Problem 7.47
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 value of the torsional spring constant
kt
so that
when the cutting arm is released from rest at
iD70ı
, it reaches
fD15ı
with zero angular speed. 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
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Dynamics 2e 1507
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Problem 7.48
The uniform thin platform
AB
of length
L
and mass
mp
is pinned both at
A
and at
D
. A uniform crate of height
h
, width
w
, and mass
mc
is placed
at the end of the platform a distance
`
from the pin at
A
. The system is at
rest when the pin at
A
breaks. Determine the angular acceleration of the
platform and crate, as well as the force on the platform due to the pin at
D
, immediately after the pin at
A
breaks. Assume that the crate and the
platform do not separate immediately after the pin fails and that friction
is sufficient to prevent slipping between the platform and crate.
Solution
The FBD and KD of the platform and crate after the pin at Abreaks are shown below.
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Dynamics 2e 1509
Computation.
Substituting the kinematic equations and the expressions for the mass moments of inertia
into the Newton-Euler equations, we have the following system of three equations in the three unknowns
Dx
,
Dy, and ˛p:
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Problem 7.49
The ladder of mass
m
and length
L
is released from rest at the angle
0
. Model
the ladder as a uniform slender bar.
If the friction at
O
between the ladder and the ground is sufficient to
prevent slipping, and the ladder is given a slight nudge from rest at
D90ı
,
determine the angular speed of the ladder when it reaches D0.
Solution
The FBD of the ladder as it falls is shown on the right, where
G
is the mass
center of the ladder.
Balance Principles.
The Newton-Euler equations corresponding to this FBD
are
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 1511
Problem 7.50
The ladder of mass
m
and length
L
is released from rest at the angle
0
. Model
the ladder as a uniform slender bar.
If the friction at
O
between the ladder and the ground is sufficient to
prevent slipping, and the ladder is given a slight nudge from rest at
D90ı
,
determine the normal and frictional forces at
O
as a function of the angle
,
and find the minimum coefficient of static friction that is compatible with this
motion.
Solution
The FBD of the ladder as it falls is shown on the right, where
G
is the mass
center of the ladder.
Balance Principles.
The Newton-Euler equations corresponding to this FBD
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Computation.
Substituting the expression for
IO
and the kinematic equations into the Newton-Euler
equations, we obtain
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Dynamics 2e 1513
Problem 7.51
The T-bar consists of two thin rods,
OA
and
BD
, each of length
LD
1:5
m and mass
mD12 kg
, that are connected to the frictionless pin at
O. The rods are welded together at Aand lie in the vertical plane.
If the rods are released from rest in the position shown, determine
the force on the pin at
O
, as well as the angular acceleration of the rods
immediately after release.
Solution
The FBD and KD of the T-bar are shown below, where
G
is the mass center of the section
OA
and
A
is the
mass center of the section BD.
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 1515
Problem 7.52
The T-bar consists of two thin rods,
OA
and
BD
, each of length
LD
1:5
m and mass
mD12 kg
, that are connected to the frictionless pin at
O. The rods are welded together at Aand lie in the vertical plane.
If, at the instant shown, the system is rotating clockwise with angular
velocity
!0D7rad=s
, determine the force on the pin at
O
, as well as the
angular acceleration of the rods.
Solution
The FBD and KD of the T-bar are shown below, where
G
is the mass center of the section
OA
and
A
is the
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