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C HAPTE R 3 9 Relativity
u
u x" !
x
# v
# 0.850c # 0.750c
this example, we would find that u " ! u # v ! # 0.850c #
!
x
(# 0.850c)(0.750c)
c2
1#
u vx
1 # c2
! # 0.977c
What If? What if the two spacecraft pass each oth
To finalize this problem, note t
1258
C HAPTE R 3 9 Relativity
contradiction due to the apparent symmetry of the observations. Which
twin has developed signs of excess aging?
The situation in our current problem is actually not symmetrical. To
resolve this apparent paradox, recall that t
1256
C HAPTE R 3 9 Relativity
1.0
Muon
movi
ng at
0.999
4c
0.5
50
100
Figure 39.9 Decay curves for
muons at rest and for muons
traveling at a speed of 0.999 4c.
150
t(s)
high in the atmosphere where they are produced. However, experiments
show that a larg
S E C T ION 3 9 . 4 Consequences of the Special Theory of
Relativity
1253
v
v
Mirror
y
y
d
O
O
O
x
O
d
O
c t
2
x
v
t
2
(a)
Active Figure 39.6 (a) A mirror is fixed to a moving vehicle, and a light pulse
is sent out by observer O" at rest in the vehicle. (
S E C T ION 3 9 . 4 Consequences of the Special Theory of
Relativity
1255
general, the proper time interval is the time interval between two
events measured by an observer who sees the events occur at the same
point in space.
If a clock is moving with res
1254
C HAPTE R 3 9 Relativity
flashlight at an angle with respect to the vertical direction. Comparing Figure
39.6a and b, we see that the light must travel farther in (b) than in (a). (Note
that neither observer knows that he or she is moving. Each is at
Our everyday experiences and
observations have to do with objects that
move at speeds much less than the speed of light. Newtonian mechanics
was formulated by observing and describing the motion of such objects,
and this formalism is very successful in de
1252
C HAPTE R 3 9 Relativity
coordinates and one time coordinate. Observers in different inertial
frames will describe the same event with coordinates that have different
values.
As we examine some of the consequences of relativity in the
remainder of th
S E C T ION 3 9 . 4 Consequences of the Special Theory of
Relativity
1251
difficulties and at the same time completely altered our notion of space and
time.3
He based his special theory of relativity on two
postulates:
1. The principle of relativity: The
S E C T ION 3 9 . 4 Consequences of the Special Theory of
Relativity
1259
For example, suppose that a meter stick moves past a stationary Earth observer
with
y speed v, as in Figure 39.11. The length of the stick as measured by an
observer in a frame atta
1260
C HAPTE R 3 9 Relativity
from the point of view of Goslo. A path through spacetime is called a
world-line. At the origin, the world-lines of Speedo and Goslo coincide
because the twins are in the same location at the same time. After Speedo
leaves on
S E C T ION 3 9 . 7 Relativistic Linear Momentum and the Relativistic Form of Newtons
Laws
1267
u"y for
Emily using Equations
39.16 and 39.17:
ux # v
u x"
!
1#
Thus, the speed of Emily as observed by David is
u" ! (ux")2 & (u"y)2 !
0 # 0.75c
u v !
! # 0.7
S E C T ION 3 9 . 4 Consequences of the Special Theory of
Relativity
1257
Example 39.2 How Long Was Your Trip?
Suppose you are driving your car on a business trip
and are traveling at 30 m/s. Your boss, who is
waiting at your destination, expects the trip
1264
C HAPTE R 3 9 Relativity
Example 39.6 Simultaneity and Time Dilation Revisited
Use the
Lorentz transformation equations in
time interval for the same two events as measured by
difference form to show that
O is nonzero, and so the events do not appear
S E C T ION 3 9 . 5 The Lorentz Transformation Equations
1263
at which the events occur does not depend on motion of the observer.
Because this is contradictory to the notion of length contraction, the
Galilean trans- formation is not valid when v approac
S E C T ION 3 9 . 6 The Lorentz Velocity Transformation Equations
1265
and u"z do not contain the parameter v in the
numerator because the relative velocity is along the x axis.
Note that u"y
When v is much
smaller than
c (the
nonrelativistic case), the
d
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C HAPTE R 3 9 Relativity
What If?
What if this trip is observed with a very
powerful telescope by a technician in Mission Control
on Earth? At what time will this technician see that
the astronaut has
the telescope. This will require a time interval
Modern
Physics
t the end of the nineteenth century, many scientists believed
that they had learned most of what there was to know about
physics. Newtons laws of motion and his theory of universal
gravitation, Maxwells theoretical work in unifying
electric
S E C T ION 3 9 . 4 Consequences of the Special Theory of
Relativity
1261
ct (m)
Front
ct (m)
Trailing
end of
Back
Leading door
pole
door
Pole is
Leading Front door
20
entirely
pole
20
10
trailing end
10
0
Back door end
10
10
x (m)
end o
in barn
of pole
T
1250
C HAPTE R 3 9 Relativity
Because v 2/c
2
( 1, we can simplify this expression by using
the following binomial expansion after dropping all terms
higher than second order:
(1 # x)n %
1 # nx
(for x ( 1) In our case, x ! v 2/c 2, and
we find that
't ! '
S E C T ION 3 9 . 2 The MichelsonMorley
Experiment
1249
The experiment was designed to determine the velocity of the Earth
relative to that of the hypothetical ether. The experimental tool used was the
Michelson interfer- ometer, which was discussed in Se
1248
C HAPTE R 3 9 Relativity
at rest with respect to the ether. The Galilean velocity transformation
equation was expected to hold for observations of light made by an observer in
any frame moving at
c
v
travels along the x axis and an observer moves wit
Problems
13.
In Figure 37.5 let L # 1.20 m and d
# 0.120 mm and assume that the slit
system is illuminated with monochromatic 500-nm light. Calculate the
phase difference
between the two wave fronts arriving at P when (a)
! #
0.500 and (b) y # 5.00 mm. (c
1196
C HAPTE R 3 7 Interference of Light Waves
Figure 37.23 The Laser Interferometer Gravitational-Wave Observatory (LIGO) near
Richland, Washington. Note the two perpendicular arms of the Michelson interferometer.
S U M MAR Y
Take a practice test
for thi
Problems
1197
QUESTIONS
1. What is the necessary condition on the path length
differ- ence between two waves that interfere (a)
constructively and (b) destructively?
illuminated with white light from above and observed
from above. Is there a dark spot or
1194
C HAPTE R 3 7 Interference of Light Waves
O
Figure 37.21 (Example 37.5) (a)
Inter- ference bands in reflected
light can be observed by
illuminating a wedge- shaped film
with monochromatic light. The
darker areas correspond to regions
where rays tend
1198
C HAPTE R 3 7 Interference of Light Waves
the same wavelength. A radio in a car traveling due
north receives the signals. (a) If the car is at the
position of the second maximum, what is the
wavelength of the signals? (b) How much farther
must the ca
S E C T ION 3 7. 7 The Michelson Interferometer
1195
Fourier
Transform
Spectroscopy (FTIR)
Infrared
Spectroscopy is the study of the wavelength distribution of radiation from a
sample that can be used to identify the characteristics of atoms or
molecules
S E C T ION 3 7. 6 Interference in Thin
Films
1193
Example 37.4
Nonreflective Coatings for Solar Cells
Solar cellsdevices that generate electricity when exposed to
silicon monoxide (SiO, n # 1.45) to minimize reflective losses
from the surface. Suppose th
Answers to Quick Quizzes
kinetic energy of the rod depends on the square
of the angular speed, the same work will result in
half of the angular speed.
10.12 (b). All of the gravitational potential energy of
the boxEarth system is transformed to kinetic
en