physics52 optics ch37

7/31/2019 Physics52 Optics Ch37

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SPECIAL RELATIVITY-Postulates of Special Relativity

-Relativity of time > time dilation-Relativity of length > lengthcontraction

2005 J. F. Becker


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C 2001 Wiley, PhysicsCutnell & Johnson 5th Ed.

An event takes place at a certain time andplace as measured in a frame of reference, or

coordinate system (x, y, z, t)

INERTIAL FRAME OF REFERENCE:

Newtons law of inertia is valid in the frame ofreference. The acceleration of a body is zero

when measured in the coordinates.

C 2005 J. F. Becker


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Two inertial frames of reference moving withconstant relative velocity v

vt C 2005 J. F. Becker


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POSTULATES (ASSUMPTIONS)of Special Relativity

1. The relativity postulate: The laws of physicsare the same in every inertial referenceframe.

2. The speed of light postulate: The speed oflight in a vacuum, measured in any inertialreference frame, always has the same value

of c, no matter how fast the source of lightand the observer are moving relative to eachother.

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Both the person on the truck and the observeron the earth measure the speed of light to be

c, regardless of the speed of the truck.

The speed of light postulate

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We will see some mind-boggling effects of thetheory of special relativity, such as:

MOVING CLOCKS TICK SLOW!(Time dilation)

MOVING RULERS APPEAR CONTRACTED!

(Length contraction)

But first, lets consider the concept ofsimultaneity

(two events happening at the same time)

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SIMULTANEITY IS RELATIVE!Whether two events are simultaneous dependson the frame of reference. Two lightning boltsstrike the railroad car and ground at each end.

u

u

u

u

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TIME DILATIONThe proper time interval to between twoevents is the time interval measured by an

observer who is at rest relative to the eventsand views them as occurring at the same place.

An observer who is in motion with respect tothe events and who views the events asoccurring at different places measures a

dilated (expanded) time interval t.

The dilated time interval is greater than theproper time interval ( t > to)according to thetime-dilation equation: t = to / (1 u2/c2)

t > to and t indicates a slow clock!


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Time dilation: t = to / (1 u2/c2)

A spacecraft speeds past Earth at a constant

speed of u = 0.92c and the astronauts measurethe time between ticks of the spacecraftclock to be 1.0 sec. ( to = 1.0 sec.).

What time interval do observers on Earthmeasure? t = ?

t = to / [1 (0.92)2]1/2 = to / 0.392 =2.6 to

t = 2.6 sec.

MOVING CLOCKS TICK SLOW!

C 2005 J. F. Becker


10/26C 2001 Wiley, PhysicsCutnell & Johnson 5th Ed.

A light clock: One tick is the time interval ittakes for the light pulse to travel the roundtrip distance / speed of light or (2 d = c x t)

time interval = 2 d / c

For stationary clock:to = (2 d) / c

For moving clock:t = (distance) / c

d

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Time dilation

u

to t

u t

u t/2

L = [ d2 + (u t/2) 2 ]

The path observed by Stanley is longer than d.

d L L

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(a) Mavis measures (proper) to (= 2 d /c).(b) Stanley measures t.

The path observed by Stanley is longer:

2 L = 2 [ (d2 + (u t / 2)2 ]

So c t = 2 L = 2 [ (c to / 2)2 + (u t / 2)2 ]

t)2 = (2 L/c)2 = (2/c)2 [ (c to/2)2 + (u t/2)2]

t)2 = [ ( to)2 + (u t/c)2]

t)2 -(u t/c)2 = [ ( to)2 ]

t)2 {1-(u /c)2 }= [ ( to)2 ]

Time dilation: t = to / {1 u2/c2}


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Experimental verification of time dilation Muons are particles observed on earth afterbeing created in the upper atmosphere whencosmic rays from the Sun collide with atomsin our atmosphere. The muon quickly decays

into an electron and a neutrino particle.

These muons travel toward earth with speedu = 0.998 c

(Lifetime of a muon at rest = 2.2 (10)-6 s.)

(a) How long does one of these muons liveaccording to an observer on earth?

(b) How far does a muon travel before itdisinte rates? C 2005 J. F. Becker


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Verification of time dilation

How long does one of these muons liveaccording to an observer on earth?

The two events are the generation anddisintegration of the muon. When the muon

is at rest the events take place at the sameplace so the lifetime is the proper time

interval to = 2.2 (10)-6 s.

The muon moves at u = 0.998 c so Earthlingsmeasure a dilated time interval t.

t = to / (1 u2/c2)

t = 35 10 -6 s. C 2005 J. F. Becker


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Verification of time dilation

How far does a muon travel before it

disintegrates?If we mistakenly neglected relativistic effects

the calculated distance would be

d = ut =0.998 [300(10)6 m/s] 2.2(10)-6s =659 m(distance too short; muons never reach earth!)

An Earthling measures a distance of

d = u t =0.998 [300(10)6 m/s] 35(10)-6s

d = u t = 10,500 m

(the distance to top of atmosphere)C 2005 J. F. Becker

h



Length contraction As measured by anEarthling the Earth-to-star distance is Lo, theproper length as measured by an observer atrest with respect to the ends of the ruler, andthe time to make the trip is t. An Astronautmeasures the distance to be L & the time to.

Voyage to a star!

Lo L

u

u u

to

t

C 2005 J. F. Becker

h d /


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Length contraction: u = 0.95 c = distance/timeThe relative velocity is: u = L / to = Lo / t

L = (Lo / t) toAnd substituting t = to / (1 u2/c2)

we get L = (Lo / t) t(1 u2/c2)

L = Lo (1 u2

/c2

)

Lo = the proper length between two points asmeasured by an observer at rest with respectto the two points, i.e., a stationary ruler.


C 2005 J. F. Becker

L h f h f


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Length contraction from the muons frame:Recall the muons speeding toward earth at u =0.998 c. In the muons reference frame it

lives to = 2.2 (10)-6 s and it measures thedistance to earth as L. But, as measured byEarthlings the distance is Lo = proper length =

10,500 m. Since moving rulers are contractedthe muon measures L = Lo (1 u2/c2)

L=10,500 m. (1 u2/c2) =632 m., a short

enough distance to earth to cover in 2.2 ms.

Lo = the proper length between two points asmeasured by an observer at rest with respectto a ruler a stationar ruler . C 2005 J. F. Becker


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Time dilation:

MOVING CLOCKS TICK SLOW!

Length contraction:


C 2005 J. F. Becker

G h h h



Graph shows howthe factor

(1 u2

/c2

)

increases as therelative speed

approaches c.c ~ 300 (10)6

km/s

c ~ 670 (10)6mi/hr

u

EQUIVALENCE OF MASS AND ENERGY


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EQUIVALENCE OF MASS AND ENERGY

Total energy of an object:E = mc2 / (1 u2/c2)

Rest energy of an object:Eo = mc2 / (1 - 0)

Eo

= mc

2

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REVIEW

C 2005 J. F. Becker

L th t ti li ht l fl t d ff


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Length contraction light pulse reflected offmirror at end of ruler


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( ) A t t ( ) t


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(a) Astronaut measures (proper) to.(b) Earthling measures t.

The path observed by Earthling is longer:

s = [ D2 + L2 ] = [ (D2 + (u t/2) 2 ]

L = u t/2

C 2001 Wil Ph C ll & J h 5th Ed

physics52 optics ch37

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