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12.91. Model:
Assume that the marble does not slip as it rolls down the track and around a looptheloop.
The mechanical energy of the marble is conserved.
Visualize:
Solve:
The ball’s center of mass moves in a circle of radius
.
R
r
−
The freebody diagram on the marble at its
highest position shows that Newton’s second law for the marble is
2
1
mv
mg
n
R
r
+=
−
The minimum height (
h
) that the track must have for the marble to make it around the looptheloop occurs when
the normal force of the track on the marble tends to zero. Then the weight will provide the centripetal
acceleration needed for the circular motion. For
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 Fall '09
 geller
 Center Of Mass, Energy, Mass

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