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PHYS 3202 Classical Mechanics II  Homework #3
Additional Hints
Problem 1. Part 1
The circle is
r
= constant
,
z
= another constant
in the cylindrical coordinate system.
See Appendix F.2 of Thornton and Marion, 5th Edition for information about the
cylindrical coordinate system. You will need Equation (F.7) for the length element
of a path.
The time it takes light to travel a tiny distance
ds
is
dt
=
ds
v
,
where
v
(
r
) =
c/n
(
r
) is the local speed of light at distance
r
from the
z
axis.
Part 2
If you get two equations involving
z
0
(
φ
) and
r
0
(
φ
), you can solve them
for
z
0
and
r
0
in terms of
r
. The resultant equation for
r
0
can then be solved by an
integral. The following formula may be useful:
Z
dr
r
√
1

B
2

A
2
r
2
=
C
+
1
√
1

B
2
ln
r
√
1

B
2
+
√
1

B
2

A
2
r
2
.
After you ﬁnd
r
(
φ
), try to simplify it as much as possible. The hyperbolic func
tions will be very useful when simplifying your results. Search for hyperbolic func
tions at http://en.wikipedia.org/ , or go to Appendix D.6 of Thornton and Marion,
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This note was uploaded on 01/22/2012 for the course PHYS 3202 taught by Professor Tan during the Spring '11 term at Georgia Institute of Technology.
 Spring '11
 Tan
 mechanics, Work

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