General Physics (PHY 2140)

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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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Lightning Review

• Last lecture
• Modern physics
• Time dilation, length contraction

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 dont 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
  2.6 108 s. If the meson moves with a speed of
  0.98c, what is
• its mean lifetime as measured by an observer on
  Earth and
• the average distance it travels before decaying
  as measured by an observer on Earth?
• 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
2.6 108 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?
Recall that the time measured by observer on
Earth will be longer then the proper time. Thus
for the lifetime

• Given
• v 0.98 c
• tp 2.6 108 s
• Find
• t ?
• d ?
• dn ?

Thus, at this speed it will travel
If special relativity were wrong, it would only
fly about
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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.
• 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
• for x ltlt 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

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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,
• what shape does it appear to have to this
  observer, and
• 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?
Recall that only the length in the direction of
motion is contracted, so

• Given
• v 0.8 c
• Lip 2.0 m
• Find
• shape
• Li?

Thus, numerically,
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Relativistic Definitions

• To properly describe the motion of particles
  within special relativity, Newtons 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
• 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 ltlt 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 1028 kg, and that of
the heavier fragment is 1.67 1027 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 1028 kg, and that of
the heavier fragment is 1.67 1027 kg. If the
lighter fragment has a speed of 0.893c after the
breakup, what is the speed of the heavier
fragment?
Momentum must be conserved, so the momenta of the
two fragments must add to zero. Thus, their
magnitudes must be equal, or
Given v1 0.8 c m12.501028
kg m21.671027 kg Find v2 ?
For the heavier fragment,
which reduces to
and yields
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26.8 Relativistic Addition of Velocities

• Galilean relative velocities cannot be applied to
  objects moving near the speed of light
• Einsteins modification is
• 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?
Since vES -VSE velocity of Earth relative to
ship, the relativistic velocity addition equation
gives
Given vSE 0.750 c vRE 0.950 c
Find vRS ?
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26.9 Relativistic Energy

• The definition of kinetic energy requires
  modification in relativistic mechanics
• KE ?mc2 mc2
• 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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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.
QUICK QUIZ

• False
• False
• True
• False

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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
• Momentum must be conserved

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26.10 General relativity Mass Inertial vs.
Gravitational

• Mass has a gravitational attraction for other
  masses
• 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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Einsteins 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
• 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
• 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 Suns 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