Problem 7.42
The driveway gate is hinged at its right end and can swing freely in
the horizontal plane. The gate is pushed open by the force
P
that
always acts perpendicular to the plane of the gate at point
A
, which
is a horizontal distance
d
from the gate hinge. The weight of the
gate is
WD215 lb
, and its mass center is at
G
, which is a distance
w=2
from each end of the gate, where
wD16 ft
. Assume that the
gate is initially at rest, and model the gate as a uniform thin bar as
shown below the photo.
Given that a force of
PD20 lb
is applied at the center of mass
of the gate (i.e.,
dDw=2
), determine the reactions at the hinge
O
after the force Phas been continuously applied for 2s.
Solution
Viewing the gate from above, which allows us to view its plane of
motion, we have the FBD of the gate as shown on the right, where
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Dynamics 2e 1497
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Problem 7.43
The driveway gate is hinged at its right end and can swing freely in
the horizontal plane. The gate is pushed open by the force
P
that
always acts perpendicular to the plane of the gate at point
A
, which
is a horizontal distance
d
from the gate hinge. The weight of the
gate is
WD215 lb
, and its mass center is at
G
, which is a distance
w=2
from each end of the gate, where
wD16 ft
. Assume that the
gate is initially at rest, and model the gate as a uniform thin bar as
shown below the photo.
Given that a force of
PD20 lb
is applied at the center of
percussion of the gate, determine the reactions at the hinge
O
after
the force Phas been continuously applied for 2s.
2CPdw
2DIG˛g;
where
˛g
is the angular acceleration of the gate and where, modeling the gate as a thin uniform bar, its mass
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Dynamics 2e 1499
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Problem 7.44
The uniform thin bar
AB
of length
L
and mass
m
is released
from rest in the horizontal position shown.
Determine the distance
`
at which the pin
O
should be located
from the end of the bar so that it has the maximum possible
angular acceleration ˛max, and find that angular acceleration.
Solution
The FBD of the bar immediately after it is released is shown at the
right, where Gis the mass center of the bar.
Balance Principles.
Based on the FBD shown, the Newton-
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1501
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Problem 7.45
The uniform thin bar
AB
of length
L
and mass
m
is released
from rest in the horizontal position shown.
Determine the distance
`
at which the pin should be located
from the end of the bar so that it has the maximum possible
angular acceleration
˛max
, and find that angular acceleration. In
addition, determine the angular acceleration
˛0
of the bar when
`D0, and then find the ratio ˛max=˛0.
Solution
The FBD of the bar immediately after it is released is shown at the
right, where Gis the mass center of the bar.
Balance Principles.
Based on the FBD shown, the Newton-
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1503
To determine the distance
`
at which
˛AB
is a maximum, we differentiate the expression for
˛AB
with respect
to `and set the result equal to zero, that is
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Problem 7.46
The cutting arm of the paper cutter is pinned about a fixed axis at
O
,
and its angle relative to the horizontal is measured by
. A linear
elastic torsional spring at
O
with constant
kt
keeps the arm from
falling when not in use. Model the cutting arm as a uniform slender
bar of length
LD20 in:
and weight
WD2:5 lb
. Neglect friction
in the pin at O.
Determine the angular speed with which the cutting arm will
reach the horizontal position if it is released from rest at
iD
70ı
with
ktD1:6 ftlb=rad
. Assume that the torsional spring is
undeformed when D90ı.
Solution
The FBD of the cutting arm is shown on the right, where
MO
is the moment
at Ocaused by the torsional spring.
Balance Principles.
The Newton-Euler equations corresponding to this
FBD are
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Dynamics 2e 1505
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