LECTURE 20
RELATIVITY III
Instructor: Kazumi Tolich
Lecture 20
2
¨
Reading chapter 39-7
¤
Relativistic energy
¤
Mass and energy
Rest energy
3
¨
Kinetic energy is defined as the work done by the net force in accelerating a particle from rest
to some final
velocity.
2
K=
¤
¨
¨
mc
(
2
1− u c
2
)
(
)
− mc 2 = γ − 1 mc 2
In the limit u << c, relativistic kinetic energy expression reduces to the classical kinetic energy expression.
The first term of the relativistic kinetic energy expression depends on the speed of the particle,
but the second term does not.
The quantity mc2 is called the rest energy, E0, of a particle.
E0 = mc 2
¤
This shows that mass is a form of energy!
Total relativistic energy
4
¨
The total relativistic energy is defined as
2
E = K + mc =
¤
¨
mc 2
(
1 − u2 c2
)
= γ mc 2
The work done by a net force on a particle at rest increases the energy from the rest energy to the
total relativistic energy.
Comparing with the relativistic momentum expression, we have a new useful expression:
u pc
=
c E
¨
Energies in atomic and nuclear physics are often expressed in eV or MeV,
1eV = 1.602 × 10-19 J, and masses of atomic particles are often given in the units of eV/c2
or MeV/c2.
Quiz: 1 & 2
Neutrinos traveling faster than c?
6
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¨
¨
¨
OPERA experiment announced that they measured neutrinos traveling faster than c in
September 2011.
Neutrinos have mass, so if this result was correct, the theory of special relativity,
which agreed with many other experimental results, is no longer valid.
In February 2012, OPERA announced that their measurement had two errors causing
their previous wrong results.
So, they do not travel faster than light! The special relativity is still working! Phew!
Energy, momentum, and rest mass
7
¨
In experiments, often momentum and energy, rather than velocity of
particles are measured. So, combine the expressions for energy and
momentum, and eliminate u.
E=
¨
mc 2
(
1 − u2 c2
p=
)
mu
(
1 − u2 c2
)
This yields a useful expression relating energy, momentum, and rest
energy.
( )
E 2 = p 2 c 2 + mc 2
2
Example 1
8
¨
The rest energy of an electron is 0.511 MeV.
The rest energy of a proton is 938 MeV.
Assume both particles have kinetic energies of
2.00 MeV.
a)
b)
c)
d)
Find the speed of the electron.
Find the speed of the proton.
By what factor does the speed of the electron
exceed that of the proton?
Repeat a) through c) assuming that both
particles have kinetic energies of 2000 MeV.
Quiz: 3
Mass and energy
10
¨
¨
¨
Energy and mass (inertia) were two completely distinct concepts before
Einstein related them with now-famous E = mc2 in special theory of
relativity.
c2 in E = mc2 is simply a conversion factor, a rather large factor. A small
mass corresponds to an enormous amount of energy.
In nuclear reactions (e.g. nuclear fission), matter, or rest mass, is
converted into other forms of energy.
¤
The fission of 1 kg of uranium decreases its rest mass by 1 gram, which is easily
measured.
Example 2
11
¨
In a nuclear power plant, the fuel rods
last 3 years before they are replaced.
The plant can transform energy at a
maximum possible rate of 1.00 GW.
Supposing it operates at 80.0 %
capacity for 3.00 years, what is the
loss of mass of the fuel?
Example 3
12
¨
The power output of Sun is
3.85 × 1026 W. By how much
does the mass of Sun decrease
each second?
Example: 4
13
¨
The energy released by the
nuclear bomb that destroyed
Hiroshima was equivalent to
12.4 kilotons of TNT. This is
equivalent to 9.0 ´ 1026 MeV.
What is the mass that was
converted into energy in this
explosion?