4–81.
Two couples act on the frame.If determine the
resultant couple moment. Compute the result by resolving
each force into xand ycomponents and (a) finding the
moment of each couple (Eq. 4–13) and (b) summing the
moments of all the force components about point B.
d=4ft,
SOLUTION
(a)
MC=©(r*F)
2ft
B
y
1ft
3ft
50 lb
80 lb
50 lb
30
30
5
4
3
3ft
d
5
3
309
Ans:
SOLUTION
4–82.
Express the moment of the couple acting on the pipe
assembly in Cartesian vector form. What is the magnitude
of the couple moment?
z
y
x
B
A
20 lb
20 lb
1 ft
C
1.5 ft
3 ft
2 ft
1 ft
310
4–83.
If ,,and ,
determine the magnitude and coordinate direction angles
of the resultant couple moment.
M3=120 lb
#
ftM2=90 lb
#
ftM1=180 lb
#
ft
SOLUTION
Since the couple moment is a free vector, it can act at any point without altering its
effect.Thus, the couple moments M1,M2,M3, and M4acting on the gear deducer can
be simplified, as shown in Fig. a. Expressing each couple moment in Cartesian
vector form,
x
z
y
2ft
2ft
2ft
3ft
150 lb ft
1ft
45
45
M
2
M
3
The resultant couple moment is given by
The magnitude of is
The coordinate angles of are
(Mc)R
(Mc)R
311
*4–84.
SOLUTION
Since the couple moment is a free vector, it can act at any point without altering its
effect.Thus, the couple moments M1,M2,M3, and M4acting on the gear deducer can
be simplified, as shown in Fig. a. Expressing each couple moment in Cartesian
vector form,
Determine the magnitudes of couple moments
so that the resultant couple moment is zero.M1,M2, and M3
x
z
y
2ft
2ft
2ft
3ft
150 lb ft
1ft
45
45
M
1
M
2
M
3
The resultant couple moment is required to be zero.Thus,
Equating the i, j, and kcomponents,
312
Ans:
4–85.
SOLUTION
T
h
e gears are su
bj
ecte
d
to t
h
e coup
l
e moments s
h
own.
Determine the magnitude and coordinate direction angles
of the resultant couple moment.
z
y
M
1
40 lb ft
M
2
30 lb ft
20
30
15
313
4–86.
Determine the required magnitude of the couple moments
and so that the resultant couple moment is zero.M3
M2
M
3
M
2
45
SOLUTION
Since the couple moment is the free vector, it can act at any point without altering
314
SOLUTION
4–87.
Determine the resultant couple moment of the two couples
that act on the assembly. Specify its magnitude and
coordinate direction angles.
30
60 lb
80 lb
x
y
z
2 in.
2 in.
4 in.
315
Ans:
*4–88.
Express the moment of the couple acting on the frame in
Cartesian vector form. The forces are applied perpendicular
to the frame.What is the magnitude of the couple moment?
Take .F=50 N
F
x
y
z
O
1.5 m
3m
30
F
SOLUTION
Ans:
4–89.
In order to turn over the frame,a couple moment is applied
as shown. If the component of this couple moment along
the xaxis is , determine the magnitude
Fof the couple forces.
Mx=5–20i6N#m
y
z
O
3m
F
SOLUTION
Ans:
4–90.
SOLUTION
Express the moment of the couple acting on the pipe in
Cartesian vector form. What is the magnitude of the couple
moment? Take F=125 N.
z
O
y
A
B
–F
600 mm
200mm
150 mm
318
4–91.
If the couple moment acting on the pipe has a magnitude of
determine the magnitude Fof the forces applied
to the wrenches.
300 N #m,
SOLUTION
z
O
y
A
B
–F
600 mm
200mm
150 mm
319
*4–92.
If , determine the magnitude and coordinate
direction angles of the couple moment. The pipe assembly
lies in the x–y plane.
F=80 N
x
z
y
300 mm
200 mm
200mm300 mm
300 mm
F
F
SOLUTION
It is easiest to find the couple moment of Fby taking the moment of For –Fabout
point Aor B, respectively,Fig. a. Here the position vectors and must be
determined first.
The coordinate angles of are
Mc
rBA
rAB
Ans:
Ans:
4–93.
If the magnitude of the couple moment acting on the pipe
assembly is , determine the magnitude of the couple
forces applied to each wrench.The pipe assembly lies in the
x–y plane.
50 N #m
x
z
y
300 mm
200 mm
200 mm 300 mm
300 mm
F
F
SOLUTION
It is easiest to find the couple moment of Fby taking the moment of either For –F
about point Aor B, respectively,Fig. a. Here the position vectors and must
be determined first.
Thus, the couple moment of Fcan be determined from
The magnitude of is given by
Mc
rBA
rAB
321
Ans:
4–94.
Express the moment of the couple acting on the rod in
Cartesian vector form. What is the magnitude of the couple
moment?
SOLUTION
Couple Moment.
MC=rAB *F
The magnitude of
MC
is
1 m
2 m
1 m 3 m
x
y
A
z
F { 4i 3j 4k} kN
322
4–95.
If , and , determine the
magnitude and coordinate direction angles of the resultant
couple moment.
SOLUTION
Couple Moment: The position vectors r1,r2,r3, and r4,Fig. a,must be determined
first.
From the geometry of Figs. band c, we obtain
Thus,
Resultant Moment: The resultant couple moment is given by
The magnitude of the couple moment is
The coordinate angles of (Mc)Rare
F
3=80 NF
2=120 NF
1=100 N
– F1
y
x
z
0.2 m
0.2 m
0.2 m
0.2 m
0.2 m
0.3 m
0.3 m
30
– F2
F1
–F4 [ 150 k] N
F4 [150 k] N
F2
F3
323
*4–96.
Determine the required magnitude of
F
1,
F
2,and
F
3
so that the resultant couple moment is
.
SOLUTION
Couple Moment: The position vectors r1,r2,r3, and r4,Fig. a, must be determined
first.
From the geometry of Figs. band c, we obtain
The force vectors F1,F2, and F3are given by
Thus,
Resultant Moment:The resultant couple moment required to equal
Equating the i,j, and kcomponents yields
[50 i–45 j–20 k] N #m
(Mc)R=
y
z
0.2 m
0.2 m
0.2 m
0.2 m
0.3 m
0.3 m
30
– F2
F1
–F4 [ 150 k] N
F4 [150 k] N
F2
324
Ans:
SOLUTION
Equivalent Resultant Force And Couple Moment At O.
As indicated in Fig. a
4–97.
Replace the force system by an equivalent resultant force
and couple moment at point O.
y
x
O
600 N
60
2.5 m 2 m
0.75 m
0.75 m
1 m
5
12
13
455 N
P
325
Ans:
4–98.
Replace the force system by an equivalent resultant force
and couple moment at point P.
SOLUTION
Equivalent Resultant Force And Couple Moment At P.
And
y
x
O
60
2.5 m 2 m
0.75 m
0.75 m
1 m
5
12
13
455 N
P
326
Ans:
4–99.
SOLUTION
Thus,
Replace the force system acting on the beam by an
equivalent force and couple moment at point A.2.5 kN 1.5 kN
3 kN
AB
4 m
3
4
5
2
m2
m
30
327
Ans:
*4–100.
Replace the force system acting on the beam by an
equivalent force and couple moment at point B.
SOLUTION
2.5 kN 1.5 kN
3 kN
AB
4 m
3
4
5
2
m2
m
30