AP Physics B Giancoli 11

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AP Physics BGiancoli 11 12

• Waves and Sound

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Assignments

• Reading 11.1-4,7-9,11-13 and 12.1,2,4-7
• Problems Waves 11.42,43,55,56
• Sound 12.4,5,10,11
• Strings/AirColumns12.29,30,35
• Interference 12.42,43
• Doppler 12.51,52
• SHM 11.4,5,20, 25
• Pendulum 11.31,32

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Preview

• What are the two categories of waves with regard
  to mode of travel?
• Mechanical
• Electromagnetic
• Which type of wave requires a medium?
• Mechanical
• An example of a mechanical wave?
• Sound

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Velocity of a Wave

• The speed of a wave is the distance traveled by a
  given point on the wave (such as a crest) in a
  given internal of time.
• v d/t
• d distance (m)
• t time (s)
• v f l
• v speed (m/s)
• l wavelength (m)
• f frequency (s-1, Hz)

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Period of a Wave

• T 1/f
• T Period (s)
• F frequency (s-1, Hz)

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Problem Sound travels at approximately 340 m/s,
and light travels at 3.0 x 108 m/s. How far away
is a lightning strike if the sound of the thunder
arrives at a location 5.0 seconds after the
lightning is seen?
Light travels almost instantaneously from strike
location to the observer. The sound travels much
more slowly d vs t (340 m/s)(5.0 s) 170m
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Problem The frequency of a C key on the piano
is 262 Hz. What is the period of this note? What
is the wavelength? Assume speed of sound in air
to be 340 m/s at 20 oC. T 1/f 1/262 s-1
0.00382 s V f l l v/f l 340 m/s / 262
/s 1.30 m
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Problem

• A sound wave traveling through water has a
  frequency of 500 Hz and a wavelength of 3 m. How
  fast does sound travel through water?
• v l f 3m (500 Hz) 1500 m/s

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Wave on a Wire

• v FT
• m / L
• v, velocity, m/s
• FT, tension on a wire, N
• m/L mass/unit length, kg/m
• m/L may be shown as m

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Problem Ex. 11-11

• A wave whose wavelength is 0.30 m is traveling
  down a 300 m long wire whose total mass is 15 kg.
  If the tension of the wire is 1000N, what are the
  speed and frequency of the wave?
• Using equation on prior slide
• v v( 1000N) / (15kg)(300m)
• 140m/s
• f v / l 140 m/s / 0.30 m 470 Hz

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Types of Waves

• A transverse wave is a wave in which particles of
  the medium move in a direction perpendicular to
  the direction which the wave moves.
• Example Waves on a guitar string
• A longitudinal wave is a wave in which particles
  of the medium move in a direction parallel to the
  direction which the wave moves. These are also
  called compression waves.
• Example Sound
• http//einstein.byu.edu/masong/HTMstuff/WaveTrans
  .html

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What are two types of wave shapes?

• Transverse
• Longitudinal

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• http//www.school-for-champions.com/science/sound.
  htm

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Transverse Wave Type
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Longitudinal Wave Type
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Longitudinal vs Transverse
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Other Waves Types Occurring in Nature

• Light electromagnetic
• Ocean waves surface
• Earthquakes combination
• Wave demos
• http//www.kettering.edu/drussell/Demos/waves/wav
  emotion.html
• http//www.kettering.edu/drussell/Demos/doppler/m
  ach1.html

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Properties of Waves

• Reflection occurs when a wave strikes a medium
  boundary and bounces back into the original
  medium.
• Those waves completely reflected have the same
  energy and speed as the original wave.

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Types of Reflection

• Fixed-end Reflection-
• The wave reflects with
• inverted phase.
• Open-end Reflection-
• The wave reflects with
• The same phase.
• www.iop.org/activity/education/Teaching_Resources

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Refraction of Waves

• Wave is transmitted
• from one medium to
• another.
• Refracted waves may
• change speed and
• Wavelength
• Almost always is accompanied
• by some reflection.
• Refracted waves do not
• change frequency.

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Sound - a longitudinal wave

• Sound travels through air about 340 m/s.
• Sound travels through other media as well, often
  much faster than 340 m/s.
• Sound waves are started by vibration of some
  other material, which starts the air vibrating.
• www.silcom.com/aludwig/musicand.htm

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Hearing Sounds

• We hear a sound as high or low pitch
  depending on the frequency or wavelength.
  High-pitched sounds have short wavelengths and
  high frequencies. Low-pitched sounds have long
  wavelengths and low frequencies. Humans hear
  from about 20 Hz to about 20,000 Hz.
• The amplitude of a sounds vibration is
  interpreted as its loudness. We measure loudness
  (also known as
  sound intensity)
  on the decibel scale, which is
  logarithmic.

http//www.allegropianoworks.com/assets/rare_compr
ess.jpg
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Doppler Effect

• The Doppler Effect is the apparent change in
  pitch of a sound as a result of the relative
  motion of an observer and the source of a sound.
  Coming toward you a car horn appears higher
  pitched because the wavelength has been
  effectively decreased by the motion of the car
  relative to you. The opposite occurs when you are
  behind the car.

http//people.finearts.uvic.ca/aschloss/course_ma
t/MU207/images/Image2.gif
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Pure Sound

• Sounds are longitudinal waves, but they can be
  shown to look like transverse waves.
• When air motion is graphed in a pure sound tone
  versus position, we get what looks like a sine or
  cosine function.
• A tuning fork produces a relatively pure tone as
  does a human whistle.

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Graphing a Sound Wave
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Complex Sounds

• Because of superposition and interference, real
  world waveforms may not appear to be pure sine or
  cosine functions.
• This is because most real world sounds are
  composed of multiple frequencies.
• The human voice and most musical instruments are
  examples.

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The Oscilloscope

• With an Oscilloscope we can view waveforms. Pure
  tones will resemble sine or cosine functions, and
  complex tones will show other repeating patterns
  that are formed from multiple sine and cosine
  functions added together. (Amplitude vs time.)

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The Fourier Transform

• The Fourier transform has long been used for
  characterizing linear systems and for identifying
  the frequency components making up a continuous
  waveform. This mathematical technique separates a
  complex waveform into its component frequencies.
• The Fourier Transforms ability to represent
  time-domain data in the frequency domain and
  vice-versa has many applications. One of the most
  frequent applications is analysing the spectral
  (frequency) energy contained in data that has
  been sampled at evenly-spaced time intervals.
  Other applications include fast computation of
  convolution (linear systems responses, digital
  filtering, correlation (time-delay estimation,
  similarity measurements) and time-frequency
  analysis.

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Fourier Transform - showing time domain and
frequency domain.
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Superposition Principle

• When two or more waves pass a particular point in
  a medium simultaneously, the resulting
  displacement of the medium at that point is the
  sum of the displacements due to each individual
  wave.
• The waves are said to interfere with each other.

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Superposition of Waves

• When two or more waves meet, the displacement at
  any point of the medium is equal to the algebraic
  sum of the displacements due to the individual
  waves.

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Types of Interference

• If the waves are in phase, when crests and
  troughs are aligned, the amplitude in increased
  and this is called constructive interference.
• If the waves are out of phase, when crests and
  troughs are completely misaligned, the amplitude
  is decreased and can even be zero. This is called
  destructive interference.

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Constructive Interference

• Crests are
• Aligned ?
• the waves are
• in phase

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Destructive Interference
Crests are aligned with troughs ? Waves are out
of phase
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Constructive Destructive Interference
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Interference Problem Draw the waveform from the
two components shown below.
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Standing Waves

• A standing wave is one which is reflected back
  and forth between fixes ends of a string or pipe.
• Reflection may be fixed or open-ended.
• Superposition of the wave upon itself results in
  a pattern of constructive and destructive
  interference and an enhanced wave. Lets see a
  simulation
• http//www.5min.com/Video/The-Rubens-Tube-Frequenc
  y-of-Fire-1858291

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Fixed-end standing waves - guitar or violin string

• Fundamental
• 1st harmonic
• l 2L
• First overtone
• 2nd harmonic
• l L
• Second Overtone
• 3rd harmonic
• l 2L/3

http//id.mind.net/zona/mstm/physics/waves/standi
ngWaves/standingWaves1/StandingWaves1.html
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Problem

• A string of length 12 m thats fixed at both ends
  supports a standing wave with a total of 5 nodes.
  What are the harmonic number and wavelength of
  this standing wave?
• L 4(1/2 l ) ? l 2L/4 4th harmonic since
  it matches ln 2L/n for n 4
• wavelength l4 2(12m) / 4 6 m

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Open-ended standing waves - flute clarinet
l 2L l L l (2/3)L

• 4L
• l (4/3)L
• l (4/5L

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physics.indiana.edu/p105_f02/standing_waves_...
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http//upload.wikimedia.org/wikibooks/en/3/32/Fhss
t_waves40.png

• open ends one end both ends
• closed closed

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Sample Problem

• 12-30. a) Determine the length of an organ pipe
  that emits middle C (262Hz). The air temp. is
  21oC.
• A) v 331m/s 0.6 m/soC(21oC) 344m/s
• A) l 2L v fl 2lf L v/2f
  344m/s/2(262/s)
• L 0.656m
• B) What are the wavelength and frequency of the
  1st harmonic?
• Frequency is 262 Hz
• Wavelength is twice the length of the pipe, 1.31
  m.
• C) What is the wavelength and frequency in the
  traveling sound wave produced in the outside air?
• They are the same because it is air that is
  resonating in the organ pipe 262Hz and 1.31 m

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Superposition of 2 sound waves http//www.ece.utex
as.edu/nodog/me379m/superposition.html
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Resonance and Beats

• Resonance occurs when a vibration from one
  oscillator occurs at a natural frequency for
  another oscillator.
• The first oscillator will cause the second to
  vibrate.
• See next slide.

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Resonance

• http//www.isd-dc.org/ISD-Wash/GIFS20Pictures20
  20Whatnots/tuningforkresonance.jpg

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Beats

• The word physicists use to describe the
  characteristic loud/soft pattern that
  characterizes two nearly matched frequencies.
• Musicians call this being out of tune.

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Beats

• When two sound waves whose frequencies are close
  but not exactly the same, the resulting sound
  modulates in amplitude changing from loud to soft
  to loud. This is called beat frequency and is
  shown by
• fbeat f 1 - f 2

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Diffraction

• Bending of a wave around a barrier
• Diffraction of waves combined with interference
  of the diffracte waves causes diffraction
  patterns.
• Here is an example using a ripple tank.
• http//www.falstad.com/ripple/

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Double-slit or multi-slit diffraction

• micro.magnet.fsu.edu/.../doubleslit/
• Remove frame 

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(No Transcript)
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Single Slit Diffraction

• n l s sin q
• n -- dark band number
• l -- wavelength (m)
• s -- slit width (m)
• q -- angle defined by central band, slit, and
  dark bank

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Sample Problem

• Light of wavelength 360 nm is passed through a
  diffraction grating that has 10,000 slits per cm.
  If the screen is 2.0 m from the grating, how far
  from the central bright band is the first order
  bright band?

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Sample Problem

• Light of wavelength 560 nm is passed through two
  slits. It is found that, on a screen 1.0 m from
  the slits, a bright spot is formed at x 0, and
  another is formed at x 0.03m. What is the
  spacing between the slits?

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Sample Problem

• Light is passed through a single slit of width
  2.1 x 10-6 m. How far from the central bright
  band do the 1st and 2nd order dark bands appear
  if the screen is 3.0 m away from the slit?

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Mathematical Description of a Traveling Wave

• Y A sin (v t k x )
• Y dependent of x and t y(x,t) or y of x t
• If the - sign is used, wave is traveling in x
  direction ?
• A is amplitude of the wave
• m (omega) is angular frequency (m 2pf)
• k angular wave (k 2 pk, k 1/l)

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Other forms

• Important features of the wave amplitude,
  frequency f (through v), period T (which is 1/f
  2p/v), wavelength (l 2p/k) and wave speed v
  (which is lf v/k)
• y Asin2pft (1/l)x or
• y Asin(2p/l)(vt x)

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Sample Problem The vertical position y of any
point on a rope that supports a transverse wave
traveling horizontally is given by the
equation y 0.1 sin (6 p t 8 p x)
Find amplitude 0.1 angular frequency v
6 p s -1 frequency f v / (2 p) (6 p s -1 )/
(2 p ) 3 Hz angular wave number k k / (2 p )
8 p m -1) / ((2 p) 4 m-1 wavelength l 1/ k
1/ (4 m-1) 0.25 m period T 1/f 1 / 3 Hz
0.33s wave speed v f l 0.25m (3 Hz)
0.75 m/s
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Assignment

• P15/MC1-4

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Sound Level

• Intensity Rate at which sound waves transmit
  energy is measured in energy per unit area
  watt/m2 or watt/cm 2
• Intensity level or loudness level, B
• B 10log I/Io where Io 1x10-12 w/m2
• 
  or 1x10-16 w/cm2

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More math

• b 10 log I
• Io 10-16 w/cm 2
• Intensity level 10 log Intensity / threshhold
  of hearing
• We all dont hear the same, so this is a
  comparative measurement in decibels

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Flow chart for b problem

• If I 4.7 x 10-10 w/cm2
• 10xlog(4.7 2nd EE -10 / 1 2nd EE -16) 66.7
  dB
• If I 2.9 x 10-3 w/cm2
• 10xlog(2.9 2nd EE -3 / 1 2nd EE -16) 135
  dB

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Problem

• Now we are going backwards from intensity level
  (dB) to intensity (w/cm2)
• If the intensity level is 83 dB, convert that to
  intensity in w/cm2.
• B 10 log I / Io get to a working eqtn
• B /10 log I / Io
• Log-1(B/10) Log-1(log I/Io)
• Log-1(B/10) I/Io

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• Lets say that the intensity level of a sound is
  25.3 dB. What is the intensity of the sound in
  w/cm2?
• B 10 log I / Io
• 25.3 dB 10 log (I/10-16 w/cm2)
• 2.53 log I log 10-16
• 2.50 log 10-16 log I
• 2.50 16 log I

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• 25.3 dB 10 log (I/10-16 w/cm2)
• 2.53 log I log 10-16
• 2.50 log 10-16 log I ? what power do you
  raise 10 to, to get 10-16?
• 2.50 16 log I
• Adding on the left ? -13.5 log I
• Raise 10 to the -13.5 power by this sequence
• 2nd 10x (-13.5) ? 3.16 x 10-14 w/cm2 I

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Doppler Effect

• Doppler Effect is the apparent change in
  frequency as a result of relative motion between
  the source of a sound and an observer.
• f frequency heard by observer
• f frequency of source
• v velocity of sound in air
• vd velocity of detector
• vs velocity of source

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Sample Problem

• A source of 4 kHz sound waves travel at 1/9 the
  speed of sound toward a detector thats moving at
  1/9 the speed of sound, toward the source.
• a. what is the frequency of the waves as theyre
  received by the detector?
• b. how does the wavelength of the detected waves
  compare to the wavelength of the emitted waves?

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• v 1/9 v x f 5/4 f 5/4 (4 kHz) 5 kHz
• v - 1/9 v
• sign on top as detector moves toward source
• sign on bottom as source moves toward det.
• Frequency is shifted up by a factor of 5/4, the l
  will shift down by the same factor.
• ld ls / 5/4 4/5 l s

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Sample

• A person yells, emitting a constant frequency of
  200 Hz, as he runs at 5m/s toward a stationary
  brick wall. When the reflected waves reach the
  person, how many beats per second will he hear?
  (Use 343 m/s for the speed of sound.)

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Determine what f will be reflected and heard by
the runner. person is source and wall is
detector fwall v f
v - vrunner Reflected sound wave
(no change in f) wall is the source and runner
is the detector. frunner v v
runner f
v Combine these two formulae frunner v
vrunner f v - vrunner
f (3435)m/s (200Hz) 206 Hz
(343-5)m/s Beat frequency is fbeat 206Hz - 200
Hz 6 Hz. p. 343 Princeton
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Doppler Effect for Light

• wps.prenhall.com/.../ch25_SWA/images/8.gif
• c 3 x 10 8 m/s
• u speed of the source and the detector.

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www.sv.vt.edu/.../class95/physics/doppler.gif
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Periodic Motion

• Motion that repeats itself over a fixed and
  reproducible period of time
• An example is a planet moving about its sun.
  This is called the period (T) or year of the
  planet
• Mechanical devices on earth that have periodic
  motion are useful timers and are called
  oscillators.

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Simple Harmonic Motion

• Attach a weight to a spring, stretch the spring
  past its equilibrium point and release. The
  weight bobs up and down with a reproducible
  period, T
• Plot position vs time to get a graph that
  resembles a sine or cosine function. The graph is
  sinusoidal, so the motion is referred to as
  simple harmonic motion.
• Springs and pendulums undergo simple harmonic
  motion and are referred to as simple harmonic
  oscillators.

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Graph Analysis
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Graph Analysis
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Oscillator Definitions

• Amplitude
• Maximum displacement from equilibrium
• Related to energy
• Period
• Length of time required for one oscillation
• Frequency
• How fast the oscillator is oscillating
• f 1/T in Hz or s-1

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Problem 11-5

• An elastic cord vibrates with a frequency of 3.0
  Hz when a mass of 0.60 kg is hung from it. What
  is its frequency if only 0.38 kg hangs from it?

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Springs

• Springs are a common type of simple harmonic
  oscillator.
• Our springs are ideal springs, which means
• They are massless
• They are both compressible and extensible.
• They will follow Hookes Law.
• F -kx

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Hookes Law Review
The force constant of a spring can be determined
by attaching a weight and seeing how far it
stretches.
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Period of a Spring
T period (s) m mass (kg) k force constant
(N/m)
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Sample Problem

• Calculate the period of a 350 g mass attached to
  an ideal spr5ing with a force constant of 35 N/m.

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Sample Problem

• A 300 g mass attached to a spring undergoes
  simple harmonic motion with a frequency of 25 Hz.
  What is the force constant of the spring?

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Sample Problem

• An 80 g mass attached to a spring hung vertically
  causes it to stretch 30 cm from its unstretched
  position. If the mass is set into oscillation on
  the end of the spring, what is its period?

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Combinations of Springs

• In parallel, springs work together
• In series, springs work independently

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What do you think?
Does this combination of springs act as parallel
or series? Answer parallel !
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Problem

• If you want to double the force constant of a
  spring, you
• A. double its length by connecting it to another
  one just like it.
• B. cut it in half.
• C. add twice as much mass.
• D. take half of the mass off.
• Answer B

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Conservation of Energy

• Springs and pendulums obey conservation of energy
• The equilibrium position has high kinetic energy
  and low potential energy.
• The positions of maximum displacement have high
  potential energy and low kinetic energy.
• Total energy of the oscillating system is
  constant.

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Problem 11-20

• A block of mass m is supported
• by 2 identical parallel vertical
• springs, each with spring stiffness
• constant k. What will be the
• frequency of vibration?

General form of a restoring force producing SHM
with spring constant of 2k
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Sample Problem
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Spring Problem
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Another spring problem
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Pendulums

• Pendulums can be thought of as simple harmonic
  oscillators.
• The displacement needs to be small for it to work
  properly.

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Conservation of Energy

• Pendulums also obey conservation of energy.
• The equilibrium position has high kinetic energy
  and low potential energy.
• The positions of maximum displacement have high
  potential energy and low kinetic energy.
• Total energy of the oscillating system is
  constant.

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Pendulum Forces
2005 Pearson Prentice Hall Fig 11-12
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Period of a Pendulum
T -- period (s) l -- length of string (m) g --
gravitational acceleration (m/s2)
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Problem

• Predict the period of a pendulum consisting of a
  500 g mass attached to a 2.5 m long string.

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Problem

• Suppose you notice that a 5 kg weight fixed to a
  string swings back and forth 5 times in 20
  seconds. How long is the string?

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Last problem!

• The period of a pendulum is observed to be T.
  Suppose you want to make the period 2T. What do
  you do to the pendulum?