Oscillations About Equilibrium

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Oscillations About Equilibrium
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• 7.1 Periodic Motion

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Periodic Motion repeat, same time, same
path Period (T) time required for one complete
cycle (seconds) or seconds/cycle Frequency (f)
the number of oscillations per second (s-1 or
hertz)

• 7.2 Simple Harmonic Motion

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• 7.2 Simple Harmonic Motion

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A form of Periodic Motion Simple Harmonic
Motion A restoring force is applied proportional
to the distance from equilibrium So Hookes Law

• 7.2 Simple Harmonic Motion

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If a graph of simple harmonic motion is
created And spread out over time We get a wave
pattern Amplitude maximum displacement

• 7.2 Simple Harmonic Motion

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• 7.3 The Period of a Mass on a Spring

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The period of a spring is given by the equation
A larger mass would have greater inertia
longer period A larger spring constant would
produce more acceleration, so a shorter
period The period is independent of amplitude

• 7.3 The Period of a Mass on a Spring

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• 7.5 The Pendulum

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A simple Pendulum The potential energy
is So potential energy is zero at
equilibrium (like SHM)

• 7.5 The Pendulum

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The period of a pendulum is given as
Independent of the mass of the bob

• 7.5 The Pendulum

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Restoring Force Forces Components A pendulum does
not act as a Simple Harmonic Oscillator,
but at small angles (lt30o) it approximates
SHM

• 7.5 The Pendulum

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• 7.7 Driven Oscillations and Resonance

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Natural Frequency depends on the variables (m,k
or L,g) of the object Forced Vibrations
caused by an external force

• 7.7 Driven Oscillations and Resonance

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Resonant Frequency the natural vibrating
frequency of a system Resonance when the
external frequency is near the natural frequency
and damping is small
Tacoma Narrow Bridge

• 7.7 Driven Oscillations and Resonance

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• 7.8 Types of Waves

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Mechanical Waves travels through a medium The
wave travels through the medium, but the medium
undergoes simple harmonic motion Wave
motion Particle motion

• 7.8 Types of Waves

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Waves transfer energy, not particles A
single bump of a wave is called a pulse A wave
is formed when a force is applied to one end
Each successive particle is moved by the one
next to it

• 7.8 Types of Waves

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Parts of a wave Transverse wave particle
motion perpenduclar to wave
motion Wavelength (l) measured in
meters Frequency (f) measured in Hertz (Hz) Wave
Velocity (v) meters/second

• 7.8 Types of Waves

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Longitudinal (Compressional) Wave Particles move
parallel to the direction of wave
motion Rarefaction where particles are
spread out Compression particles are
close

• 7.8 Types of Waves

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Earthquakes S wave Transverse P wave
Longitudinal Surface Waves can travel along
the boundary Notice the circular motion of the
particles

• 7.8 Types of Waves

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• 7.9 Reflection and Transmission of Waves

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When a wave comes to a boundary (change in
medium) at least some of the wave is
reflected The type of reflection depends
on if the boundary is fixed (hard) -
inverted

• 7.9 Reflection and Transmission of Waves

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When a wave comes to a boundary (change in
medium) at least some of the wave is
reflected Or movable (soft) in phase

• 7.9 Reflection and Transmission of Waves

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For two or three dimensional we think in terms of
wave fronts A line drawn perpendicular to the
wave front is called a ray When the waves get far
from their source and are nearly straight, they
are called plane waves

• 7.9 Reflection and Transmission of Waves

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Law of Reflection the angle of reflection
equals the angle of incidence Angles are always
measured from the normal

• 7.9 Reflection and Transmission of Waves

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• 7.10 Characteristics of Sound

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Sound is a longitudinal wave Caused by the
vibration of a medium The speed of sound
depends on the medium it is in, and the
temperature For air, it is calculated as

• 7.10 Characteristics of Sound

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Loudness sensation of intensity Pitch
sensation of frequency Range of human hearing
20Hz to 20,000 Hz ultrasonic higher than human
hearing dogs hear to 50,000 Hz, bats to
100,000 Hz infrasonic lower than human
hearing

• 7.10 Characteristics of Sound

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Often called pressure waves Vibration produces
areas of higher pressure These changes in
pressure are recorded by the ear drum

• 7.10 Characteristics of Sound

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• 7.11 Intensity of Sound

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Loudness sensation Relative to surrounding and
intensity Intensity power per unit area Humans
can detect intensities as low as 10-12
W/m2 The threshold of pain is 1 W/m2

• 7.11 Intensity of Sound

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• Sound intensity is usually measured in decibels
  (dB)
• Sound level is given as
• I intensity of the sound
• I0 threshold of hearing (10-12 W/m2)
• sound level in dB
• Some common relative intensities

Source of Sound Sound Level (dB)
Jet Plane at 30 m 140
Threshold of Pain 120
Loud Rock Concert 120
Siren at 30 m 100
Auto Interior at 90 km/h 75
Busy Street Traffic 70
Conversation at 0.50 m 65
Quiet Radio 40
Whisper 20
Rustle of Leaves 10
Threshold of Hearing 0

• 7.11 Intensity of Sound

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• 7.12 The Ear

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Steps in sound transmission

• 7.12 The Ear

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• 7.13 Sources of Sound Strings and Air Columns

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Vibrations in strings Fundamental
frequency Next Harmonic

• 7.13 Sources of Sound

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Vibrations in strings Next Harmonic Strings
produce all harmonics all whole number
multiples of the fundamental frequency

• 7.13 Sources of Sound

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Vibrations in an open ended tube (both
ends) Fundamental frequency Next Harmonic

• 7.13 Sources of Sound

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Vibrations in open ended tubes Next
Harmonic Open ended tubes produce all
harmonics all whole number multiples of the
fundamental frequency Examples include organ
pipes and flutes.

• 7.13 Sources of Sound

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Vibrations in an closed end tube (one
end) Fundamental frequency Next Harmonic

• 7.13 Sources of Sound

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Vibrations in open ended tubes Next
Harmonic Closed end tubes produce only odd
harmonics Examples include reeded wind
instruments and brass instruments

• 7.13 Sources of Sound

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• 7.14 Interference of Sound Waves Beats

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If waves are produced by two identical sources A
pattern of constructive and destructive
interference forms
Applet

• 7.14 Interference of Sound Waves Beats

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• 7.15 The Doppler Effect

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Doppler Effect the change in pitch due to the
relative motion between a source of sound and the
receiver Applies to all wave phenomena Objects
moving toward you have a higher apparent
frequency Objects moving away have a lower
apparent frequency
Doppler Effect
Light Doppler

• 7.15 The Doppler Effect

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If an object is stationary the equation for the
wave velocity is Sound waves travel outward
evenly in all directions If the object moves
toward the observed, the waves travel at the same
velocity, but each new vibration is created
closer to the observer
Doppler Applet

• 7.15 The Doppler Effect

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The general equation is The values of Vo
(speed of observer) and Vs (speed of source) is
positive when they approach each other
Radar Gun

• 7.15 The Doppler Effect

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• 7.16 Interference

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Interference two waves pass through the same
region of space at the same time The waves pass
through each other Principle of Superposition
at the point where the waves meet the
displacement of the medium is the algebraic sum
of their separate displacements

• 7.16 Interference

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Phase relative position of the wave crests If
the two waves are in phase Constructive
interference If the two waves are out of
phase Destructive Interference

• 7.16 Interference

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For a wave (instead of a single
phase) Interference is calculated by adding
amplitude In real time this looks like

• 7.16 Interference