Lecture 20 Relativistic Effects

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Lecture 20Relativistic Effects
Chapter 26.6 ? 26.10
Outline

• Relativity of Time
• Time Dilation
• Length Contraction
• Relativistic Momentum and Addition of Velocities

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Basis for Special Relativity

• Einsteins relativity
• The laws of physics are the same in any inertial
  frame of reference.
• The speed of light (c 300,00 km/s) is the same
  for all observers, irrespective of their relative
  speeds.

• Consequences
• Relativity of Time
• Time Dilation
• Length Contraction

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Relativity of Time
4
Explanation of Time Relativity
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Simultaneity in Relativity

• Conclusions from the box car experiment
• Two simultaneous events in one reference frame
  can be non-simultaneous in another reference
  frame, moving with respect to the first one.
• Simultaneity is not absolute, but depends on the
  observers state of motion.
• There is no preferred inertial reference frame.

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Time Dilation
c?t/2
d
Experiment
v?t/2
?tp 2d/c ? time interval for light to reach the
mirror at the stationary reference frame (proper
time) ?t ? this interval in the moving reference
frame Speed of light c is equal in both frames
c?t 2 v ?t 2 ? ? d2 2
2
2d ?t ???? c?(1 ?v2/c2)
?t ??tp
?
? 1/?(1 ?v2/c2)
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Conclusions from Time Dilation
?t ??tp
? is always grater than 1
The time interval between two events measured by
an observer moving with respect to the clock is
longer than that between the same two events
measured by an observer at rest with the
clock. A clock moving past an observer runs
slower than an identical clock at rest with the
observer. Proper time is the time interval
between two events as measured by an observer who
sees them at same position.
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Twin Paradox
Twin paradox is a consequence of time dilation.
The age of the traveling twin is indeed smaller
than that of the non-traveling one, because the
former experiences acceleration and lives in a
non-inertial reference frame. Only the
non-traveling twin can apply Einsteins
relativity and determine the time dilation.
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Length Contraction
The proper length Lp is the length of an object
measured at rest with it. In a moving reference
frame, the length L is shorter. L Lp/? Lp ?(1
?v2/c2) Length contraction occurs only in the
direction of motion.
Example A right triangle with the sides of 10 m
each at the right angle. How would the angles
change if the triangle moves at 0.9 c along one
of the short sides?
Solution l 10 ?(1 ? 0.92) 1.9 m tan ?'
1.9/10 0.19 ? ?' 10.8?
? 45?
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Length Contraction
?'
?
10
10
10
1.9
v 0.9 c
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Other Effects of Relativity
Momentum is the product of mass and velocity
p(rest) mv
p(relativistic) ?mv
Relativistic momentum obeys the law of momentum
conservation even at velocities, close to c.
Relativistic addition of velocities (works for
parallel or antiparallel paths)
vx ? speed of object x vy? speed of object y vxy
? relative speed of objects x and y
vx vy vxy ????? 1 (vx vy)/c2
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Relativistic Addition of Velocities
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Equivalence of Mass and Energy
Kinetic energy of a non-relativistic object
KEmv2/2
For a relativistic object KE ?mc2 mc2
The term independent of velocity is the rest
energy
ER mc2
Total energy, defined as E ?mc2, is the sum of
the kinetic and rest energies.
Any object has energy by virtue of its mass
alone. Mass can be transferred into energy and
back.
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Summary

• There are two postulates of special relativity
• The laws of physics are the same in any inertial
  frame of reference.
• The speed of light is the same for all observers,
  irrespective of their relative speeds.
• The consequences (relativity of time, time
  dilation, length contraction, velocity addition)
  seem unusual for our experience, but can be
  verified by experiments or observations of
  distant objects.