Problem 5.123
Alternative equilibrium equations:
Consider the FBD shown where
F1
,
F2
,
etc., are the forces acting on the body (including reaction forces from supports),
and
M1
,
M2
, etc., are the moments acting on the body (including reaction
moments from supports, if any). Let the total number of forces and moments
be nand m, respectively. Consider the equilibrium equations
XE
FDE
0W
n
X
iD1E
FiDE
0; (1)
XE
MADE
0W
n
X
iD1ErAi E
FiC
m
X
iD1E
MiDE
0; (2)
XE
MBDE
0W
n
X
iD1ErBi E
FiC
m
X
iD1E
MiDE
0; (3)
where
ErAi
is a position vector from point
A
to anywhere on the line of action of
E
Fi
, and
ErBi
is a position vector from point
B
to the same point on the line of
action of
E
Fi
. Equations (1) and (2) are the “standard” equilibrium equations as
given in Eq. (5.16) on p. 322, while Eqs. (2) and (3) are alternative equilibrium
equations that may be used instead. Show that satisfying Eqs. (2) and (3) means
that Eq. (1) is also satisfied. Hint: Note that
ErBi D ErBA C ErAi
where
ErBA
is the
position vector from point
B
to point
A
, combine Eqs. (2) and (3), and after
simplification and imposition of necessary restrictions on the locations of points
Aand B, show that Eq. (1) results.
Problem 5.124
The transmission tower
ABC
shown supports forces from three electrical wires
at points
C
,
D
, and
E
. The tower is supported at point
A
by a concrete
foundation, and this foundation is normally adequate to support the tower
without the guy wire
BF
. For towers that are sited in poor soil, or when the
horizontal forces from the electrical wires are high due to a sharp turn, an
optional guy wire
BF
may also be used to help support the tower. Neglect the
weight of the tower.
(a) If guy wire BF is not used, determine the support reactions at A.
(b)
If guy wire
BF
is used, determine the force it must support so that the
moment reaction at Ais zero.
(c)
If guy wire
BF
is used, determine the smallest force it must support so
that the moment reaction at Ais no larger than 500 kNm.
Problem 5.125
The propellers of the twin engine airplane shown rotate in the same direction,
and each propeller exerts a moment
MD1:3 kNm
on the wings of the plane.
To equilibrate this moment, trim tabs on the vertical stabilizer are used to
produce trim forces
T
. Determine the value of
T
, assuming the trim forces are
vertical.
Problem 5.126
In Prob. 5.125, in what direction do the propellers rotate? Specify clockwise or
counterclockwise with respect to the view shown in Fig. P5.125, and explain
your reasoning.
Problem 5.127
A bracket is supported by a loose-fitting pair of rollers at points
A
and
B
, and
another loose-fitting pair at
C
and
D
, and a frictionless pin at
F
. The forces at
E
and
G
are horizontal and vertical, respectively. Determine the reactions at
the pin and each of the four rollers.
Problem 5.128
A semicircular geared bracket is subjected to a vertical
80 N
force at point
C
.
The bracket is supported by frictionless pins at
A
and
B
and a gear at
D
. The
pins and gear are fixed to plate
E
, and the gear at
D
is not allowed to rotate.
Determine the tangential force supported by the gear at Dand the reactions at
pins Aand B.
(4)
Equation (3) is solved to obtain
B
, and then Eq. (4) is solved
Problem 5.129
A frame supports three frictionless pulleys that guide a tape that runs at constant
speed. Determine the reactions at support Aif ˛DˇD0ı.
Problem 5.130
Repeat Prob. 5.129 if ˛D30ıand ˇD0ı.
Problem 5.131
Repeat Prob. 5.129 if ˛D30ıand ˇD45ı.
Problem 5.132
A horizontal wooden beam is used to display iron pots and pans in a store. The
wooden beam is suspended from the ceiling by one cable whose ends are pinned
at points
A
and
I
, and which wraps around seven frictionless pulleys. The beam
is also supported by a vertical bar
JK
. Points
B
,
D
,
F
,
H
, and
I
have equal
horizontal spacing, and each of these has a pot or pan hanging immediately
below it. The straight cable segments are oriented at 30
ı
from the vertical (as
shown for segment
AB
). The wooden beam weighs
60 lb
with center of gravity
immediately below the bearing of pulley
F
. Determine the forces supported by
the cable and bar JK.
Problem 5.133
The ramp shown is used on the back of a trailer for loading and unloading
equipment. The lowered position is shown by points
A
,
B
,
C
, and
D
, and the
raised position is shown by points
A
,
B0
,
C0
, and
D0
. The ramp has
800
N
weight with center of gravity at point
C
, and in the raised position, point
C0
is
immediately above point
A
. To raise the ramp, the user applies a vertical force
P
at handle
D
. To lower the ramp, the user applies a horizontal force
Q
to
handle
B0
. To make raising and lowering the ramp easier, there is a torsional
spring at pin
A
. Determine the stiffness
kt
of the torsional spring and the
pretwist
0
it must have when in the raised position so that
PD100
N and
QD60 N.
Problem 5.134
A momentary-on electrical toggle switch is shown. To operate the switch, the
user applies a horizontal force
Q
. The switch has a torsional spring that causes
lever
AB
to make contact at point
C
when
QD0
(this is the off position).
When
Q
is sufficiently large, lever
AB
moves to position
A0B
and contact is
made at point
D
(this is the on position). Determine the stiffness
kt
of the
spring and the pretwist
0
it must have so that
QD3lb
will cause contact to
be broken at point
C
, and
QD4lb
will cause contact to be made at point
D
.
Note: The pretwist
0
is the amount the torsional spring is twisted when the
lever is in the AB position.
Problem 5.135
For the toggle switch described in Prob. 5.134, let the stiffness and pretwist
of the torsional spring be
ktD0:6 in:lb=rad
and
0D10 rad
. Determine the
force
Q
as a function of
˛
over the full range of motion for lever
AB
(i.e.,
0˛60ı
). Plot
Q
versus
˛
, and determine the value of
Q
needed to break
contact at point C, and the value of Qneeded to make contact at point D.
Problem 5.136
The trigger of a high-pressure washer gun is shown. The torsional spring at
point
A
has stiffness
ktD1100 Nmm=rad
and is prewound by
6ı
when it is
installed (i.e., when the trigger makes contact at point
G
). The axial spring
DE
has stiffness
kaD0:1 N=mm
and has
40 mm
unstretched length. The
trigger operates the washer on low pressure when
0ı< 15ı
, and when
15ı< 25ı
, the trigger operates the washer on high pressure. Assume force
Fremains horizontal with the same line of action for all trigger positions.
(a) Determine the force Fthat causes the trigger to begin movement.
(b)
Determine the force
F
that causes the trigger to first make contact with
the plunger at C(D15ı).
(c)
Determine the force
F
required to fully pull the trigger (
D25ı
).
Assume the plunger at
C
contacts the back of the trigger at a right angle
when D25ı.
Problem 5.137
A truck pulling a trailer with a tractor on it is shown in Fig. P5.137(a). The trailer’s mass is
500 kg
and
the tractor’s mass is
900 kg
, with centers of gravity at points
E
and
G
, respectively. A model for the
trailer is shown in Fig. P5.137(b) where point
A
is idealized as a pin and the tires are modeled by vertical
springs with stiffness
kD300 kN=m
. Assuming the trailer is in static equilibrium, determine the vertical
deflection of point
D
due to the weight of the trailer and tractor. Also determine the reactions at
A
and the
force supported by each spring.
