Special Relativity: Equivalence of Energy and Mass

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PH103
Special Relativity Equivalence of Energy and Mass
Dr. James van Howe Lecture 15
April 21, 2008
Some slides courtesy of Prof. Vogel AKA Ms.
Relativity
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Is this picture of a 9 ft long, 646 pound catfish
for real?
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Imagine a short-lived particle traveling towards
Earth near light-speed. I measure the lifetime of
the particle to be ________ than the particle
measures its own lifetime. When we both measure
the length of the trip it takes, the particle
measures __________ trip length than I do.

• Longer, longer
• Longer, shorter
• Shorter, longer
• Shorter, shorter

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True or False
A particle at rest has zero energy
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True or False
A particle of light has zero rest energy
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Classical Momentum and Kinetic Energy

• Classical momentum pmv
• Classical kinetic energy K½mv2
• As long as vltltc, we can use these equations and
  find the energy and momentum are conserved
• If v is closer to c, these quantities not
  conserved

What good are energy and momentum if not
conserved?
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Relativistic Momentum

• mo is rest mass
• measured when object is at rest
• v is objects velocity
• p is the objects momentum.
• This quantity is conserved in all collisions,
  reactions, etc

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Fma?

• What is mass?
• Mass tells how hard object is to accelerate.
• F ma is a classical equation
• F is proportional to Dp
• a is proportional to Dv
• Classically Dpm Dv -- same force, same Dv
• Not at high speed!
• As speed gets close to c, object gets harder and
  harder to accelerate
• Can get close to c, but never reach it!

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What is mass?

• Rest mass (mo) is mass measured when at rest
• property of the object
• what you look up in back of text
• Relativistic mass (m) shows how hard it is to
  accelerate the object
• increases with speed

• p mv
• if use relativistic mass

Can then define relativistic force using general
force equation
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Energy

• Momentum increases with speed, so does energy.
• E mc2

Conserved in a closed system

• Kinetic energy
• is zero when v0
• K mc2 - moc2

• When v 0
• Energy is not zero
• rest energy moc2

Only conserved in elastic collisions
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Momentum, Mass, and Energy Units
Convenient Units
SI Units
Mass
Momentum
Energy
An 1.0 electron-volts (eV) is the amount of
energy it takes to move one electron across one
volt of potential
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Momentum Change
To increase the speed of electron

• from 0.1c to 0.2 c

• from 0.8c to 0.9 c

Same change in speed, but seven times as much
force needed.
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Energy Change
To increase the speed of electron

• from 0.1c to 0.2 c

• from 0.8c to 0.9 c

Same change in speed, forty times as much energy
needed!
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Muon Decay
Demo Muon detector
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Names______________________________________
PH103, Muon Lifetime
1. Muons are born 15 km above sea level. They
travel this distance before I stop them in the
scintillator detector. Heres where we make an
assumption. We say that they must be traveling
pretty close to the speed of light. A) So how
long do we time them to make the trip to my
detector on earth? Now that we have the
approximate time it takes for them to get to my
detector. Lets calculate more precisely how fast
they are really going. I know that in my detector
they take 2.2 ms to decay. This is the lifetime
of the muon. This is the time the muon would
measure its own life, and the time I measure its
life once it is stopped in my detector and we are
in the same frame. B) For the trip to Earth
then, who has the proper time for the lifetime of
the muon? Now calculate the speed that the muons
must be going. The following equations derived
from time dilation and length contraction may
help C) Now that you have a more precise
speed, lets calculate the length the muon thinks
it is going on the trip to earth. D)
Finally, if I know the total energy for the muon
is nearly 4 GeV before I stop it, what is its
rest energy in units of GeV/c2.
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Solution
A) So how long do we time the muons to make the
trip to my detector on earth? B) For the trip
to Earth then, who has the proper time for the
lifetime of the muon?. Now calculate the speed
that the muon must be going. C) Now that you
have a more precise speed, lets calculate the
length the muon thinks it is going on the trip to
earth. D) Finally, if I know the total energy
for the muon is nearly 4 GeV before I stop it,
what is its rest energy in units of GeV/c2.
The muon, it is stationary to its own clock which
measures 2.2 ms
Real rest mass for muon
Not bad!
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Names_________________________________
PH103, Conserved Quantities and Invariant
Quantities
Relativistic energy and momentum are conserved
just like they were in classical mechanics. That
is, they are the same before and after a process.
However, they are not invariant (the same value
in all inertial frames). Oddly enough, mass is
not conserved in relativistic systems. However it
is invariant. The following exercise should help
explain. Imagine two lumps of clay of mass, m,
collide head-on at 0.6c and end up sticking
together to form a new system with mass M. A)
What is the new mass M of the stuck-together clay
in terms of m? Hint momentum conservation
should hold pbefore pafter. Energy
conservation should also hold Ebefore Eafter.
Recall, that sign of velocity is important. B)
Where did the extra mass come from?
Before
After