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Giancoli 6th edition
Vibrations & Waves
Oscillation – vibrations back and forth
Periodic motion
Same path
Same time
Equilibrium position -- No force is exerted on the osculating object
Restoring Force (F): Force generated by elastic material (spring) when a mass is moved away from the equilibrium position
F = -kx
if the force is exerted by the spring
Negative because the force is always applied opposite the displacement of the mass
F = +kx
if the force is generated external to the spring
x -- displacement
amplitude (A)
maximum displacement
cycle -- one complete back and forth motion
mass leaves a point and then arrives back at the same point moving in the same direction
period (T) time needed to complete 1 cycle
frequency (f) number of cycles per time period
cycles/sec = hertz
T = 1/f
f = 1/T
Simple Harmonic Motion (SHM) a vibrating system where the restoring force is directly proportional to the negative of the
displacement
Restoring Force is always trying to pull the mass back to the equilibrium position
Simple Harmonic Oscillator (SHO) a mechanism that demonstrates SHM
Energy of a SHO
Energy = PE + KE = ½ kx2 + ½ mv2
Energy total = ½ kA2 + ½ m02 = ½ kA2
Energy total = ½ k02 + ½ mv2max = ½ mv2max
11-3 The Period and Sinusoidal Nature of SHM
 FAVORITES, physics, Flash animation, classical animations, SHM
 www.splung.com/content/sid/2/page/shm
 Tom Altman and SHM at the Exploriatorium
go to google

www.youtube.com/watch?v=egPZUDAMJZ0
 http://www.youtube.com/watch?v=Ce9FWd1LX9Y
 Animation http://www.youtube.com/watch?v=5L_vuX0xeJY

https://www.google.com/search?q=simple+harmonic+motion&rlz=1C2EODB_enUS
549US549&tbm=isch&imgil=pdk4xsCxYLw8EM%253A%253Bhttps%253A%252F%2
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3KPtM%253Bhttp%25253A%25252F%25252Fwww.gailruby.com%25252FSHMGraph
withrelations.htm&source=iu&usg=__GNPdGFVabA77QZJ0aTNnMaROPms%3D&s
a=X&ei=Qm77UrHO4e8rQGppICADw&sqi=2&ved=0CLABEP4dMA8&biw=1600&bih=799#facrc=_&im
gdii=_&imgrc=pdk4xsCxYLw8EM%253A%3B551I9zauF3KPtM%3Bhttp%253A%252
F%252Fwww.gailruby.com%252FPicture4.png%3Bhttp%253A%252F%252Fwww.gai
lruby.com%252FSHMGraphwithrelations.htm%3B1275%3B653
If we look at the projection onto the x axis of an object moving in a circle of radius A at a
constant speed vmax, we find that the x component of its velocity varies as:
This is identical to SHM.
Therefore, we can use the period and frequency of a particle moving in a circle to find the
period and frequency:
similar triangles for
We can similarly find the position as a function of time:
The top curve is a graph of the previous equation.
The bottom curve is the same, but shifted ¼ period so that it is a sine function rather
than a cosine.
Pendulum
T = 2π (L/g)1/2
`
Video Pendulum effect http://www.youtube.com/watch?v=BtoMIKelvN0
SHM No loss of energy - NOT PRACTICAL
Energy (force) must be applied to maintain the constant amplitude
Damped Harmonic Motion
Something interferes with normal SHM.
 The interference is due to something removing the energy from the
SHM.
 Or the driving force (energy) is removed
Video dampened harmonic motion: http://www.youtube.com/watch?v=hv30JMdpx4o
Example: an auto shock absorber
Resonance
Natural Frequency: Frequency at which an object vibrates at naturally.
Example
Bell
Resonance is when an object vibrates because an outside source of waves (energy)
hit this object with a frequency that matches its natural frequency.
A prolonged resonance can cause a large increase in the amplitude of the object’s
vibrations.
Forced Frequency: when an object vibrates at the frequency of an outside source
of vibrational waves that are striking it. These outside vibrations due not match
the natural frequency of the object.
Wave Motion
Wave classification
A. By medium type
1. Mechanical waves—require a medium to travel through
2. Electro magnetic waves --- can travel through a vacuum. A partial
vacuum will reduce its velocity
B. By wave shape http://www.youtube.com/watch?v=7cDAYFTXq3E
1. transverse
https://www.youtube.com/watch?v=RJnfMY4iT_g
2. longitudinal (compression)
https://www.youtube.com/watch?v=9wwpPBe_L-4
3. surface wave - ----- combination of transverse and longitudinal
water wave http://www.youtube.com/watch?v=oFhMrwQHvFo
https://www.youtube.com/watch?v=Nw6UavK488Q
Pulse ------ a single wave formed by a single disturbance
https://www.youtube.com/watch?v=t-e66Ds8rW8
Periodic wave (continuous wave) ----- a series of waves caused by a series of disturbances
https://www.youtube.com/watch?v=FtgkrgTQdAc
 Source is often oscillating vibration
 If the source is SMH the wave will appear as a sine wave.
o Resting point (line)
o Positive amplitude
o Negative amplitude
o Period
Wave Characteristics
 Amplitude—maximum height or depth a crest or trough moves from a resting point
or line
o Crest --- positive distance from resting point (compare long vs. transverse)
o Trough --- negative distance from a resting point
 Wavelength --- distance from a point on a wave to the same point on an adjacent
wave (often measured crest to crest)
 Explain the difference between longitudinal and transverse wave
length)
 Frequency --- number of waves that pass a fixed point in a time unit (often seconds
which are expressed as hertz))
 Period --- time required for a one complete wave to pass a fixed pint (time from
crest to crest)

Velocity of a wave v = fλ waves will have specific velocity in a specific medium
Velocity of a mechanical wave
Vel = [elastic factor /inertia factor]1/2
In liquid
velocity = [β/ρ]1/2
In solid
Velocity = [Y/ρ]1/2
In air
Velocity = 331m/s + 0.6 Tc m/s
Velocity = 331[1 + Tc/273]1/2 m/s
On string
Velocity =
[FT/liner density]1/2
linear density = m/L
Length of a string is a multiple of the wave’s ½ wave length’s
L= n1/2λn
n = 1,2,3,….
Fundamental frequency
Overtones
1st Harmonic
2nd Harmonic
3rd Harmonic
Energy in a Wave:
Larger the amplitude the more energy a wave has
Energy is proportional to the square of the amplitude
E ≈ A2
Intensity of a wave (I)
Power in a given segment of a wave (m2)
I = Power/area
area of a sphere = 4πr2
From Wikipedia, the free encyclopedia
For other uses, see Intensity (disambiguation).
In physics, intensity is the power transferred per unit area. In the SI system, it has units watts per metre squared
(W/m2). It is used most frequently with waves (e.g. sound or light), in which case the average power transfer over
one period of the wave is used. Intensity can be applied to other circumstances where energy is transferred. For
example, one could calculate the intensity of the kinetic energy carried by drops of water from a garden sprinkler.
The word "intensity" as used here is not synonymous with "strength", "amplitude", "magnitude", or "level", as it
sometimes is in colloquial speech.
Intensity can be found by taking the energy density (energy per unit volume) at a point in space and multiplying it by
the velocity at which the energy is moving. The resulting vector has the units of power divided by area.
Inverse Square Law
Let the total power radiated from a point source be P. At large distances from
the source (compared to the size of the source), this power is distributed over
larger and larger spherical surfaces as the distance from the source increases.
Since the surface area of a sphere of radius r is A = 4πr 2, then I (power per unit
area) of radiation at distance r is
Carrying the energy of the initial disturbance away from the source
As the sphere enlarges the energy becomes increasingly more dilute
I≈
1/r2
I2/I1 = r12/r22
WAVE at the BOARDER
Reflects or Transmits
Reflection a wave that remains in the original medium after striking the boundary of
its medium
Transmits wave leaves original medium and inters a new medium
Refraction bending of a wave as it enters a new medium
Diffraction the bending of a wave around an obstacle that is in the wave’s medium
Reflection and Transmission of Waves: When a wave strikes the boundary of a new medium
1. some of the energy is reflected off of the boundary and remains in the original medium
2. some of the energy is transmitted through the boundary into the new medium
Inverted and erect reflections dependent on the density of the new medium
More dense--- inverted reflection, (OUT OF PHASE) the reflected wave’s
profile will be the inverse of the incident wave
Less dense ---- erect reflection,(IN PHASE ) the reflected wave’s profile will be
the same as the incident wave
Video http://www.youtube.com/watch?v=0mZk2vW5rWU
Law of Reflection the angle of the incident wave will equal the angle of the reflected
wave.
Principle of Superposition when waves of the same type meet and merge the resultant amplitude
of the waves is the algebraic sum of their separate displacements.
Constructive interference the crests or troughs of two or more waves combine to
create a larger crest or trough
Destructive interference a crest combines with a trough with the result wave
having a reduced amplitude
Wave interference does not result in a permentantly altered amplitude in either wave
IN PHASE when a series of waves are aliened crest to crest and trough to trough.
Both waves must have the same speed & frequency This results in constructive
interference.
OUT OF PHASE when a series of waves are aliened crest to trough. This results in
destructive interference.
Can be some percentage of phase 30% out of phase
Stop video to show in phase and out of phase and percentage of phase
http://www.youtube.com/watch?v=ic73oZoqr70
Standing wave
Video http://www.youtube.com/watch?v=ic73oZoqr70
Standing Waves Two waves traveling in the same medium in opposite directions
will form standing waves when they have similar amplitudes and the same
frequency. This is the result of a combination of constructive and destructive
interference
Conditions for standing wave
1.
2.
3.
4.
5.
Similar wave types
Travel in opposite directions in same medium
Have the same velocity
Have same frequency
Have same wavelength
Nodes points of total destructive interference in a standing wave
Combination of 1 crest and 1 trough
Antinodes points of total constructive interference in a standing wave
Combination 2 crests or 2 troughs
Wave interference ORIGINAL DOUBLE SLIT EXPERIMENT
https://www.youtube.com/watch?v=Iuv6hY6zsd0 TIME 4:35