ch03-rigid bodi (equivalent systems of forces)
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VECTOR MECHANICS FOR ENGINEERS:
STATICS
Ninth Edition
Ferdinand P. Beer
E. Russell Johnston, Jr.
Lecture Notes:
J. Walt Oler
Texas Tech University
CHAPTER
2010 The McGraw-Hill Companies, Inc. All rights reserved.
3Rigid Bodies:Equivalent Systems
of Forces
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2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: StaticsNinth
Edition
Contents
3 - 2
Introduction
External and Internal Forces
Principle of Transmissibility: Equivalent
Forces
Vector Products of Two Vectors
Moment of a Force About a PointVarigons Theorem
Rectangular Components of the
Moment of a Force
Sample Problem 3.1
Scalar Product of Two Vectors
Scalar Product of Two Vectors:
Applications
Mixed Triple Product of Three Vectors
Moment of a Force About a Given Axis
Sample Problem 3.5
Moment of a Couple
Addition of Couples
Couples Can Be Represented By Vectors
Resolution of a Force Into a Force at Oand a Couple
Sample Problem 3.6
System of Forces: Reduction to a Force
and a Couple
Further Reduction of a System of Forces
Sample Problem 3.8
Sample Problem 3.10
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Vector Mechanics for Engineers: StaticsNinth
Edition
Introduction
3 - 3
Treatment of a body as a single particle is not always possible. In
general, the size of the body and the specific points of application of theforces must be considered.
Most bodies in elementary mechanics are assumed to be rigid, i.e., the
actual deformations are small and do not affect the conditions of
equilibrium or motion of the body.
Current chapter describes the effect of forces exerted on a rigid body and
how to replace a given system of forces with a simpler equivalent system.
moment of a force about a point
moment of a force about an axis
moment due to a couple
Any system of forces acting on a rigid body can be replaced by an
equivalent system consisting of one force acting at a given point and one
couple.
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Vector Mechanics for Engineers: StaticsNinth
Edition
External and Internal Forces
3 - 4
Forces acting on rigid bodies are
divided into two groups:- External forces
- Internal forces
External forces are shown in a
free-body diagram.
If unopposed, each external force
can impart a motion of
translation or rotation, or both.
NE
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Vector Mechanics for Engineers: StaticsNinth
Edition
Principle of Transmissibility: Equivalent Forces
3 - 5
Principle of Transmissibility-
Conditions of equilibrium or motion arenot affected by transmittinga force
along its line of action.
NOTE: Fand F are equivalent forces.
Moving the point of application of
the force Fto the rear bumper
does not affect the motion or the
other forces acting on the truck.
Principle of transmissibility may
not always apply in determining
internal forces and deformations.
NE
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Vector Mechanics for Engineers: StaticsNinth
Edition
Vector Product of Two Vectors
3 - 6
Concept of the moment of a force about a point is
more easily understood through applications ofthe vector product orcross product.
Vector product of two vectors Pand Qis defined
as the vector Vwhich satisfies the following
conditions:
1. Line of action of Vis perpendicular to plane
containing Pand Q.
2. Magnitude of Vis
3. Direction of Vis obtained from the right-hand
rule.
sinQPV
Vector products:
- are not commutative,
- are distributive,
- are not associative,
QPPQ
2121 QPQPQQP
SQPSQP
NE
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Vector Mechanics for Engineers: StaticsNinth
Edition
Vector Products: Rectangular Components
3 - 7
Vector products of Cartesian unit vectors,
0
0
0
i i j i k k i j
i j k j j k j i
i k j j k i k k
r rr r r r r r
r r r r r r rv
Vector products in terms of rectangularcoordinates
x y z x y zV P i P j P k Q i Q j Q k r r r r r
y z z y z x x z
x y y x
P Q P Q i P Q P Q j
P Q P Q k
r
x y z
x y z
i j k
P P P
Q Q Q
r r
NE
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Vector Mechanics for Engineers: StaticsNinth
Edition
Moment of a Force About a Point
3 - 8
A force vector is defined by its magnitude and
direction. Its effect on the rigid body also dependson it point of application.
The momentof Fabout Ois defined as
FrMO
The moment vector MOis perpendicular to theplane containing Oand the force F.
Any force Fthat has the same magnitude and
direction as F, is equivalentif it also has the same line
of action and therefore, produces the same moment.
Magnitude of MOmeasures the tendency of the force
to cause rotation of the body about an axis along MO.
The sense of the moment may be determined by the
right-hand rule.
FdrFMO sin
f SNE
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Vector Mechanics for Engineers: StaticsNinth
Edition
Moment of a Force About a Point
3 - 9
Two-dimensional structureshave length and breadth but
negligible depth and are subjected to forces contained inthe plane of the structure.
The plane of the structure contains the point Oand the
force F. MO, the moment of the force about Ois
perpendicular to the plane.
If the force tends to rotate the structure clockwise, the
sense of the moment vector is out of the plane of the
structure and the magnitude of the moment is positive.
If the force tends to rotate the structure counterclockwise,
the sense of the moment vector is into the plane of the
structure and the magnitude of the moment is negative.
V t M h i f E i St tiNE
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Vector Mechanics for Engineers: StaticsNinth
Edition
Varignons Theorem
3 - 10
The moment about a give point Oof theresultant of several concurrent forces is equal
to the sum of the moments of the various
moments about the same point O.
Varigons Theorem makes it possible to
replace the direct determination of the
moment of a force Fby the moments of two
or more component forces of F.
1 2 1 2r F F r F r F
r r r
L L
V t M h i f E i St tiNE
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Vector Mechanics for Engineers: StaticsNinth
Edition
Rectangular Components of the Moment of a Force
3 - 11
O x y z
x y z
z y x z y x
M M i M j M k
i j k
x y z
F F F
yF zF i zF xF j xF yF k
rr r
rr r
The moment of Fabout O,
,O
x y z
M r F r xi yj zk
F F i F j F k
r r
rr r r
V t M h i f E i St tiNE
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Vector Mechanics for Engineers: StaticsNinth
Edition
Rectangular Components of the Moment of a Force
3 - 12
The moment of FaboutB,
/B A BM r F
r
/A B A B
A B A B A B
x y z
r r r
x x i y y j z z k
F F i F j F k
rr r
rr r r
B A B A B A Bx y z
i j k
M x x y y z zF F F
r r
r
V t M h i f E i St tiNE
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Vector Mechanics for Engineers: StaticsNinth
Edition
Rectangular Components of the Moment of a Force
3 - 13
For two-dimensional structures,
O y zO Z
y z
M xF yF k
M M
xF yF
O A B y A B z
O Z
A B y A B z
M x x F y y F k
M M
x x F y y F
V t M h i f E i St tiNE
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Vector Mechanics for Engineers: StaticsNinth
Edition
Sample Problem 3.1
3 - 14
A 100-lb vertical force is applied to the end of a
lever which is attached to a shaft at O.
Determine:
a) moment about O,
b) horizontal force atA which creates the samemoment,
c) smallest force at A which produces the same
moment,
d) location for a 240-lb vertical force to produce
the same moment,
e) whether any of the forces from b, c, and d is
equivalent to the original force.
V t M h i f E i St tiNE
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Vector Mechanics for Engineers: StaticsNinth
Edition
Sample Problem 3.1
3 - 15
a) Moment about O is equal to the product of the
force and the perpendicular distance between theline of action of the force and O. Since the force
tends to rotate the lever clockwise, the moment
vector is into the plane of the paper.
in.12lb100
in.1260cosin.24
O
O
M
dFdM
inlb1200 OM
V t M h i f E i St tiN
Ed
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Vector Mechanics for Engineers: StaticsNinth
Edition
Sample Problem 3.1
3 - 16
c) Horizontal force atAthat produces the same
moment,
in.8.20
in.lb1200
in.8.20in.lb1200
in.8.2060sinin.24
F
F
FdM
d
O
lb7.57F
V t M h i f E i St tiNi
Ed
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Vector Mechanics for Engineers: StaticsNinth
Edition
Sample Problem 3.1
3 - 17
c) The smallest forceAto produce the same moment
occurs when the perpendicular distance is amaximum or whenFis perpendicular to OA.
in.42
in.lb1200
in.42in.lb1200
F
F
FdMO
lb50F
Vector Mechanics for Engineers StaticsNi
Ed
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Vector Mechanics for Engineers: Staticsinth
dition
Sample Problem 3.1
3 - 18
d) To determine the point of application of a 240 lb
force to produce the same moment,
in.5cos60
in.5lb402
in.lb1200
lb240in.lb1200
OB
d
d
FdMO
in.10OB
Vector Mechanics for Engineers: StaticsNi
Ed
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Vector Mechanics for Engineers: Staticsinth
dition
Sample Problem 3.1
3 - 19
e) Although each of the forces in parts b), c), and d)produces the same moment as the 100 lb force, none
are of the same magnitude and sense, or on the same
line of action. None of the forces is equivalent to the
100 lb force.
Vector Mechanics for Engineers: StaticsNi
Ed
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Vector Mechanics for Engineers: Staticsnth
dition
Sample Problem 3.4
3 - 20
The rectangular plate is supported by
the brackets atAandBand by a wire
CD. Knowing that the tension in the
wire is 200 N, determine the moment
aboutAof the force exerted by the
wire at C.
SOLUTION:
The momentMAof the forceFexerted
by the wire is obtained by evaluating
the vector product,
A C A
M r F r
Vector Mechanics for Engineers: StaticsNi
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Vector Mechanics for Engineers: Staticsnth
dition
Sample Problem 3.4
3 - 21
SOLUTION:
0.3 0 0.08
120 96 128
A
i j k
M
r r
r
7.68 N m 28.8 N m 28.8 N mAM i j k v
0.3 m 0.08 mC A C Ar r r i j r r r
A C AM r F
r
200 N
0.3 m 0.24 m 0.32 m200 N
0.5 m
120 N 96 N 128 N
C D
C D
rF F
r
i j k
i j k
rr
rr r
rr r
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Vector Mechanics for Engineers: Staticsnth
dition
Scalar Product of Two Vectors
3 - 22
Thescalar productor dot productbetween
two vectorsP
andQ
is defined as cos scalar resultP Q PQ
Scalar products:
- are commutative,
- are distributive,- are not associative,
P Q Q P
1 2 1 2P Q Q P Q P Q
r r r
undefinedP Q S r rr
Scalar products with Cartesian unit components,
1 1 1 0 0 0i i j j k k i j j k k i v
x y z x y zP Q P i P j P k Q i Q j Q k r r r r r
2 2 2 2
x x y y z z
x y z
P Q P Q P Q P Q
P P P P P P
r r
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Vector Mechanics for Engineers: Staticsnth
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Scalar Product of Two Vectors: Applications
3 - 23
Angle between two vectors:
cos
cos
x x y y z z
x x y y z z
P Q PQ P Q P Q P Q
P Q P Q P Q
PQ
Projection of a vector on a given axis:
cos projection of along
cos
cos
OL
OL
P P P OL
P Q PQ
P QP P
Q
rr
rr
cos cos cos
OL
x x y y z z
P P
P P P
For an axis defined by a unit vector:
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Vector Mechanics for Engineers: Staticsnth
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Mixed Triple Product of Three Vectors
3 - 24
Mixed triple product of three vectors,
scalar resultS P Q r rr
The six mixed triple products formed from S, P, and
Qhave equal magnitudes but not the same sign,
S P Q P Q S Q S P
S Q P P S Q Q P S
r r r r r r r r
x y z z y y z x x z
z x y y x
x y z
x y z
x y z
S P Q S P Q PQ S PQ P Q
S P Q P Q
S S S
P P P
Q Q Q
Evaluating the mixed triple product,
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Moment of a Force About a Given Axis
3 - 25
Moment MOof a force Fapplied at the point A
about a point O,
OM r F
r
Scalar momentMOLabout an axis OL is the
projection of the moment vector MOonto the
axis,
OL OM M r F r
Moments of Fabout the coordinate axes,
xyz
zxy
yzx
yFxFM
xFzFMzFyFM
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Moment of a Force About a Given Axis
3 - 26
Moment of a force about an arbitrary axis,
BL B
A B
A B A B
M M
r F
r r r
r rr
r r r
The result is independent of the pointB
along the given axis.
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Sample Problem 3.5
3 - 27
a) aboutA
b) about the edgeAB and
c) about the diagonalAGof the cube.d) Determine the perpendicular distance
betweenAGandFC.
A cube is acted on by a force Pas
shown. Determine the moment of P
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Sample Problem 3.5
3 - 28
Moment of PaboutA,
2 2 2
2
A F A
F A
A
M r P
r ai a j a i j
P P i j P i j
M a i j P i j
r
r r r r r
r r r r r
r r r r r
2AM aP i j k r r r
Moment of PaboutAB,
2AB A
M i M
i aP i j k
rr r r
2aPMAB
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Sample Problem 3.5
3 - 29
Moment of Pabout the diagonalAG,
1
3 3
2
1
3 2
1 1 1
6
AG A
A G
A G
A
AG
M M
r ai aj ak i j k
r a
aP
M i j k
aPM i j k i j k
aP
rr rrrr r r
rr r r
r rr r r r
6
aPMAG
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Vector Mechanics for Engineers: Staticsthtion
Sample Problem 3.5
3 - 30
Perpendicular distance betweenAGandFC,
1 0 1 12 3 6
0
P PP j k i j k
r rrr r r r
Therefore,Pis perpendicular toAG.
PdaP
MAG 6
6
ad
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Vector Mechanics for Engineers: Staticsthtion
Moment of a Couple
3 - 31
Two forces Fand -F having the same magnitude,
parallel lines of action, and opposite sense are saidto form a couple.
Moment of the couple,
sin
A B
A B
M r F r F
r r F
r F
M rF Fd
r r rr r
rr r
rr
The moment vector of the couple isindependent of the choice of the origin of the
coordinate axes, i.e., it is afree vectorthat can
be applied at any point with the same effect.
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Vector Mechanics for Engineers: Staticsthtion
Moment of a Couple
3 - 32
Two couples will have equal moments if
2211 dFdF
the two couples lie in parallel planes, and
the two couples have the same sense orthe tendency to cause rotation in the same
direction.
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Vector Mechanics for Engineers: Staticsthtion
Addition of Couples
3 - 33
Consider two intersecting planesP1and
P2 with each containing a couple
1 1 1
2 2 2
in plane
in plane
M r F P
M r F P
r
r rr
Resultants of the vectors also form a
couple
1 2M r R r F F r r
By Varigons theorem
1 2
1 2
M r F r F
M M
r r
r r
Sum of two couples is also a couple that is equal
to the vector sum of the two couples
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Vector Mechanics for Engineers: Staticsthtion
Couples Can Be Represented by Vectors
3 - 34
A couple can be represented by a vector with magnitudeand direction equal to the moment of the couple.
Couple vectorsobey the law of addition of vectors.
Couple vectors are free vectors, i.e., the point of applicationis not significant.
Couple vectors may be resolved into component vectors.
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Vector Mechanics for Engineers: Staticsthtion
Resolution of a Force Into a Force at Oand a Couple
3 - 35
Force vector Fcan not be simply moved to Owithout modifying its
action on the body.
Attaching equal and opposite force vectors at Oproduces no net
effect on the body.
The three forces may be replaced by an equivalent force vector and
couple vector, i.e, aforce-couple system.
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Vector Mechanics for Engineers: Staticsthion
Resolution of a Force Into a Force at Oand a Couple
3 - 36
Moving FfromAto a different point Orequires the
addition of a different couple vector MO
'OM r F
r
The moments of Fabout O and Oare related,
' 'O
O
M r F r s F r F s F
M s F
r r r r r
r rr
Moving the force-couple system from Oto Orequires the
addition of the moment of the force at Oabout O.
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Vector Mechanics for Engineers: Staticshion
Sample Problem 3.6
3 - 37
Determine the components of thesingle couple equivalent to the
couples shown.
SOLUTION:
Attach equal and opposite 20 lb forces inthe +xdirection atA, thereby producing 3
couples for which the moment components
are easily computed.
Alternatively, compute the sum of the
moments of the four forces about an
arbitrary single point. The pointDis a
good choice as only two of the forces will
produce non-zero moment contributions..
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Vector Mechanics for Engineers: Staticshion
Sample Problem 3.6
3 - 38
Attach equal and opposite 20 lb forces in
the +xdirection atA
The three couples may be represented by
three couple vectors,
in.lb180in.9lb20
in.lb240in.12lb20
in.lb540in.18lb30
z
y
x
M
M
M
540 lb in. 240lb in.
180 lb in.
M i j
k
r
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Vector Mechanics for Engineers: Staticshion
Sample Problem 3.6
3 - 39
Alternatively, compute the sum of the
moments of the four forces aboutD.
Only the forces at CandEcontribute to
the moment aboutD.
18 in. 30 lb
9 in. 12 in. 20 lb
DM M j k
j k i
rr r
540 lb in. 240lb in.
180 lb in.
M i j
k
r
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Vector Mechanics for Engineers: StaticshonSystem of Forces: Reduction to a Force and Couple
3 - 40
A system of forces may be replaced by a collection of
force-couple systems acting a given point O
The force and couple vectors may be combined into a
resultant force vector and a resultant couple vector,
ROR F M r F r
The force-couple system at Omay be moved to Owith the addition of the moment of Rabout O,
'
R R
O OM M s R
r
Two systems of forces are equivalent if they can be
reduced to the same force-couple system.
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Vector Mechanics for Engineers: StaticshonFurther Reduction of a System of Forces
3 - 41
If the resultant force and couple at Oare mutually
perpendicular, they can be replaced by a single force actingalong a new line of action.
The resultant force-couple system for a system of forces
will be mutually perpendicular if:
1) the forces are concurrent,
2) the forces are coplanar, or3) the forces are parallel.
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Vector Mechanics for Engineers: StaticshonFurther Reduction of a System of Forces
3 - 42
System of coplanar forces is reduced to a
force-couple system that ismutually perpendicular.
and ROR M
System can be reduced to a single force
by moving the line of action of until
its moment about Obecomes ROM
R
In terms of rectangular coordinates,R
Oxy MyRxR
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Vector Mechanics for Engineers: StaticshonSample Problem 3.8
3 - 43
For the beam, reduce the system of
forces shown to (a) an equivalent
force-couple system atA, (b) an
equivalent force couple system atB,
and (c) a single force or resultant.
Note: Since the support reactions are
not included, the given system will
not maintain the beam in equilibrium.
SOLUTION:
a) Compute the resultant force for the
forces shown and the resultant
couple for the moments of the
forces aboutA.
b) Find an equivalent force-couple
system atBbased on the force-
couple system atA.
c) Determine the point of application
for the resultant force such that itsmoment aboutAis equal to the
resultant couple atA.
Vector Mechanics for Engineers: StaticsNinth
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Vector Mechanics for Engineers: StaticshonSample Problem 3.8
3 - 44
SOLUTION:
a) Compute the resultant force and theresultant couple atA.
150 N 600 N 100 N 250 N
R F
j j j j
r r r r
600 NR j
1.6 600 2.8 100
4.8 250
R
AM r F
i j i j
i j
r
r r r r
r r
1880 N mRAM k
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Vector Mechanics for Engineers: StaticshonSample Problem 3.8
3 - 45
b) Find an equivalent force-couple system atB
based on the force-couple system atA.The force is unchanged by the movement of the
force-couple system fromAtoB.
600 NR j
The couple atBis equal to the moment aboutB
of the force-couple system found atA.
1880 N m 4.8 m 600 N
1880 N m 2880 N m
R R
B A B AM M r R
k i j
k k
r
r r r
r r
1000 N mRBM k
Vector Mechanics for Engineers: StaticsNinth
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Vector Mechanics for Engineers: StaticshonSample Problem 3.10
3 - 46
Three cables are attached to thebracket as shown. Replace the
forces with an equivalent force-
couple system atA.
SOLUTION:
Determine the relative position vectors
for the points of application of the
cable forces with respect toA.
Resolve the forces into rectangular
components.
Compute the equivalent force,
R F
Compute the equivalent couple,
RAM r F r
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2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: StaticsonSample Problem 3.10
3 - 47
SOLUTION:
Determine the relative position
vectors with respect toA.
0.075 0.050 m
0.075 0.050 m
0.100 0.100 m
B A
C A
D A
r i k
r i k
r i j
r
rrr
r rr
Resolve the forces into rectangular
components.
700 N
75 150 50
175
0.429 0.857 0.289
300 600 200 N
B
E B
E B
B
F
r i j k
r
i j k
F i j k
rr rrr
rr r
rr r r
1200 N cos60 cos30
600 1039 N
DF i j
i j
r r
1000 N cos 45 cos 45
707 707 N
CF i j
i j
r r
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ecto ec a cs o g ee s Stat csonSample Problem 3.10 Compute the equivalent force,
300 707 600
600 1039
200 707
R F
i
j
k
r
r
r
1607 439 507 NR i j k
Compute the equivalent couple,
0.075 0 0.050 30 45
300 600 200
0.075 0 0.050 17.68
707 0 707
0.100 0.100 0 163.9
600 1039 0
RA
B A B
C A c
D A D
M r F
i j k
r F i k
i j k
r F j
i j k
r F k
v r
rr r
rr rr
rr r
r rr
rr r
rrr
30 17.68 118.9R
AM i j k
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