Ch03-Eqm of Particle and Moment of Force
-
Upload
ankit-vyas -
Category
Documents
-
view
18 -
download
0
description
Transcript of Ch03-Eqm of Particle and Moment of Force
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 1/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i on
2 - 1
Equilibrium of a Particle
andConcept of MOMENT of
FORCE
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 2/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i on
Condition for the Equilibrium of a Particle
2 - 2
OBJECTIVE Introduction of free-body diagram (FBD) for a particle and
to solve particle equilibrium problems
Necessary and Sufficient condition: = 0To apply it, account for ALL the Known & Unknown forces that
act ON the particle.
Done by ISOLATING the particle from surroundings : FBD
Springs Cables & Pulleys Smooth Contact
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 3/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i on
Procedure for drawing FBD for a Particle
2 - 3
Draw Outlined Shape
Show All Forces
Identify Each Force
Ex.
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 4/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: StaticsNi n t h
Ed i t i on
Procedure for Analysis- Co planar Forces
2 - 4
Ex.
NE
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 5/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: StaticsNi n t h
Ed i t i on
Procedure for Analysis- 3D Forces
2 - 5NE
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 6/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: StaticsNi n t h
Ed i t i on
External and Internal Forces
3 - 6
• 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
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 7/59© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: StaticsNi n t h
Ed i t i on
Forces
2 - 7
Principle of Transmissibility: A force may be moved anywhere along its
line of action without changing its external
effects (state & support reactions) on a rigid
body.
Concurrent Forces NE
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 8/59© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: StaticsNi n t h
Ed i t i on
Principle of Transmissibility: Equivalent Forces
3 - 8
• Principle of Transmissibility -
Conditions of equilibrium or motion arenot affected by transmitting a force
along its line of action.
NOTE: F and F’ are equivalent forces.
• Moving the point of application of
the force F to 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.
f SNE
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 9/59© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: StaticsNi n t h
Ed i t i on
Vector Product of Two Vectors
3 - 9
• Concept of the moment of a force about a point is
more easily understood through applications ofthe vector product or cross product .
• Vector product of two vectors P and Q is defined
as the vector V which satisfies the following
conditions:
1. Line of action of V is perpendicular to plane
containing P and Q .
2. Magnitude of V is
3. Direction of V is obtained from the right-hand
rule.
sinQ P V
• Vector products:
- are not commutative,
- are distributive,
- are not associative,
Q P P Q
2121 Q P Q P Q Q P
S Q P S Q P
V t M h i f E i St tiNE
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 10/59© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: StaticsNi n t h
Ed i t i on
Vector Products: Rectangular Components
3 - 10
• Vector products of Cartesian unit vectors,
0
0
0
k k ik j jk i
i jk j jk ji
jik k i jii
• Vector products in terms of rectangularcoordinates
k Q jQiQk P j P i P V z y x z y x
k Q P Q P
jQ P Q P iQ P Q P
x y y x
z x x z y z z y
z y x
z y x
QQQ
P P P
k ji
V t M h i f E i St tiNE
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 11/59© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: StaticsNi n t h
Ed i t i on
Force System Resultants
2 - 11
OBJECTIVE
Concept of the moment of
a force and finding it about
a specified axis
Define moment of a couple
Find the resultant effect of
a non-concurrent force
system
To indicate how to reduce asimple distributed loading
to a resultant force acting
at a specified location
V t M h i f E i St tiNE
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 12/59© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: StaticsNi n t h
Ed i t i on
Moment of a Force About a Point
3 - 12
• A force vector is defined by its magnitude and
direction. Its effect on the rigid body also dependson it point of application.
• The moment of F about O is defined as
F r M O
• The moment vector M O is perpendicular to the plane containing O and the force F .
• Any force F’ that has the same magnitude and
direction as F , is equivalent if it also has the same line
of action and therefore, produces the same moment.
• Magnitude of M O
measures the tendency of the force
to cause rotation of the body about an axis along M O .
The sense of the moment may be determined by the
right-hand rule.
Fd rF M O sin
Moment Arm
Line of
Actionof Force
O: an arbitrary point, called
MOMENT CENTER
V t M h i f E i St tiNE
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 13/59© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: StaticsNi n t h
Ed i t i on
Moment of a Force About a Point
3 - 13
• Two-dimensional structures have 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 O and the
force F . M O , the moment of the force about O is
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
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 14/59© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: StaticsNi n t h
Ed i t i on
Principle of Moments
3 - 14
• The moment about a given point O of the
resultant of several concurrent forces is equalto the sum of the moments of the various
moments about the same point O.
• Varignon’s Theorem makes it possible to
replace the direct determination of the
moment of a force F by the moments of
two or more component forces of F .
Forces , ,
, must be
concurrent, say
at point A.
VARIGNON’s
THEOREM
= × = × + × + ⋯⇒ = × + + ⋯
Ex. (i) Findmomentof forceaboutpt. C
(ii) Find ‘d’
V t M h i f E i St tiNE
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 15/59© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: StaticsNi n t h
Edi t i on
Rectangular Components of the Moment of a Force
3 - 15
k yF xF j xF zF i zF yF
F F F
z y x
k ji
k M j M i M M
x y z x y z
z y x
z y xO
The moment of F about O,
k F j F i F F
k z j yi xr F r M
z y x
O
,
V t M h i f E i St tiN
E d
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 16/59© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: StaticsNin t h
Edi t i on
Rectangular Components of the Moment of a Force
3 - 16
The moment of F about B,
F r M B A B
/
k F j F i F F
k z z j y yi x x
r r r
z y x
B A B A B A
B A B A
/
z y x
B A B A B A B
F F F z z y y x x
k ji
M
V t M h i f E i St tiNi
E d
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 17/59© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: StaticsNin t h
Edi t i on
Rectangular Components of the Moment of a Force
3 - 17
For two-dimensional structures,
z y
Z O
z yO
yF xF
M M
k yF xF M
z B A y B A
Z O
z B A y B AO
F y y F x x
M M
k F y y F x x M
Vector Mechanics for Engineers StaticsNi
E d
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 18/59© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticsin t h
di t i on
Sample Problem 3.1
3 - 18
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 at A 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.
Vector Mechanics for Engineers: StaticsNi
E d
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 19/59© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticsin t h
di t i on
Sample Problem 3.1
3 - 19
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
d Fd M
inlb1200 O M
Vector Mechanics for Engineers: StaticsNi
E d
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 20/59© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticsn t h
di t i on
Sample Problem 3.1
3 - 20
c) Horizontal force at A that produces the same
moment,
in.8.20in.lb1200
in.8.20in.lb1200
in.8.2060sinin.24
F
F
Fd M
d
O
lb7.57 F
Vector Mechanics for Engineers: StaticsNi
E d
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 21/59© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticsn t h
di t i on
Sample Problem 3.1
3 - 21
c) The smallest force A to produce the same moment
occurs when the perpendicular distance is amaximum or when F is perpendicular to OA.
in.42in.lb1200
in.42in.lb1200
F
F
Fd M O
lb50 F
Vector Mechanics for Engineers: StaticsNi n
E d
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 22/59© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticsn t h
di t i on
Sample Problem 3.1
3 - 22
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
Fd M O
in.10OB
Vector Mechanics for Engineers: StaticsNi n
E d
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 23/59© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticsn t h
di t i on
Sample Problem 3.1
3 - 23
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 n
E d
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 24/59© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticsn t h
di t i on
Sample Problem 3.4
3 - 24
The rectangular plate is supported by
the brackets at A and B and by a wire
CD. Knowing that the tension in the
wire is 200 N, determine the moment
about A of the force exerted by the
wire at C .
SOLUTION:
The moment M A of the force F exerted
by the wire is obtained by evaluating
the vector product,
F r M AC A
Vector Mechanics for Engineers: StaticsNi n
E d
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 25/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticsn t h
i t i on
Sample Problem 3.4
3 - 25
SOLUTION:
12896120
08.003.0
k ji
M A
k ji M A
m N8.82m N8.82m N68.7
jir r r AC AC
m08.0m3.0
F r M AC A
k ji
k jir
r F F
DC
DC
N128 N69 N120
m5.0
m32.0m0.24m3.0 N200
N200
Vector Mechanics for Engineers: StaticsNi n
E d
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 26/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticsn t h
it i on
Scalar Product of Two Vectors
3 - 26
• The scalar product or dot product between
two vectorsP
andQ
is defined as resultscalar cos PQQ P
• Scalar products:
- are commutative,
- are distributive,- are not associative,
P QQ P
2121 Q P Q P QQ P
undefined S Q P
• Scalar products with Cartesian unit components,
000111 ik k j jik k j jii
k Q jQiQk P j P i P Q P z y x z y x
2222 P P P P P P
Q P Q P Q P Q P
z y x
z z y y x x
Vector Mechanics for Engineers: StaticsNi n
E d i
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 27/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticsn t h
it i on
Scalar Product of Two Vectors: Applications
3 - 27
• Angle between two vectors:
PQ
Q P Q P Q P
Q P Q P Q P PQQ P
z z y y x x
z z y y x x
cos
cos
• Projection of a vector on a given axis:
OL
O L
P P Q
Q P
PQQ P
OL P P P
cos
cos
alongof projectioncos
z z y y x x
OL
P P P
P P
coscoscos
• For an axis defined by a unit vector:
Vector Mechanics for Engineers: StaticsNi n
E d i
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 28/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticsn t h
it i on
Mixed Triple Product of Three Vectors
3 - 28
• Mixed triple product of three vectors,
resultscalar Q P S
• The six mixed triple products formed from S , P , and
Q have equal magnitudes but not the same sign,
S P QQS P P QS
P S QS Q P Q P S
z y x
z y x
z y x
x y y x z
z x x z y y z z y x
QQQ
P P P
S S S
Q P Q P S
Q P Q P S Q P Q P S Q P S
• Evaluating the mixed triple product,
Vector Mechanics for Engineers: StaticsNi n
E d i
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 29/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticsn t h
t i on
Moment of a Force About a Given Axis (1)
3 - 29
Vector Mechanics for Engineers: StaticsNi n
E d i
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 30/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Statics t h t i on
Moment of a Force About a Given Axis (2)
3 - 30
• Moment M O
of a force F applied at the point A
about a point O ,
F r M O
• Scalar moment M OL about an axis OL is the
projection of the moment vector M O
onto the
axis,
F r M M OOL
• Moments of F about the coordinate axes,
x y z
z x y
y z x
yF xF M
xF zF M
zF yF M
Given a force and an arbitrary line. What is the moment
of force about this line??
= = . ×
: unit vector
along the axis OL
= + +
Vector Mechanics for Engineers: StaticsNi n
E d i t
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 31/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Statics t h ti on
Moment of a Force About a Given Axis (3)
3 - 31
• Moment of a force about an arbitrary axis,
B A B A
B A
B BL
r r r
F r
M M
• The result is independent of the point B
along the given axis.
Vector Mechanics for Engineers: StaticsNi n t
E d i t
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 32/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticsth ti on
Sample Problem 3.5
3 - 32
a) about A
b) about the edge AB and
c) about the diagonal AG of the cube.d) Determine the perpendicular distance
between AG and FC .
A cube is acted on by a force P asshown. Determine the moment of P
Vector Mechanics for Engineers: StaticsNi n t
E d i t
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 33/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticsth ti on
Sample Problem 3.5
3 - 33
• Moment of P about A,
ji P jia M
ji P ji P P
jia jaiar
P r M
A
A F
A F A
2
222
k jiaP M A
2
• Moment of P about AB,
k jiaP i M i M A AB
2
2aP M AB
Vector Mechanics for Engineers: StaticsNi n t
E d i t
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 34/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticsth ti on
Sample Problem 3.5
3 - 34
• Moment of P about the diagonal AG,
1116
23
12
3
1
3
aP
k jiaP
k ji M
k jiaP
M
k jia
k a jaia
r
r
M M
AG
A
G A
G A
A AG
6
aP M AG
Vector Mechanics for Engineers: StaticsNi n t
E d i t
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 35/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticsth ti on
Sample Problem 3.5
3 - 35
• Perpendicular distance between AG and FC,
0
11063
1
2
P
k jik j P
P
Therefore, P is perpendicular to AG.
Pd aP
M AG 6
6
ad
Vector Mechanics for Engineers: StaticsNi n t
E d i t
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 36/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticsth ion
Moment of a Couple
3 - 36
• A force has two effects on a rigid body: translational due to
the force itself and rotation due to the moment of the force.Couple, on the other hand, is PURE ROTATIONAL effect as
it has NO resultant force. Two forces F and -F having the
same magnitude, parallel lines of action, and opposite sense
are said to form a couple.
• Moment of the couple,
Fd rF M
F r
F r r
F r F r M
B A
B A
sin
• The moment vector of the couple is independent
of the choice of the origin of the coordinate axes,
i.e., it is a free vector that can be applied at any
point with the same effect.
Vector Mechanics for Engineers: StaticsNi n t
E d i t
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 37/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticshion
Moment of a Couple
3 - 37
Two couples will have equal moments if
•2211 d F d F
• the two couples lie in parallel planes, and
• the two couples have the same sense orthe tendency to cause rotation in the same
direction.
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 38/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticshion
Addition of Couples
3 - 38
• Consider two intersecting planes P 1 and
P 2 with each containing a couple
222
111
planein
planein
P F r M
P F r M
• Resultants of the vectors also form a
couple
21 F F r Rr M
• By Varigon’s theorem
21
21
M M F r F r M
• Sum of two couples is also a couple that is equal
to the vector sum of the two couples
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 39/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticshion
Couples Can Be Represented by Vectors
3 - 39
• A couple can be represented by a vector with magnitudeand direction equal to the moment of the couple.
• Couple vectors obey 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.
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 40/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticshon
Resolution of Couples
2 - 40
• Couples do not require the condition to have acommon point of application (Why? Free Vecto rs )
• This is in contrast to the addition of forces and for
addition of moments of forces about points, which
can be added only if they are concurrent.
• : Moment of Force and : Couple
• Resolution of Couples is no different than the
resolution of moments of forces; for ex. moment
of a couple about axis AB is:
•
= . (No subscript on
??)
Ex.
(i) Find the
corresponding
couple-vector.
(i) Find the moment of
the couple about the
axis GH .
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 41/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticshon
Resolution of a Force into a Force at appoint + Couple
3 - 41
The couple of transfer is equal to the moment of the original force (acting at A)
about the transfer point B.
The couple of transfer: = ×
Demonstrates “Principle of Transmissibility”
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i o
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 42/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticshon
Resolution of a Force Into a Force at O and a Couple
3 - 42
• Force vector F can not be simply moved to O without modifying its
action on the body.
• Attaching equal and opposite force vectors at O produces no neteffect on the body.
• The three forces may be replaced by an equivalent force vector and
couple vector, i.e, a force-couple system.
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i o
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 43/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticshon
Example
2 - 43
The weights of the two traffic lights (shown above) can be
replaced by their equivalent force = + +
a couple moment = + at the support O.
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i o
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 44/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticshon
Example
2 - 44
(i) Find an equivalent force-couple
system with the force acting atpoint B.
(ii) Find an equivalent twohorizontal forces, one acting atpoint B and the other acting on
point C.
Ans (i) Ans (ii)
Work out the algebra and you should get the following:
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i o
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 45/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticshon
Resolution of a Force into a Force at O and a Couple
3 - 45
• Moving F from A to a different point O’ requires the
addition of a different couple vector M O’
F r M O
'
• The moments of F about O and O’ are related,
F s M
F s F r F sr F r M
O
O
''
• Moving the force-couple system from O to O’ requires the
addition of the moment of the force at O about O’ .
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i o
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 46/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticshon
Sample Problem 3.6
3 - 46
Determine the components of thesingle couple equivalent to the
couples shown.
SOLUTION:
• Attach equal and opposite 20 lb forces inthe + x direction at A, 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 point D is a
good choice as only two of the forces will
produce non-zero moment contributions..
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i o
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 47/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Staticson
Sample Problem 3.6
3 - 47
• Attach equal and opposite 20 lb forces
in the + x direction at A
• 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
k
ji M
in.lb180
in.lb240in.lb540
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i o
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 48/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
ecto ec a cs o g ee s Stat cson
Sample Problem 3.6
3 - 48
• Alternatively, compute the sum of the
moments of the four forces about D.
• Only the forces at C and E contribute to
the moment about D.
ik j
k j M M D
lb20in.12in.9
lb30in.18
k
ji M
in.lb180
in.lb240in.lb540
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i o
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 49/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
gon
System of Forces: Reduction to a Force and Couple
3 - 49
• 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,
F r M F R R
O
• The force-couple system at O may be moved to O’ with the addition of the moment of R about O’ ,
R s M M R
O
R
O
'
• Two systems of forces are equivalent if they can be
reduced to the same force-couple system.
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i o
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 50/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
gon
2 - 50
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i o
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 51/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
gon
2 - 51
Trick!!Chose a point O. Bring theequivalent of these forces at
O, i.e. . In addition to that there is a resultant moment
of couples, given by .
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i o
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 52/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
gn
2 - 52
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i on
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 53/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
gn
Reduction to a Wrench
2 - 53
Fig. (i)
Fig. (ii)
Fig. (iii)
In general, a 3D force + couple-moment system an
equiv.
acting at O +
.
These two are NOT necessarily perpendicular to each
other. Resolve into ∥ and ⊥. . ⊥ can be replaced if we move to point P , a distance
d from O, given by = ⊥ . ∥ being a free-vector, can be moved to point P , thereby
reducing it to (∥ + ) acting at point P . This combination of a
resultant + a collinear
couple moment ∥ will
tend to translate and rotate
the body about its axis and
is referred to as a
WRENCH or SCREW.
A wrench is the simplest system that can represent any general force and couple moment system
acting on a body.
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i on
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 54/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
gn
Sample Problem 3.8
3 - 54
For the beam, reduce the system of
forces shown to (a) an equivalent
force-couple system at A, (b) an
equivalent force couple system at B,
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 about A.
b) Find an equivalent force-couple
system at B based on the force-
couple system at A.
c) Determine the point of application
for the resultant force such that itsmoment about A is equal to the
resultant couple at A.
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i on
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 55/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
gn
Sample Problem 3.8
3 - 55
SOLUTION:
a) Compute the resultant force and theresultant couple at A.
j j j j
F R
N250 N100 N600 N150
j R
N600
ji
ji ji
F r M R A
2508.4
1008.26006.1
k M R A
m N1880
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i on
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 56/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
gn
Sample Problem 3.8
3 - 56
b) Find an equivalent force-couple system at B
based on the force-couple system at A.The force is unchanged by the movement of the
force-couple system from A to B.
j R
N600
The couple at B is equal to the moment about B
of the force-couple system found at A.
k k
jik
Rr M M A B
R
A
R
B
m N2880m N1880
N600m8.4m N1880
k M R B
m N1000
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i on
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 57/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
gn
Sample Problem 3.10
3 - 57
Three cables are attached to the bracket as shown. Replace the
forces with an equivalent force-
couple system at A.
SOLUTION:
• Determine the relative position vectors
for the points of application of the
cable forces with respect to A.
• Resolve the forces into rectangular
components.
• Compute the equivalent force,
F R
• Compute the equivalent couple,
F r M R A
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i on
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 58/59
© 2010 The McGraw-Hill Companies, Inc. All rights reserved.
gn
Sample Problem 3.10
3 - 58
SOLUTION:
• Determine the relative position
vectors with respect to A.
m100.0100.0
m050.0075.0
m050.0075.0
jir
k ir
k ir
A D
AC
A B
• Resolve the forces into rectangular
components.
N 200600300
289.0857.0429.0
175
5015075
N700
k ji F
k ji
k ji
r
r
F
B
B E
B E
B
N 1039600
30cos60cos N1200
ji
ji F D
N 707707
45cos45cos N1000
ji
ji F C
Vector Mechanics for Engineers: StaticsNi n t h
E d i t i on
7/21/2019 Ch03-Eqm of Particle and Moment of Force
http://slidepdf.com/reader/full/ch03-eqm-of-particle-and-moment-of-force 59/59
gn
Sample Problem 3.10• Compute the equivalent force,
k
j
i
F R
707200
1039600
600707300
N 5074391607 k ji R
• Compute the equivalent couple,
k
k ji
F r
jk ji
F r
k i
k ji
F r
F r M
D A D
c AC
B A B
R
A
9.163
01039600
0100.0100.0
68.17
7070707
050.00075.0
4530
200600300
050.00075.0
k ji M R A
9.11868.1730