PROBLEM 3.78
Replace the two couples shown with a single equivalent
couple, specifying its magnitude and the direction of its axis.
SOLUTION
PROBLEM 3.79
Solve Prob. 3.78, assuming that two 10-N vertical forces
have been added, one acting upward at C and the other
downward at B.
PROBLEM 3.78 Replace the two couples shown with a
single equivalent couple, specifying its magnitude and the
direction of its axis.
SOLUTION
PROBLEM 3.79 (Continued)
123
(3.84 2.88 ) 2.40 ( 1.92 1.6 )
(1.92 N m) (1.44 N m) (1.28 N m)
MM M M= + + = + − +− −
=− ⋅+ ⋅+ ⋅
j k j ik
i jk
PROBLEM 3.80
Shafts A and B connect the gear box to the wheel
assemblies of a tractor, and shaft C connects it to the
engine. Shafts A and B lie in the vertical yz plane,
while shaft C is directed along the x axis. Replace
the couples applied to the shafts by a single
equivalent couple, specifying its magnitude and the
direction of its axis.
SOLUTION
PROBLEM 3.81
A 500-N force is applied to a bent plate as shown. Determine
(a) an equivalent force-couple system at B, (b) an equivalent
system formed by a vertical force at A and a force at B.
SOLUTION
(a) Force–couple system at B
(500 N)sin 30 (500 N) cos30
(250.0 N) (433.01 N)
= −
= −
F ik
Fik
PROBLEM 3.82
The tension in the cable attached to the end
C of an adjustable boom ABC is 560 lb.
Replace the force exerted by the cable at C
with an equivalent force–couple system
(a) at A, (b) at B.
SOLUTION
=F
MM T dΣ=°
(a) Based on
: 560 lb
A
FFTΣ==
or
560 lb
A
=F
20.0°
: ( sin 50 )( )
AA A
MM T dΣ=°
PROBLEM 3.83
A dirigible is tethered by a cable attached to its cabin at B.
If the tension in the cable is 1040 N, replace the force
exerted by the cable at B with an equivalent system formed
by two parallel forces applied at A and C.
SOLUTION
Require the equivalent forces acting at A and C be parallel and at an
angle of
a
with the vertical.
Then for equivalence,
: (1040 N)sin 30 sin sin
x AB
F FF
aa
Σ °= +
(1)
PROBLEM 3.84
A 30–lb vertical force P is applied at A to the bracket shown, which is held
by screws at B and C. (a) Replace P with an equivalent force-couple
system at B. (b) Find the two horizontal forces at B and C that are
equivalent to the couple obtained in part a.
SOLUTION
(a)
PROBLEM 3.85
A worker tries to move a rock by applying a 360-N
force to a steel bar as shown. (a) Replace that force
with an equivalent force-couple system at D. (b) Two
workers attempt to move the same rock by applying a
vertical force at A and another force at D. Determine
these two forces if they are to be equivalent to the
single force of Part a.
SOLUTION
(a)
(b)
PROBLEM 3.85 (Continued)
or
(0.90933 ) 230.45
A
F=kk
253.42 N
A
F=
or 253 N
A
=F
PROBLEM 3.86
A worker tries to move a rock by applying a 360-N
force to a steel bar as shown. If two workers attempt
to move the same rock by applying a force at A and a
parallel force at C, determine these two forces so that
they will be equivalent to the single 360–N force
shown in the figure.
SOLUTION
: –360sin40 = sin sin (1)
X AC
F FF
aa
Σ −−
: –360cos 40 cos F cos (2)
Y AC
FF
aa
Σ=−
PROBLEM 3.87
The shearing forces exerted on the cross section of a steel channel can be
represented by a 900-N vertical force and two 250–N horizontal forces as
shown. Replace this force and couple with a single force F applied at
Point C, and determine the distance x from C to line BD. (Point C is
defined as the shear center of the section.)
SOLUTION
Replace the 250–N forces with a couple and move the 900-N force to Point C such that its moment about
H is equal to the moment of the couple
PROBLEM 3.88
A force and a couple are applied as shown to the end of a
cantilever beam. (a) Replace this system with a single force
F applied at Point C, and determine the distance d from C to
a line drawn through Points D and E. (b) Solve part a if the
directions of the two 360-N forces are reversed.
SOLUTION
(a) We have
: (360 N) (360 N) (600 N)Σ= − −FF j j k
or
(600 N)= −Fk
PROBLEM 3.89
Three control rods attached to a lever ABC exert on it the
forces shown. (a) Replace the three forces with an
equivalent force–couple system at B. (b) Determine the
single force that is equivalent to the force–couple system
obtained in Part a, and specify its point of application on
the lever.
SOLUTION
(a) First note that the two 20–lb forces form A couple. Then
48 lb=F
θ
PROBLEM 3.90
A rectangular plate is acted upon by the force and couple
shown. This system is to be replaced with a single equivalent
force. (a) For
a
= 40°, specify the magnitude and the line of
action of the equivalent force. (b) Specify the value of
a
if the line
of action of the equivalent force is to intersect line CD 300 mm
to the right of D.
SOLUTION
(a) The given force-couple system (F, M) at B is
48 N=F
PROBLEM 3.91
While tapping a hole, a machinist applies the horizontal
forces shown to the handle of the tap wrench. Show that
these forces are equivalent to a single force, and specify, if
possible, the point of application of the single force on the
handle.
SOLUTION
Since the forces at A and B are parallel, the force at B can be replaced with the sum of two forces with one
of the forces equal in magnitude to the force at A except with an opposite sense, resulting in a force-
couple.
PROBLEM 3.92
A hexagonal plate is acted upon by the force P and the couple shown.
Determine the magnitude and the direction of the smallest force P for
which this system can be replaced with a single force at E.
SOLUTION
From the statement of the problem, it follows that
for the given force-couple system. Further,
PROBLEM 3.93
An eccentric, compressive 250–kN force P is applied to the end
of a column. Replace P with an equivalent force-couple system
at G.
SOLUTION
PROBLEM 3.94
A 2.6–kip force is applied at Point D of the cast iron post
shown. Replace that force with an equivalent force-couple
system at the center A of the base section.
SOLUTION
(12 in.) (5 in.) ; 13.00 in.DE DE=−− =jk
(2.6 kips) DE
DE
=F
PROBLEM 3.95
Replace the 150–N force with an equivalent force-couple
system at A.
SOLUTION
Equivalence requires
: (150 N)( cos 35 sin 35 )
(122.873 N) (86.036 N)
Σ = − °− °
=−−
FF j k
jk