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Physics 409/Classical Mechanics
Test #3
19 October 2010
Name
ANSWERS
Answer all parts of each of the two questions.
25
points apiece. 50 points total.
Begin the
answer to each question on a new sheet of paper. Clearly define all coordinates and variables.
Be sure to include sufficient detail in each answer so that the logic you are using is clear.
(1)
Consider the twobody conservative central force problem in polar coordinates
r
and
θ
.
The
effective radial equation of motion is
.
Here,
l
is the constant angular
23
(/ ) (
)
rl
r
f
r
μμ
=+
±±
momentum,
μ
is the reduced mass, and
f
(
r
)
is the central force, derivable from a potential
V
(
r
).
For the attractive potential
V
(
r
) =
!
k
/
r
, the equation of the orbit has the form
r
=
c/
(1 +
e
cos
θ
), where
c
=
l
2
/(
μk
),
k
is a positive constant, and
e
is a nonnegative constant
known as the eccentricity.
The energy of any possible orbit with angular momentum
l
and
eccentricity
e
can be expressed generally as
.
E
k
l
e
=
⎛
⎝
⎜
⎞
⎠
⎟ −
μ
2
1
2
2
()
(a)(10 pts.)
Add
l
2
/(2
μr
2
) to
V
(
r
) to obtain the effective onedimensional potential
for this
′
V
problem.
Draw a sketch of
versus
r
, and indicate
E
on it for the four cases:
(i)
e
>1, (ii)
e=
1,
′
V
(iii) 1>
e
> 0, and (iv)
e
=0 . Qualitatively describe the type of motion possible for
each
of these
cases. Indicate the allowable ranges for the radial coordinate. It should be possible to have a
circular orbit for your effective potential. Is this circular orbit stable or unstable? Explain.
ANS:
See Figures 3.3, 3.4, 3.6, and 3.8 in Goldstein, either edition.
(b)(3 pts.)
Find the radius
r
0
and period
τ
0
for the stable circular orbit.
Express
τ
0
in terms of
r
0
,
μ
, and
k
.
ANS:
From the circular orbit condition,
f
ef
=
!
d
V
N
(
r
)
/dr =
0, we have
l
2
/(
μr
0
3
)
!
k
/
r
0
2
=
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This note was uploaded on 12/19/2010 for the course PHYSICS ph 409 taught by Professor Wilemski during the Fall '10 term at Missouri S&T.
 Fall '10
 Wilemski
 mechanics

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