Particle Suspended on a Sti
ff
Fiber next to an Attractive Wall
Because the motion of the mass point is con
fi
ned to a plane and the
fi
ber
has constant length
L
, the system has only one degree of freedom.
In plane
polar coordinates a suitable generalized coordinate is the polar angle
θ
, which
will be measured from a
y
axis that points downwards.
I’ll imagine that the
charged wall is to the left of the particle, so that the constant force
F
acts in
the negative
x
direction (to the left), and
θ
will be positive when the mass point
is displaced to the left from its rest position.
Note that
x
= 0
corresponds to
the particle hanging vertically (
θ
= 0
).
The kinetic energy of the mass point is then
T
=
m
2
(
·
x
2
+
·
y
2
)
.
(1)
In the polar coordinate system we have
x
=
−
L
sin
θ ,
y
=
L
cos
θ .
(2)
The minus sign is needed because
θ
is positive for displacements in the negative
x
direction. From this it follows that
·
x
=
−
L
·
θ
cos
θ ,
·
y
=
−
L
·
θ
sin
θ ,
(3)
and Eq.(1) can then be written as
T
=
1
2
mL
2
·
θ
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 Fall '10
 Wilemski
 mechanics, Cartesian Coordinate System, Mass, Potential Energy, Cos, Coordinate system, Polar coordinate system

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