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B. Balachandran
ENME 361
Fall 2008
Solution to Quiz #4
(October 9, 2008; Duration: 15 minutes)
1) Given a vector velocity
v
of the form
( )
cos
sin
xb
b
θ
θθ
=+
+
±±
±
vi
j
where
x
=
x
(
t
),
=
(
t
), and
b
is a constant, construct
()
1
2
T
=
⋅
vv
and determine an expression for
dT
dt
∂
⎛⎞
⎜⎟
∂
⎝⎠
±
Solution
:
22
2
11
() (
2 c
o
s
)
Tx
b
x
b
==
+
+
v.v
2
cos
T
bx
b
∂
∂
±
±
±
2
cos
sin
b
b
x
dt
∂
−
∂
±
±
±
2) Consider vertical oscillations and determine the kinetic energy for the system shown in the
figure below, where an unbalance of
mass
m
o
rotates with an angular speed
ω
,
and this mass is
located a fixed distance
ε
from the center of rotation
O
.
Note that
M
does not include the
unbalance
m
o
.
M
m
o
k
c
t
X
Y
i
j
O
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View Full DocumentSolution
:
To determine the kinetic energy associated with the unbalanced mass, we first write the position
vector from point
O’
, the origin of the inertial reference frame, to mass
m
o
as
o
m
r(
c
o
s
)
(
s
i
n
)
ht
yt
ε
ωε
ω
=+
++
i
j
where
h
is the horizontal offset of point
O
from point
O’
and
y
is measured in the vertical
direction from the staticequilibrium position. Therefore, the absolute velocity of the unbalanced
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