Department of Physics
Prof. R. Bundschuh
The Ohio State University
Fifth Problem Set for Physics 664 (Theoretical Mechanics)
Spring quarter 2004
Important dates:
May 11 9:30am10:18am midterm exam,
May 31 no class, Jun 9 9:30am11:18am final exam
Due date:
Thursday, May 6
12. Shortest path on a volcano
12 points
You are hiking on a volcano given by the
equation
z
= 1

q
x
2
+
y
2
.
You are at point (

1
,
0
,
0) and want to get to
the other side, namely to the point (1
,
0
,
0),
along the shortest possible path.
x
y
z
a) Express the infinitesimal line element d
s
=
√
d
x
2
+ d
y
2
+ d
z
2
in cylindrical coordinates
(
r, φ, z
) defined by
x
=
r
cos
φ
and
y
=
r
sin
φ
.
b) Write down the length of an arbitrary path (specified by
r
(
φ
), why?) on the volcano as
an integral over
φ
.
c) Find a differential equation for the path
r
(
φ
) of minimal length.
d) Solve the differential equation for your hiking problem.
[Hint: in order to solve the
differential equation, it is helpful to reformulate it in terms of the variable
u
(
φ
)
≡
1
/r
(
φ
).]
e) Calculate the length of your path and sketch the path. [Hint:
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 Fall '04
 BUNDSCHUH
 Physics, mechanics, Mass, General Relativity

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