Ch03-Eqm of Particle and Moment of Force

7/21/2019 Ch03-Eqm of Particle and Moment of Force

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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





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





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.





Ed i t i on

Procedure for Analysis- Co planar Forces

2 - 4

Ex.

NE





Ed i t i on

Procedure for Analysis- 3D Forces

2 - 5NE





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


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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




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




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




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





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





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





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.





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’





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



Vector Mechanics for Engineers: StaticsNin t h

Edi t i on


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



Vector Mechanics for Engineers: StaticsNin t h

Edi t i on


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



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



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


E d



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


E d




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




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


E d




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.


E d




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


E d





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


E d





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


E d i





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:


E d i





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,


E d i





t i on

Moment of a Force About a Given Axis (1)

3 - 29


E d i




Vector Mechanics for Engineers: Statics t h t i on


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

= + +


E d i t




Vector Mechanics for Engineers: Statics t h ti on


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




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


E d i t





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


E d i t





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


E d i t





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


E d i t




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.


E d i t




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.


E d i t i





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


E d i t i





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.


E d i t i




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 .


E d i t i





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”


E d i t i o





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.


E d i t i o





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.


E d i t i o





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:


E d i t i o





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’ .


E d i t i o





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..


E d i t i o




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


E d i t i o




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


E d i t i o




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.


E d i t i o




gon

2 - 50


E d i t i o




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 .


E d i t i o




gn

2 - 52


E d i t i on




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.


E d i t i on




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.


E d i t i on




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


E d i t i on




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


E d i t i on




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


E d i t i on




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


E d i t i on



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

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Ch03-Eqm of Particle and Moment of Force

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