11/14/2011
Time Dilation
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The train is moving to the
right at speed v
An observer, O’, at rest in this
system, with laser emits a
pulse of light directed at the
mirror (event 1) and the
pulse arrives back after being
reflected (event 2)
Observer O’ carries a clock
She uses it to measure the time
between the events (∆tp)
 The p stands for proper
 She observes the events to
occur at the same place
 ∆tp = distance/speed = (2d)/c
Time Dilation-- Stationary Observer
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Can relate times with geometry
and ∆
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∆
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t 
2 /
1v
where  
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 t p
2
c2
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1
1v
2
c2
Greek letter gamma “ ” > 1
Observer O measures a longer time interval than
observer O’
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Identifying Proper Time
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∆
t p
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Time Dilation Verification – Muon Decays
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Muons are unstable particles
that have the same charge as
an electron, but a mass 207
times more than an electron
Muons have a half-life of ∆tp =
2.2µs when measured in a
reference frame at rest with
respect to them (a)
Relative to an observer on
earth, muons should have a
lifetime of  ∆tp (b)
Observer O
observes the mirror
and O’ to move
with speed v
By the time the
light from the laser
reaches the mirror,
the mirror has
moved to the right
The light must
travel farther with
respect to O than
with respect to O’
Both observers must measure c for speed of light
The light travels farther for O
The time interval, ∆t, for O is longer than the time
interval for O’, ∆tp
The time interval ∆tp is called the proper time
 The proper time is the time interval between events
as measured by an observer who sees the events
occur at the same position
The view of O’ that O is really the one moving with
speed v to the left and O’s clock is running more slowly
is just as valid as O’s view that O’ was moving
The principle of relativity requires that the views of the
two observers in uniform relative motion must be
equally valid and capable of being checked
experimentally
All physical processes slow down relative to a clock
when those processes occur in a frame moving with
respect to the clock-can be chemical and biological as
well as physical
Time dilation is a very real phenomena that has been
verified by various experiments
The Twin Paradox
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A thought experiment involving a set of twins, Speedo
and Goslo
Speedo travels to Planet X, 20 light years from earthhis ship travels at 0.95c
 After reaching planet X, he immediately returns to
earth at the same speed, keeping his eyes shut for
the whole trip
When Speedo returns, he has aged 13 years, but Goslo
has aged 42 years
Goslo’s perspective is that he was at rest while
Speedo went on the journey
Speedo thinks he was at rest and Goslo and the earth
raced away from him on a 6.5 year journey and then
headed back toward him for another 6.5 years
The paradox – which twin is the traveler and which is
really older?
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11/14/2011
The Twin Paradox – The Resolution
Today’s Lab
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Relativity applies to reference frames moving at
uniform speeds
The trip in this thought experiment is not
symmetrical since Speedo must experience a series
of accelerations during the journey
Therefore, Goslo can apply the time dilation
formula with a proper time of 42 years
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This gives a time for Speedo of 13 years and this agrees
with the earlier result
There is no true paradox since Speedo is not in an
inertial frame
Using a Tuning Fork to Produce a
Sound Wave
Producing a Sound Wave
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Sound waves are longitudinal waves traveling
through a medium
A tuning fork can be used as an example of
producing a sound wave
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Tuning Fork continues to Vibrate
A tuning fork will produce a pure musical
note
As the tines vibrate, they disturb the air
near them
As the tine swings to the right, it forces the
air molecules near it closer together
This produces a high density area in the air
 This is an area of compression
As the tine moves toward the left, the air
molecules to the right of the tine spread
out
This produces an area of low density
 This area is called a rarefaction
Speed of Sound in Air
m
T
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v   331 
s  273 K
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a succession of compressions and rarefactions
spread out from the fork
A sinusoidal curve can be used to represent
the longitudinal wave
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331 m/s is the speed of sound at
0° C
T is the absolute temperature
Crests correspond to compressions and troughs to
rarefactions
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11/14/2011
Standing Waves
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When a traveling wave reflects back on itself, it creates
traveling waves in both directions
The wave and its reflection interfere according to the
superposition principle
With exactly the right frequency, the wave will appear
to stand still
 This is called a standing wave
A node occurs where the two traveling waves have the
same magnitude of displacement, but the
displacements are in opposite directions
 Net displacement is zero at that point
 The distance between two nodes is ½λ
An antinode occurs where the standing wave vibrates
at maximum amplitude
Examples of Resonance
Name some…
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Child being pushed on a swing
Shattering glasses
Tacoma Narrows Bridge collapse due
to oscillations by the wind
Upper deck of the Nimitz Freeway
collapse due to the Loma Prieta
earthquake
Resonance in an Air Column
Closed at One End
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A system with a driving force will
force a vibration at its frequency
When the frequency of the driving
force equals the natural frequency
of the system, the system is said
to be in resonance
Standing Waves in Air Columns
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If one end of the air column is closed, a node must
exist at this end since the movement of the air is
restricted
If the end is open, the elements of the air have
complete freedom of movement and an antinode exists
Open
Closed
Standing waves on a Surface
The closed end must be a node
The open end is an antinode
fn  n
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Forced Vibrations
v
 nƒ1
4L
n  1, 3, 5, 
There are no even multiples of the
fundamental harmonic
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