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66. We make the unconventional choice of
clockwise
sense as positive, so that the
angular velocities (and angles) in this problem are positive. Mechanical energy
conservation applied to the particle (before impact) leads to
mgh
mv
v
gh
=
¡
=
1
2
2
2
for its speed right before undergoing the completely inelastic collision with the rod. The
collision is described by angular momentum conservation:
mvd
I
md
=+
rod
2
c
h
ω
where
I
rod
is found using Table 102(e) and the parallel axis theorem:
IM
d
M
d
Md
rod
F
H
G
I
K
J
=
1
12
2
1
3
2
2
2
.
Thus, we obtain the angular velocity of the system immediately after the collision:
22
2
(/
3
)
md
gh
M
dm
d
=
+
which means the system has kinetic energy
()
rod
/2
Im
d
+
which will turn into
potential energy in the final position, where the block has reached a height
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 Spring '08
 Any
 Physics, Energy

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