Goldstein 814 (3
rd
ed. 8.24)
This view is looking down on the cylinder from above.
The
z
axis is coming up out of the page
through the center of the circular cylinder. The
x
axis is fixed in space. The
x
±
axis is fixed in the
cylinder. The angle
α
measures the angular position of the particle on the cylinder.
The angle
θ
measures the angular position of the particle in the fixed frame. The angle
φ
measures the angular
displacement of the cylinder in the fixed frame. Initially, the particle was located on the
x
axis,
which then coincided with the
x
±
axis.
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View Full DocumentGoldstein 814 (3
rd
ed
8.24)
Start by writing the kinetic energy as the sum of contributions from the
cylinder of mass
M
and from the particle of mass
m
,
T
=
T
cyl
+
T
m
,
(1)
and express
T
cyl
and
T
m
using the generalized coordinates (angles) measured
from a reference axis
f
xed in space, as illustrated in the drawing on the previous
page. We have
T
cyl
=
1
2
I
·
φ
2
,
(2)
where
I
=
Ma
2
/
2
is the moment of inertia of the cylinder about its rotation
axis, the
z
axis in this case. The particle’s kinetic energy has two components,
T
m
=
m
2
[
a
2
·
θ
2
+
·
z
2
]
.
(3)
The potential energy of the system is simply
V
=
mgz .
(4)
The vertical displacement
z
of the particle is measured from its initial position
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 Fall '10
 Wilemski
 mechanics, Angular Momentum, Kinetic Energy, Particle, Lagrangian mechanics, Goldstein, Eqs.

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