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HW #4 Problem 3
The Rolling Constraint:
A small circular hoop of radius
R
and mass
m
rolls without slipping up the inside of the parabola,
y = ax
2
.
Initially, point
1
is in contact with the origin (
0
) of the parabola. After the hoop has
rolled up the parabola to the position shown in the left figure, the arc lengths from the current
point of contact (point
T
, indicated by the dashed radial line) to point
1
on the hoop (arc
1T
) and
along the parabola from the origin to the current point of contact (arc
0T
)
must be equal because
slipping is not allowed.
From the figure, if we call
s
the arc length(
0T
), you can see that this
condition may be expressed mathematically as
s
=
R
(
φ
+
θ
), where the angle
φ
(>0) is measured
from the solid vertical line to the dashed line and the angle
θ
(>0) is the angle that measures the
hoop’s angular displacement from the inertial (always vertical) reference axis. You could pick
another axis as the inertial reference axis, but this axis must always have the same absolute
orientation as the hoop rolls on the parabola.
It is important to put the rolling constraint in terms of
θ
because this angle is a good choice for
one of the generalized coordinates of this system.
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 Fall '10
 Wilemski
 mechanics, Mass

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