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C
Goldstein Problem 2.17 (3
rd
ed. # 2.18)
The geometry of the problem:
A particle of mass
m
is constrained to move on a circular hoop of radius
a
that is vertically
oriented and forced to rotate about the vertical symmetry axis with angular frequency
ω
. The
only external force is that of gravity.
The symmetry of this problem strongly suggests the use of
spherical coordinates (
r
,
θ
,
φ
), defined in the figure, as the generalized coordinates:
r
is the radial
displacement of the particle from the origin,
θ
is the polar angle, and
φ
is the azimuthal angle.
You could also use cylindrical coordinates (
ρ
,
φ
, z
), where
ρ
2
=
x
2
+
y
2
, but the analysis is then a
little more complicated.
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View Full Document Goldstein 217 (3
rd
ed. #2.18)
The Lagrangian of this system is essentially that of the spherical pendulum,
which we worked out in Problem GPS 1.19. One di
f
erence is that here we
will let the polar (
z
) axis point upwards (see the
f
gure) so that (1) the polar
angle
θ
di
f
ers by
π
from the earlier de
f
nition and (2) the potential energy
V
will be opposite in sign to that of the earlier problem. The radial coordinate
r
is, of course, constant and equals the radius of the hoop
a
. Hence,
r
is not
a generalized coordinate, i.e., there is no radial degree of freedom. The other
principal di
f
erence between this problem and the spherical pendulum is that
the azimuthal angle
ϕ
is not a generalized coordinate in this problem because
·
ϕ
is prescribed as
·
ϕ
=
ω
.A
sare
su
l
t
,
ϕ
=
ω
t
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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, Force, Mass

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