Since the ballast is modeled as a particle at a distance
B
from the spin axis, applying the parallel axis theorem,
Since
B
is modeled as a uniform bar and its mass center
E
is at a distance
eD1
2`b
from the spin axis,
applying the parallel axis theorem, we have
Force Laws. All forces are accounted for on the FBD.
Kinematic Equations.
The ballast is attached to the body. Hence, its angular velocity about the spin axis
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1839
Problem 8.108
A cord, which is wrapped around the inner radius of the spool
of mass
mD35 kg
, is pulled vertically at
A
by a constant force
PD120
N (the cord is pulled in such a way that it remains
vertical), causing the spool to roll over the horizontal bar BD. The
inner radius of the spool is
RD0:3
m, and the center of mass of
the spool is at
G
, which also coincides with the geometric center
of the spool. The spool’s radius of gyration is
kGD0:18
m.
Assuming that the spool starts from rest, that the cord’s inertia and
extensibility can be neglected, and that the spool rolls without slip,
determine the speed of the spool’s center
3
s after the application
of the force. In addition, determine the minimum static friction
coefficient for rolling without slip to be maintained during the time
interval in question.
Solution
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permission of McGraw-Hill, is prohibited.
where we have set tDt2t1. Eliminating the time integral of Fand solving for .vGx/2, we have
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1841
Problem 8.109
A spool has weight
WD450 lb
, outer and inner radii
RD6ft
and
⇢D4:5 ft
, respectively, center of mass
G
coinciding with its
geometric center, and radius of gyration
kGD4:0 ft
. The spool is
at rest when it is pulled to the right as shown. The cable wrapped
around the spool can be modeled as being inextensible and of
negligible mass. Assume that the spool rolls without slip relative
to both the cable and the ground. If the cable is pulled with a force
PD125 lb
, determine the speed of the center of the spool after
2
s
and the minimum value of the static friction coefficient between
the spool and the ground necessary to guarantee rolling without
slip.
Solution
Recalling that
RD6ft
,
⇢D4:5 ft
,
PD125 lb
,
tD2
s,
mSD450 lb=g
,
gD32:2 ft=s2
, and
kGD4ft
,
we can evaluate vGˇˇtD2sDj.vGx/2j, to obtain
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1843
Problem 8.110
The wind turbine in the figure consists of three equally spaced
blades that are rotating as shown about the fixed point
O
with an
angular velocity
!0D30 rpm
. Suppose that each
38;000 lb
blade
can be modeled as a narrow uniform rectangle of length
bD182 ft
,
width
aD12 ft
, and negligible thickness, with one of its corners
coinciding with the center of rotation
O
. The orientation of each
blade can be controlled by rotating the blade about an axis going
through the center
O
that coincides with the blade’s leading edge.
Neglecting aerodynamic forces and any source of friction, and
assuming that the turbine is freely rotating, determine the turbine’s
angular velocity
!
f
after each blade has been rotated
90ı
about
its own leading edge.
Photo credit: © Martin Child/Getty Images RF
Kinematic Equations. Following the problem statement, we have
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1845
Problem 8.111
Cars
A
and
B
collide as shown. Neglecting the effect of friction, what would be the angu-
lar velocity of
A
and
B
immediately after impact if
A
and
B
were to form a single rigid
body as a result of the collision? In solving the problem, let
C
and
D
be the mass cen-
ters of
A
and
B
, respectively, and use the following data: the weight of
A
is
WAD3130 lb
,
the radius of gyration of
A
is
kCD34:5 in:
, the speed of
A
right before impact is
vAD12 mph
,
the weight of
B
is
WBD3520 lb
, the radius of gyration of
B
is
kDD39:3 in:
, the speed of
B
right before
impact is
vBD15 mph
,
dD19 in:
, and
`D144 in:
Finally, assume that while
A
and
B
form a single
rigid body right after impact, the mass center of the rigid body formed by
A
and
B
coincides with the mass
center of the A–Bsystem right before impact.
Solution
Note:
Several intermediate numerical results are reported during the solution of this problem. While results
where
ID
and
IC
are the mass moments of inertia of
A
and
B
about their respective mass centers,
E!AD!AO
k
and E!BD!BO
kare the angular velocities of Aand B, respectively, and where ErC=D, given by
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1847
last two of Eqs. (10) along with Eqs. (11)–(13) form a system of
5
equations in the five unknowns
.vCx/2
,
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permission of McGraw-Hill, is prohibited.