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GPS 519 or Goldstein 513
This problem is distantly related to the SpringPendulum Problem from Ch.
1, but here, the spring is constrained to oscillate vertically and the mass point
is expanded into a rigid rod of mass
M
and length
2
l
that swings in a vertical
plane.
One end of the rod is attached to the spring. Because the motion of
the bar is con
f
ned to a plane, the system has only two degrees of freedom: the
vertical displacement of the spring,
y
, and the angular displacement of the rod
from the vertical,
θ
. The angle
θ
will be measured from a
y
axis that is positive
in the downward direction so that
θ
is positive when the mass point is displaced
away from its rest position. The
x
axis is perpendicular to the
y
axis and is
positive to the right.
Let’s
f
rst consider the potential energy. The contribution from the spring
is simply
1
2
ky
2
. The gravitational contribution can be worked out as follows.
Imagine that the rod is at an angle
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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, Mass

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