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Physics 210
Classical Mechanics
Autumn, 2003
TakeHome Final Exam
Due by noon Thursday, December 11
Openbook, but no collaboration allowed.
Please address all questions to me.
1. (Exercise 331) By solving the orbital equation, show that for a repulsive central force
inversely proportional to the cube of the distance,
3
)
(
r
k
r
f
=
0
k
,
the orbits are of the form
)]
(
cos[
1
0
f
f
b

=
A
r
.
Express
b
in terms of
k, E,
l
,
and the mass
m
of the incident particle.
Show that the cross section for scattering of a particle subject to this force is
( 29
Θ
⋅
Θ

Θ
Θ

⋅
=
Ω
sin
1
2
2
2
2
2
p
p
p
s
E
k
d
d
.
2. A double pendulum consists of two simple pendula, with the second pendulum
suspended from the mass of the first.
The pendula both are of the same length
l
but
different masses
m
and
m
2
.
Find the normal modes and frequencies of small
oscillations about equilibrium for this system, assuming the lower pendulum to have
the larger mass.
If you wish, you may start with the equations of motion that you
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