Environmental Engineering Reference
In-Depth Information
and we have put R
=
6400
km. Thus the GPS clock gains
10
−
10
60
2
5
.
3
×
×
24
×
=
46
µs per day.
In contrast, time dilation slows down time on the satellite by a factor
3
.
9
10
8
2
10
3
×
1
2
≈
+
γ
1
.
(14.54)
3
×
10
−
11
the GPS clock loses about 7 µs every day. The two
effects are similar in magnitude with the gravitational effect the larger of the two.
The net effect is a 39 µs per day speeding up.
Thus since γ
−
1
≈
8
.
5
×
As a final remark, we shall discuss one direct manifestation of the speeding up
of time which occurs as one increases altitude in a uniform gravitational field. The
time intervals we have been discussing could be the inverse of the frequency of a
light wave. Thus Eq. (14.47) becomes
1
1
f
B
.
1
f
A
=
gh
c
2
+
(14.55)
The upshot is that light emitted from
B
(which is at the lower altitude) is observed
at
A
to have a lower frequency, i.e. it is red-shifted.
PROBLEMS 14
14.1 The three-force is defined to satisfy
d
p
d
t
.
f
=
Show that, for the motion of a particle of mass
m
in one dimension, this
equation can be re-written as
γ(u)
3
m
d
u
f
=
d
t
.
14.2 A particle of mass
m
is moving in the laboratory with a speed
u(t)
and
it is subjected to a retarding force of magnitude
γ(u)κm
where
γ(u)
=
u
2
/c
2
)
−
1
/
2
(
1
−
and
κ
is a constant. Given that
u(
0
)
=
c/
2 determine the
time at which the particle is at rest.
14.3 A particle of mass
m
moves along the
x
-axis under an attractive force to the
origin of magnitude
mc
2
L/x
2
where
L
is constant. Initially it is at rest at
x
L
. Show that its motion is simple harmonic with a period 2
πL/c
.
14.4 At the CERN Large Electron-Positron Collider (LEP), electrons travelled
around a circular particle accelerator of circumference 27 km. Assuming that
the electrons had total energy of 45 GeV, determine their proper accelera-
tion as they travel around the accelerator and compare it with non-relativistic
expectations.
=
