L 22

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L 22 Vibrations and Waves 2

• resonance ?
• clocks pendulum ?
• springs ?
• harmonic motion ?
• mechanical waves
• sound waves
• musical instruments

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simple harmonic oscillatormass and spring on a
frictionless surface
Equilibrium position
frictionless surface
k
spring that can be stretched or compressed
A
A
0
k is the spring constant, which measures
the stiffness of the spring in Newtons per meter
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Some terminology

• the maximum displacement of an object from
  equilibrium is called the AMPLITUDE A
• the time that it takes to complete one full cycle
  (A ? B ? C ? B ? A ) is called the PERIOD T of
  the motion
• if we count the number of full cycles the
  oscillator completes in a given time, that is
  called the FREQUENCY f of the oscillator
• frequency f 1 / period 1 / T

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follow the mass position vs. time
A
- A
T
T
T
http//www.phys.hawaii.edu/teb/java/ntnujava/shm/
shm.html
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? Period of the mass-spring system

• This formula is good for a mass on a hanging
  spring or for a horizontal spring on the air
  track
• If the mass is quadrupled, the period is doubled

? Period of a pendulum of length L
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Energy in the simple harmonic oscillator

• a compressed or stretched spring has elastic
  potential energy
• this elastic potential energy is what drives the
  system
• if you pull the mass from equilibrium and let go,
  this elastic PE changes into kinetic energy.
• when the mass passes the equilibrium point, the
  KE goes back into PE
• if there is no friction the energy keeps sloshing
  back and forth but it never decreases

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springs ? amazing devices!
the harder I pull on a spring, the harder it
pulls back
stretching
the harder I push on a spring, the harder
it pushes back
compression
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Springs obey Hookes Law
spring force (N)
elastic limit of the spring
amount of stretching or compressing in meters

• the strength of a spring is measured by how much
• force it provides for a given amount of
  stretch
• we call this quantity k, the spring constant in
  N/m
• magnitude of spring force k ? amount of
  stretch

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simple harmonic oscillator

• the period of oscillation is longer (takes more
  time to complete a cycle) if a bigger mass (m)
  is used
• the period gets smaller (takes less time to
  complete a cycle) if a stronger spring (larger k)
  is used
• Period T in seconds
• the time to complete a full cycle does not depend
  on where the oscillator is started (period is
  independent of amplitude)

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Resonance effects

• all systems have certain natural vibration
  tendencies
• the mass/spring system oscillates at a certain
  frequency determined by its mass, m and the
  spring stiffness constant, k

When you push a child on a swing you are using
resonance to make the child go higher and higher.
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How resonance works

• resonance is a way of pumping energy into a
  system to make it vibrate
• in order to make it work the energy must be
  pumped in at a rate (frequency) that matches one
  of the natural frequencies that the system likes
  to vibrate at.
• you pump energy into the child on the swing by
  pushing once per cycle
• The Tacoma Narrows bridge was set into resonance
  by the wind blowing over it

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resonance examples

• mass on spring
• two tuning forks
• shattering the glass

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Waves

• What is a wave? A disturbance that moves through
  something ? rather vague!
• The wave - people stand up then sit down, then
  the people next to them do the same until the
  standing and sitting goes all around the stadium.
• the standing and sitting is the disturbance
• notice that the people move up and down but the
  disturbance goes sideways !

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Homer trips and creates a longitudinal wave
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Why are waves important?

• a mechanical wave is a disturbance that moves
  through a medium ( e.g. air, water, strings)
• ? waves carry energy ?
• they provide a means to transport energy from one
  place to another
• electromagnetic waves (light, x-rays, UV rays,
  microwaves, thermal radiation) are disturbances
  that propagate through the electromagnetic field,
  even in vacuum (e.g. light from the Sun)

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Mechanical waves

• a disturbance that propagates through a medium
• waves on strings
• waves in water
• ocean waves
• ripples that move outward when a
• stone is thrown in a pond
• sound waves pressure waves in air

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transverse wave on a string

• jiggle the end of the string to create a
  disturbance
• the disturbance moves down the string
• as it passes, the string moves up and then down
• the string motion in vertical but the wave moves
  in the horizontal (perpendicular) direction?
  transverse wave
• this is a single pulse wave (non-repetitive)
• the wave in the football stadium is a
  transverse wave

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How fast does it go?

• The speed of the wave moving to the right is not
  the same as the speed of the string moving up and
  down. (it could be, but that would be a
  coincidence!)
• The wave speed is determined by
• the tension in the string
• ? more tension ? higher speed
• the mass per unit length of the string (whether
  its a heavy rope or a light rope)
• ? thicker rope ? lower speed

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Harmonic waves keep jiggling the end of the
string up and down
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Slinky waves

• you can create a longitudinal wave on a slinky
• instead of jiggling the slinky up and down, you
  jiggle it in and out
• the coils of the slinky move along the same
  direction (horizontal) as the wave

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SOUND WAVES

• longitudinal pressure disturbances in a gas
• the air molecules jiggle back and forth in the
  same direction as the wave

the diaphragm of the speaker moves in and out
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Sound a longitudinal wave
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The pressure waves make your eardrum vibrate

• we can only hear sounds between
• 30 Hz and 20,000 Hz
• below 30 Hz is called infrasound
• above 20,000 is called ultrasound

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I cant hear you!
Since sound is a disturbance in air, without air
(that is, in a vacuum) there is no sound.
There is no sound in outer space!
vacuum pump