proportional to the displacement from equilibrium
• The force that pushes or pulls the mass back to
  its original equilibrium position is called the
  restoring force

Hookes Law
Spring force - (spring constant x displacement)
63
Hookes Law Example
Example 1 If a mass of 0.55kg attached to a
vertical spring stretches the spring 2 cm from
its equilibrium position, what is the spring
constant? Given m 0.55 kg x -0.02
m g -9.8 m/s2 Solution Fnet 0
Felastic Fg 0 - kx mg or,
kx mg k mg/x (0.55 g)(-9.8
m/s2)/(-0.02 m) 270 N/m
Fel
Fg
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Energy

• What kind of energy does a springs has when it is
  stretched or compressed?
• Elastic Potential energy
• Elastic Potential energy can be converted into
  other forms of energy
• i.e. Bow and Arrow

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The Simple Pendulum

• Consists of a mass, which is called a bob, which
  is attached to a fixed string
• Assumptions
• Mass of the string is negligible
• Disregard friction

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The Simple Pendulum

• The restoring force is proportional to the
  displacement
• The restoring force is equal to the x component
  of the bobs weight
• When the angle of displacement is gt15o, a
  pendulums motion is simple harmonic

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The Simple Pendulum

• In the absence of friction, Mechanical energy is
  conserved

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Simple Harmonic motion
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Chapter 12 Vibration and Waves

• 12.2 Measuring simple harmonic motion

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Amplitude, Period and Frequency

• Amplitude The maximum displacement from the
  equilibrium position
• Period (T) The time it takes to execute a
  complete cycle of motion
• Frequency (f) the number of cycles/vibrations
  per unit time

71
Period and Frequency

• If the time it takes to complete one cycle is 20
  seconds
• The Period is said to be 20s
• The frequency is 1/20 cycles/s or 0.05 cycles/s
• SI unit for frequency is s-1 a.k.a hertz (Hz)

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Measures of simple harmonic motion
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The period of a simple pendulum

• Changing mass does not change the period
• Has larger restoring force, but needs larger
  force to get the same acceleration
• Changing the amplitude also does not change the
  period (for small amplitudes)
• Restoring force increases, acceleration is
  greater, but distance also increases

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The Period of a simple pendulum

• LENGTH of a pendulum does affect its period
• Shorter pendulums have a smaller arc to travel
  through, while acceleration is the same
• Free-fall acceleration also affects the period of
  a pendulum

75
The Period of a mass-spring system

• Restoring force
• Not affected by mass
• Increasing mass increases inertia, but not
  restoring force --gt smaller acceleration

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The Period of a mass-spring system

• A heavier mass will take more time to complete a
  cycle --gt Period increases
• The greater the spring constant, the greater the
  force, the greater the acceleration, which causes
  a decrease in period

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Chapter 12

• 12.3 Properties of Waves

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Wave Motion

• Lets say we drop a pebble into water
• Waves travel away from disturbance
• If there is an object floating in the water, it
  will move up and down, back and forth about its
  original position
• Indicates that the water particles move up and
  down

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Wave Motion

• Water is the medium
• Material through which the disturbance travels
• Mechanical wave
• A wave that propagates through a deformable,
  elastic medium
• i.e. sound - cannot travel through outer space
• Electromagnetic wave
• Does not require a medium
• i.e. visible light, radio waves, microwaves, x
  rays

80
Types of Waves

• Pulse Wave Single nonperiodic disturbance
• Periodic Wave A wave whose source is some form
  of periodic motion
• Sine Wave A wave whose source vibrates with
  simple harmonic motion
• Every point vibrates up and down

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Types of Waves

• Transverse wave A wave whose particles vibrate
  perpendicularly to the direction of wave motion

Note The distance between the adjacent crests
and troughs are the same

• Longitudinal wave A wave whose particles vibrate
  parallel to the direction of wave motion. i.e.
  sound

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Period, Frequency, and Wave speed

• Period is the amount of time it takes for a
  complete wavelength to pass a given point

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Waves and Energy

• Waves carry a certain amount of energy
• Energy transfers from one place to another
• Medium remains essentially in the same place
• The greater the amplitude of the wave, the more
  energy transfered

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Chapter 12

• 12.4 Wave Interactions

85
Wave Interference

• Waves are not matter, but displacements of matter
• Two waves can occupy the same space at the same
  time
• Forms an interference pattern
• Superposition Combination of two overlapping
  waves

86
Constructive interference

• Individual displacements on the same side of the
  equilibrium position are added together to form a
  resultant wave

87
Destructive Interference

• Individual displacements on opposite sides of the
  equilibrium position are added together to form
  the resultant wave

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Reflection

• When a wave encounters a boundary, it is
  reflected
• If it is a free boundary/reflective surface the
  wave is reflected unchanged
• If it is a fixed boundary, the wave is reflected
  and inverted

89
Standing Waves

• A wave pattern that results when two waves of the
  same frequency travel in opposite directions and
  interfere
• Nodes point in standing wave that always
  undergoes complete destructive interference and
  is stationary
• Antinode Point in standing wave, halfway between
  two nodes, with largest amplitude

90
Chapter 13 - Sound

• 13.1 Sound Waves

91
The Production of Sound Waves
92
The Production of Sound Waves

• Compression the region of a longitudinal wave in
  which the density and pressure are greater than
  normal
• Rarefaction the region of a longitudinal wave in
  which the density and pressure are less than
  normal
• These compressions and rarefactions expand and
  spread out in all directions (like ripples in
  water)

93
The Production of Sound Waves
94
Characteristics of Sound Waves

• The average human ear can hear frequencies
  between 20 and 20,000 Hz.
• Below 20Hz are called infrasonic waves
• Above 20,000 Hz are called ultrasonic waves
• Can produce images (i.e. ultrasound)
• f 10 Mhz, v 1500m/s, wavelengthv/f 1.5mm
• Reflected sound waves are converted into an
  electric signal, which forms an image on a
  fluorescent screen.

95
Characteristics of Sound Waves

• Frequency determines pitch - the perceived
  highness or lowness of a sound.

96
Speed of Sound

• Depends on medium
• Travels faster through solids, than through
  gasses.
• Depends on the transfer of motion from particle
  to another particle.
• In Solids, molecules are closer together
• Also depends on temperature
• At higher temperatures, gas particles collide
  more frequently
• In liquids and solids, particles are close enough
  together that change in speed due to temperature
  is less noticeable

97
Speed of Sound
98
Propagation of Sound Waves

• Sound waves spread out in all directions (in all
  3 dimensions)
• Such sound waves are approximately spherical

99
Propagation of Sound Waves
100
The Doppler Effect

• When an ambulance passes with sirens on, the
  pitch will be higher as it approaches you and
  lower as it moves away
• The frequency is staying the same, but the pitch
  is changing

101
The Doppler Effect
The wave fronts reach observer A more often
thanobserver B because of the relative motion of
the car
The frequency heard by observer A is higher
thanthe frequency heard by observer B
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HW Assignment

• Section 13-1 Concept Review

103
Chapter 13 - Sound

• 13.2 - Sound intensity and resonance

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Sound Intensity

• When you play the piano
• Hammer strikes wire
• Wire vibrates
• Causes soundboard to vibrate
• Causes a force on the air molecules
• Kinetic energy is converted to sound waves

105
Sound Intensity

• Sound intensity is the rate at which energy flows
  through a unit area of the plane wave
• Power is the rate of energy transfer
• Intensity can be described in terms of power
• SI unit W/m2

106
Sound Intensity

• Intensity decreases as the distance from the
  source (r) increases
• Same amount of energy spread over a larger area

107
Intensity and Frequency
Human Hearing depends both on frequency and
intensity
108
Relative Intensity

• Intensity determines loudness (volume)
• Volume is not directly proportional to intensity
• Sensation of loudness is approximately
  logarithmic
• The decibel level is a more direct indication of
  loudness as perceived by the human ear
• Relative intensity, determined by relating the
  intensity of a sound wave to the intensity at the
  threshold of hearing

109
Relative Intensity

• When intensity is multiplied by 10, 10dB are
  added to the decibel level
• 10dB increase equates to sound being twice as loud

110
Forced Vibrations

• Vibrating strings cause bridge to vibrate
• Bridge causes the guitars body to vibrate
• These forced vibrations are called sympathetic
  vibrations
• Guitar body cause the vibration to be transferred
  to the air more quickly
• Larger surface area

111
Resonance

• In Figure 13.11, if a blue pendulum is set into
  motion, the others will also move
• However, the other blue pendulum will oscillate
  with a much larger amplitude than the red and
  green
• Because the natural frequency matches the
  frequency of the first blue pendulum
• Every guitar string will vibrate at a certain
  frequency
• If a sound is produced with the same frequency as
  one of the strings, that string will also vibrate

112
The Human Ear
The basilar membrane has different
natural Frequencies at different positions
113
Chapter 13 - Sound

• 13.3 - Harmonics

114
Standing Waves on a Vibrating String

• Musical instruments, usually consist of many
  standing waves together, with different
  wavelengths and frequencies even though you hear
  a single pitch
• Ends of the string will always be the nodes
• In the simplest vibration, the center of the
  string experiences the most displacement
• This frequency of this vibration is called the
  fundamental frequency

115
The Harmonic Series
Fundamental frequency or first harmonic Wavelength
is equal to twice the string length
Second harmonic Wavelength is equal to the string
length
116
Standing Waves on a Vibrating String

• When a guitar player presses down on a string at
  any point, that point becomes a node

117
Standing Waves in an Air Column

• Harmonic series in an organ pipe depends on
  whether the reflecting end of the pipe is open or
  closed.
• If open - that end becomes and antinode
• If closed - that end becomes a node

118
Standing waves in an Air Column
The Fundamental frequency can be changed by
changing the vibrating air column
119
Standing Waves in an Air Column
Only odd harmonics will be present
120
Standing Waves in an Air Column

• Trumpets, saxophones and clarinets are similar to
  a pipe closed at one end
• Trumpets Players mouth closes one end
• Saxophones and clarinets reed closes one end
• Fundamental frequency formula does not directly
  apply to these instruments
• Deviations from the cylindrical shape of a pipe
  affect the harmonic series

121
Harmonics account for sound quality, or timbre

• Each instrument has its own characteristic
  mixture of harmonics at varying intensities
• Tuning fork vibrates only at its fundamental,
  resulting in a sine wave
• Other instruments are more complex because they
  consist of many harmonics at different
  intensities

122
Harmonics account for sound quality, or timbre
123
Harmonics account for sound quality, or timbre

• The mixture of harmonics produces the
  characteristic sound of an instrument timbre
• Fuller sound than a tuning fork

124
Fundamental Frequency determines pitch

• In musical instruments, the fundamental frequency
  determines pitch
• Other harmonics are sometimes referred to as
  overtones
• An frequency of the thirteenth note is twice the
  frequency of the first note

125
Fundamental Frequency determines pitch
126
Beats

• When two waves differ slightly in frequency, they
  interfere and the pattern that results is an
  alternation between loudness and softness - Beat
• Out of phase complete destructive interference
• In Phase - complete constructive interference

127
Beats