Transcript Document

General Physics (PHY 2140)
Lecture 26
 Modern Physics
Relativity
Relativistic momentum, energy, …
General relativity
http://www.physics.wayne.edu/~apetrov/PHY2140/
Chapter 26
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send me an email request at
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Lightning Review
Last lecture:
1. Modern physics
 Time dilation, length contraction
t 
t p
1  v2 c2
L  Lp 1  v 2 c 2
Review Problem: A planar electromagnetic wave is propagating through
space. Its electric field vector is given by E = Eo cos(kz – wt) x, where x is a unit
vector in the positive direction of Ox axis. Its magnetic field vector is
1. B = Bo cos(kz – wt) y
2. B = Bo cos(ky – wt) z
3. B = Bo cos(ky – wt) x
4. B = Bo cos(kz – wt) z
where y and z are unit vectors in the positive directions of Oy and Oz axes
respectively.
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Reminder (for those who don’t read syllabus)
Reading Quizzes (bonus 5%):
It is important for you to come to class prepared, i.e. be familiar with the
material to be presented. To test your preparedness, a simple five-minute
quiz, testing your qualitative familiarity with the material to be discussed in
class, will be given at the beginning of some of the classes. No make-up
reading quizzes will be given.
There could be one today…
… but then again…
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Problem: relativistic pion
The average lifetime of a p meson in its own frame of reference (i.e., the
proper lifetime) is 2.6 × 10–8 s. If the meson moves with a speed of
0.98c, what is
(a) its mean lifetime as measured by an observer on Earth and
(b) the average distance it travels before decaying as measured by an
observer on Earth?
(c) What distance would it travel if time dilation did not occur?
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The average lifetime of a p meson in its own frame of reference (i.e., the proper
lifetime) is 2.6 × 10–8 s. If the meson moves with a speed of 0.98c, what is
(a) its mean lifetime as measured by an observer on Earth and (b) the average
distance it travels before decaying as measured by an observer on Earth?
(c) What distance would it travel if time dilation did not occur?
Given:
v = 0.98 c
tp = 2.6 × 10–8 s
Recall that the time measured by observer on Earth
will be longer then the proper time. Thus for the
lifetime
t
tp
1  v2 c2
 1.3 107 s
Thus, at this speed it will travel
Find:
t=?
d=?
dn =?
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


d  vt   0.98  3 108 m s 1.3 10 7 s  38m
If special relativity were wrong, it would only fly about



d  vt p   0.98  3 108 m s 2.6 108 s  7.6m
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Problem: space flight
In 1963 when Mercury astronaut Gordon Cooper orbited Earth 22 times,
the press stated that for each orbit he aged 2 millionths of a second
less than if he had remained on Earth.
(a) Assuming that he was 160 km above Earth in a circular orbit,
determine the time difference between someone on Earth and the
orbiting astronaut for the 22 orbits. You will need to use the
approximation
1
x
 1
2
1 x
for x << 1
(b) Did the press report accurate information? Explain.
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Length Contraction
The measured distance between
two points depends on the frame of
reference of the observer
The proper length, Lp, of an object
is the length of the object measured
by someone at rest relative to the
object
The length of an object measured
in a reference frame that is moving
with respect to the object is always
less than the proper length

This effect is known as length
contraction
2
LP
v
L
 LP 1  2

c
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Problem: weird cube
A box is cubical with sides of proper lengths L1 = L2 = L3= 2 m, when
viewed in its own rest frame. If this block moves parallel to one of its
edges with a speed of 0.80c past an observer,
(a) what shape does it appear to have to this observer, and
(b) what is the length of each side as measured by this observer?
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A box is cubical with sides of proper lengths L1 = L2 = L3= 2 m, when viewed in its
own rest frame. If this block moves parallel to one of its edges with a speed of
0.80c past an observer, (a) what shape does it appear to have to this
observer, and (b) what is the length of each side as measured by this
observer?
Given:
Recall that only the length in the
direction of motion is contracted, so
v = 0.8 c
Lip = 2.0 m
Find:
(a) shape
(b) Li=?
Thus, numerically,
L1  L1 p 1  v 2 c 2   2m  1   0.8   1.2m
2
L2  L2 p  2m
L3  L3 p  2m
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Relativistic Definitions
To properly describe the motion of particles within
special relativity, Newton’s laws of motion and the
definitions of momentum and energy need to be
generalized
These generalized definitions reduce to the classical
ones when the speed is much less than c
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26.7 Relativistic Momentum
To account for conservation of momentum in all inertial
frames, the definition must be modified
p


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mv
1 v c
2
2
 mv
v is the speed of the particle, m is its mass as measured by an
observer at rest with respect to the mass
When v << c, the denominator approaches 1 and so p
approaches mv
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Problem: particle decay
An unstable particle at rest breaks up into two fragments of unequal
mass. The mass of the lighter fragment is 2.50 × 10–28 kg, and that of the
heavier fragment is 1.67 × 10–27 kg. If the lighter fragment has a speed of
0.893c after the breakup, what is the speed of the heavier fragment?
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An unstable particle at rest breaks up into two fragments of unequal mass. The
mass of the lighter fragment is 2.50 × 10–28 kg, and that of the heavier fragment is
1.67 × 10–27 kg. If the lighter fragment has a speed of 0.893c after the breakup,
what is the speed of the heavier fragment?
Given:
v1 = 0.8 c
m1=2.50×10–28 kg
m2=1.67×10–27 kg
Momentum must be conserved, so the momenta of
the two fragments must add to zero. Thus, their
magnitudes must be equal, or
2.50  10


28
p2  p1   1 m 1v1

kg  0.893c
1  0.893
2


 4.96  1028 kg c
For the heavier fragment,
Find:
v2 = ?
1.67  10
27
1  v c
which reduces to
and yields
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
kg v
2


 4.96  1028 kg c
3.37 v c  1  v c
2
v  0.285c
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26.8 Relativistic Addition of Velocities
Galilean relative velocities cannot be applied to objects moving near
the speed of light
Einstein’s modification is
v ab

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v ad  v db

v ad v db
1
c2
The denominator is a correction based on length contraction and time
dilation
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Problem: more spaceships…
A spaceship travels at 0.750c relative to Earth. If the spaceship fires a
small rocket in the forward direction, how fast (relative to the ship) must it
be fired for it to travel at 0.950c relative to Earth?
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A spaceship travels at 0.750c relative to Earth. If the spaceship fires a small rocket
in the forward direction, how fast (relative to the ship) must it be fired for it to travel
at 0.950c relative to Earth?
Given:
vSE = 0.750 c
vRE = 0.950 c
Find:
vRS = ?
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Since vES = -VSE = velocity of Earth relative to ship, the
relativistic velocity addition equation gives
vRS
vRE  vES

v v
1  RE 2 ES
c
0.950 c   0.750 c 

 0.696c
0.950 c  0.750 c 

1
c2
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26.9 Relativistic Energy
The definition of kinetic energy requires modification in
relativistic mechanics
KE = mc2 – mc2


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The term mc2 is called the rest energy of the object and is
independent of its speed
The term mc2 is the total energy, E, of the object and depends
on its speed and its rest energy
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Relativistic Energy – Consequences
A particle has energy by virtue of its mass alone

A stationary particle with zero kinetic energy has an energy
proportional to its inertial mass
E = mc2
The mass of a particle may be completely convertible to
energy and pure energy may be converted to particles
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Energy and Relativistic Momentum
It is useful to have an expression relating total energy, E, to the
relativistic momentum, p

E2 = p2c2 + (mc2)2
When the particle is at rest, p = 0 and E = mc2
Massless particles (m = 0) have E = pc

This is also used to express masses in energy units
mass of an electron = 9.11 x 10-31 kg = 0.511 MeV
Conversion: 1 u = 929.494 MeV/c2
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QUICK QUIZ
A photon is reflected from a mirror. True or false:
(a) Because a photon has a zero mass, it does not exert a force on
the mirror.
(b) Although the photon has energy, it cannot transfer any energy to
the surface because it has zero mass.
(c) The photon carries momentum, and when it reflects off the mirror,
it undergoes a change in momentum and exerts a force on
the mirror.
(d) Although the photon carries momentum, its change in momentum
is zero when it reflects from the mirror, so it cannot exert a
force on the mirror.
(a)
(b)
(c)
(d)
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False
False
True
False
p  Ft
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Example 1: Pair Production
An electron and a positron
are produced and the photon
disappears

A positron is the antiparticle
of the electron, same mass
but opposite charge
Energy, momentum, and
charge must be conserved
during the process
The minimum energy
required is 2me = 1.04 MeV
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Example 2: Pair Annihilation
In pair annihilation, an
electron-positron pair
produces two photons

The inverse of pair
production
It is impossible to create a
single photon

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Momentum must be
conserved
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26.10 General relativity: Mass – Inertial vs.
Gravitational
Mass has a gravitational attraction for other masses
Fg  G
mgm'g
r2
Mass has an inertial property that resists acceleration

Fi = mi a
The value of G was chosen to make the values of mg and mi equal
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Einstein’s Reasoning Concerning Mass
That mg and mi were directly proportional was evidence
for a basic connection between them
No mechanical experiment could distinguish between the
two
He extended the idea to no experiment of any type could
distinguish the two masses
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Postulates of General Relativity
All laws of nature must have the same form for
observers in any frame of reference, whether
accelerated or not
In the vicinity of any given point, a gravitational field is
equivalent to an accelerated frame of reference without a
gravitational field

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This is the principle of equivalence
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Implications of General Relativity
Gravitational mass and inertial mass are not just proportional, but
completely equivalent
A clock in the presence of gravity runs more slowly than one where
gravity is negligible
The frequencies of radiation emitted by atoms in a strong
gravitational field are shifted to lower frequencies

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This has been detected in the spectral lines emitted by atoms in
massive stars
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More Implications of General Relativity
A gravitational field may be “transformed away” at any
point if we choose an appropriate accelerated frame of
reference – a freely falling frame
Einstein specified a certain quantity, the curvature of
time-space, that describes the gravitational effect at
every point
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Testing General Relativity
General Relativity predicts that a light ray passing near the Sun should be
deflected by the curved space-time created by the Sun’s mass
The prediction was confirmed by astronomers during a total solar eclipse
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Black Holes
If the concentration of mass becomes great enough, a
black hole is believed to be formed
In a black hole, the curvature of space-time is so great
that, within a certain distance from its center, all light and
matter become trapped
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