Homework 12
−
Solution
CHANG
1 Consider the disk, the rod, and the sphere as one system and use the energy
equation with
U
e
0 (no spring) and
W
1
−
2
′
0 (no friction):
K
1
U
g
1
K
2
U
g
2
. The
gravitational potential energy is for the entire system, including all three parts. However, since
the disk is pivoted at the center, its center of mass does not move and we only need to consider
the vertical positions for the rod and the sphere to calculate
U
g
. The kinetic energy is due to
the rotation of the assembly, and hence
K
2
1/2
I
2
while
K
1
0 because the system is
initially at rest. The total moment of inertia for the assembly is calculated to be
I
I
d
I
r
I
s
1
2
m
d
R
2
1
3
m
r
L
2
m
s
r
2
1
2
15
0.4
2
1
3
8
0.6
2
4
0.6
2
3.6 Kgm
2
With the reference chosen through the pivot,
U
g
1
0 since the initial vertical positions for
the center of mass of the rod and the sphere are zero (on the datum). When the rod swings
down to the vertical position, we have
y
cm
2
−
L
/2 for the rod and
y
s
2
−
L
for the sphere.
Note that both are negative because they are below the reference line. The final angular
velocity can then be found using the energy equation:
0
0
1
2
I
2
m
r
gy
cm
2
m
s
gy
s
2
0
1
2
3.6
2
8
9.8
−
0.3
4
9.8
−
0.6
5.11 rad/s
CHANG
2 Consider the block and the pulley together as one system. Take the starting point
as position1 and position 2 represents the configuration after 5 revolutions of pulley’s rotation.
The energy equation is written as
K
1
U
e
1
W
′
1
−
2
K
2
U
e
2
. Note that
U
e
1
0 (no
stretching in the spring) and the gravitational potential energy is not included because the
pulley is stationary and the block slides horizontally. The kinetic energy includes the
translational kinetic energy from the block and the rotational kinetic energy from the pulley,
and
K
1
0 since the system starts from rest. The work term
W
′
1
−
2
has two parts, the work
done by friction,
−
k
Ns
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 Spring '08
 TSChang
 Physics, Angular Momentum, Energy, Friction, Kinetic Energy, Potential Energy, Work, 4 m/s, 5 m, 8 m, 15 m/s

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