316
Rotation of a Rigid Object About a Fixed Axis
P10.79
(a)
∆∆
∆
KK
U
rot
trans
++
=
0
Note that initially the center of mass of the sphere is a
distance
hr
+
above the bottom of the loop; and as the
mass reaches the top of the loop, this distance above
the reference level is 2
Rr
−
.
The conservation of
energy requirement gives
mg h r
mg R r
mv
I
+=
+
+
a
f
a
f
2
1
2
1
2
22
−
ω
r
m
h
R
P
FIG. P10.79
For the sphere
Im
r
=
2
5
2
and
vr
=
so that the expression becomes
gh
gr
gR
v
+
7
10
2
(1)
Note that
hh
=
min
when the speed of the sphere at the top of the loop satisfies the condition
Fm
g
mv
∑
==
−
2
a
f
or
vg
R
r
2
=−
af
Substituting this into Equation (1) gives
hR
r
R
r
min
+
−
20
7
0
0
.
or
r
R
min
..
=
270
(b)
When the sphere is initially at
=
3 and finally at point
P
, the conservation of energy
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This note was uploaded on 12/14/2011 for the course PHY 203 taught by Professor Staff during the Fall '11 term at Indiana State University .
 Fall '11
 Staff
 Physics, Center Of Mass, Mass

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