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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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
ff
erence is that here we
will let the polar (
z
) axis point upwards (see the
fi
gure) so that (1) the polar
angle
θ
di
ff
ers by
π
from the earlier de
fi
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
ff
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
. As a result,
ϕ
=
ω
t
+
ϕ
0
, and we lose another degree
of freedom.
Thus, we see that this system has only one degree of freedom that
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 Fall '10
 Wilemski
 mechanics, Force, Mass, Potential Energy, Cos, Coordinate system, Spherical coordinate system, Polar coordinate system, Coordinate systems

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