EM Ch-4

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Mechanical Engineering Dept. CEME NUST 1

Ch-4: Force System Resultant

Book:

Engineering Mechanics Statics and Dynamics by R. C. Hibbeler

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Force System Resultant

Moment of a ForceScalar Formulation

When a force is applied to a body it will produce a tendency for the body to

Rotate About A Point that is not on the line of action of the force

Torque / Moment of a force / Moment.

o A force is applied to the handle of the wrench it will tend to

turn the bolt about point O (or thez-axis)

o Magnitude of the moment is directly proportional to the

magnitude ofFand the Perpendicular Distance orMoment

Arm d

o Larger the Force or the longer the Moment Arm, the

greater the Moment orTurning Effect

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Torque / Moment of a force / Moment..contd...

If the force Fis applied at an angle 90oit

will be more difficult to turn the bolt since the

moment arm d= dsinwill be smaller than d

IfFis applied along the wrench,its moment

arm will be zero since the line of action ofF

will intersect point O (the z axis themoment ofFabout Ois also zero and no

turning can occur

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Moment Moabout point O, or about an axis passing

through Oand perpendicular to the plane, is a Vector

Quantity since it has a specified Magnitude and

Direction

Magnitude Mois:

d=Moment arm orPerpendicular Distance from the

axis at point Oto the line of action of the force

Units of Moment: N.morlb.ft

Direction:

o Direction of Mo is defined by its Moment Axis

perpendicular to the plane that contains the force F

and its moment arm d

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Direction: Right Hand Rule

o Natural Curl of the fingers of the Right Hand, as drawn

towards the palm, represent the tendency for rotationcaused by the Moment Thumb of the Right Hand will

give the directional sense ofMo

o In 3D: Moment Vectoris represented by a Curl around an

arrow

o In 2D: Moment Vectoris represented only by a Curl

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Resultant Moment

o For 2D-problems, the resultant moment about point

O (the z axis) can be determined by finding the

algebraic sum of the moments caused by all the

forces in the systemo Counterclockwise are taken as Positive Moments

they are directed along the positive z-axis (out of the

page)

oClockwise are taken as Negative Moments

oResultant Moment is:

If the numerical result of this sum is a Posit ive Scalar, (MR)owill be a counterclockwise

moment (Out of the Page); and if the result is Negative, (MR)owill be a clockwise

moment (Into the Page)

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Example 4.1



For each case illustrated in Fig., determine the moment of the force about point O.

(a)

(c)

(d)

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Determine the resultant moment of the four forces acting on the rod shown in Fig.

about point O.

Example 4.2

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Ability to remove the nail will require the moment

of FH about point O>Moment of the force FNabout Othat is needed to pull the nail out

oMoment of a force does not always cause a rotation

oForce Ftends to rotate the beam clockwise about its support at Awith a

moment F.dA

oActual rotation would occur if the support at Bwere removed.

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Cross Product

Cross Product is used to find the moment of force

Cross Product of two vectors A and Byields the vectorC

Cequals A cross B

Direction Vector C has a direction perpendicular (or

Orthogonal) to the plane containing A and B, such that Cis

specified by the Right-hand Rule

o Right-hand Rule:Curling the fingers of the right hand

from vectorA (cross) to vectorB

Magnitude o fCis the product of the magnitudes ofA and B

and the sine of the angle between their tails :

ucdefines the direct ion o fC

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Cross Product

Laws of Operation

Commutat ive Lawis not valid:

Cross product BA yields a vector that has the same

magnitude but acts in the opposite direction to C

Associat ive Lawis valid:

Dist ribut ive Law of Ad di t ionis valid:

magnitude of the resultant |a |ABsinvector and its direction

are the same in each case

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Cross Product

Cartesian Vector Formulation

Cross Product of any pair ofCartesian Unit Vectors

=

resultant vector points in the +kdirection

o Crossing two unit vectors in a Counterc loc kwis e Fashionaround the circle yields the Posi t ive Third Uni tVector

o Crossing two unit vectors in a Clockw ise Fashionaroundthe circle yields the Negative Third Un itVector

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Cross Product

Cartesian Vector Formulationcontd--

Cross product of two general vectors A and B

To find the Cross Product of any two Cartesianvectors A and B, expand a determinant whosefirst row of elements consists of the unit vectors

i,j, and k and whose second and third rows

represent the x, y, z components of the two

vectors A and B, respectively

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Moment of a ForceVector Formulation

Moment of a force F about point O, can be expressedusing the vector cross product

rrepresents a position vector directed from Oto any point on the

line of action ofF

Magnitude

angle is measured between the tails ofrand F

Direction (Right-hand Rule):Curling the Right-hand Fingers from r toward F(r

cross F) the Thumb is directed upward or

perpendicular to the plane containing rand Fand

this is in the same direction as Mo

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Any position vectorrmeasured from point Oto any point on the line of action of

the force F



Principle of Transmissibility

Since Fcan be applied at any point alongits line of action and still create this samemoment about point O F can be

considered a Sliding Vector

This property is called the pr inc ip leof Transm issib i l i ty of a Force

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Cartesian Vector Formulation

If we establish x, y, zcoordinate axes, then the

position vectorrand force Fcan be expressed

as Cartesian Vectors

Represent x, y, zcomponents of the Posi t ionVectordrawn from

point Oto any point on the line of action of the forcerepresent the x, y, zcomponents of the Force Vecto r

Mowill always be perpendicular to the shaded plane containing vectors rand F

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Resultant Moment of a System of Forces

Resultant Moment of any number of

forces about point O can be

determined by vector addition of themoment of each force

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Example 4.3

Determine the moment produced by the

force F in Fig. aboutpoint O. Express the

result as a Cartesian vector.

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Principle of Moments

Principle of Moments----Verignons Theorem

Moment of a Force about a point is equal to the sum of the

moments of the components of the force about the point

oconsider the Moments of the Force F and two of its

components about point O

Since,

For2D problems, Pr incip le of Momentscan

be used by resolving the force into its

Rectangular Components and then

determine the moment using a scalar

analysis

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Principle of Moments

Moment of the applied force F

about point Ois easy to determine

if we use the principle of moments

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Example 4.4

Determine the moment of the force in Fig. about point O.

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Example 4.5

Determine the angle

(0o

180

o

) of the force Fso that it produces a maximummoment and a minimum moment about point A . Also, what are the magnitudes of

these maximum and minimum moments?

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Serious neck injuries can occur

when a football player is struck in

the face guard of his helmet in

the manner shown. Determine the

moment of the knee force P = 50

lbabout point A. What would be

the magnitude of the neck force F

so that it gives the

counterbalancing moment about

A?


Example 4.6

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Example 4.5

In order to raise the lamp post from the position

shown, the force F on the cable must create a

counterclockwise moment of 1500 lb.ft about point

A. Determine themagnitude of F that must be

applied to the cable.

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Example 4.7

Determine the resultant moment

produced by forces FB and FC

about point O. Express the result

as a Cartesian Vector.

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Moment of a Couple

Coupleis defined as two parallel forces that havethe same magnitude, but opposite directions, and

are separated by a perpendicular distance d

Since the Resultant Force is zero, the only effect of a

couple is to produce a rotation ortendency of rotation

in a specified direction E.g, Steering Wheel of a Car

Couple Moment: Moment produced

by a couple is called a Couple

Moment

F-F

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Moment of a Couple

Couple Moment determined about O

Couple Moment is a free vector, i.e., it can act at

any point since Mdepends only upon the positionvectorrdirected between the forces and not the

position vectors rA and rB directed from the

arbitrary point Oto the forces

Value of Couple Moment can be determinedby finding the sum of the moments of both

couple forces about any arbitrary point

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Moment of a Couple

Moment of a couple, M is defined as having a magni tude of

Scalar Formulation

F= Magnitude of one of the forces

d=Perpendicular Distance orMoment Arm between the

forces

Direction and Sense of the couple moment are

determined by the Right-hand Rule

oThumb indicates this direction when the Fingers are curled with the

sense of rotation caused by the Couple Forces

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Moment of a Couple

Vector Formulation

Moment of a Couple can also be expressed by the Vector Cross Product

If Moments are taken about point A, Moment of

Fis zero about this point, and the moment ofFis

defined as M = rF

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Moment of a Couple

Equivalent Couples

If two couples produce a moment with the same magnitude and direction, then

these two couples are Equivalent

oLarger forces are required in the second case to create the same turning effect

because the hands are placed closer together

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Moment of a Couple

Resultant Couple MomentMoments are vectors, their resultant can be determined by Vector Addition

Consider the couple moments M1and M2acting on the pipe

Since each couple moment is a Free Vecto r, their

tails cab be joined at any arbitrary point to find

the resultant couple moment

For more than two couple moments, generalized

form of the vector resultant is:

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Example 4.8

Determine the resultant couple moment of the three couples acting on the plate inFig.

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Example 4.9

Determine the couple moment acting on the pipe shown in Fig. 432. Segment

AB is directed 30 below the xy plane.

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Replace the two couples acting on the pipe column in Fig. by a resultant couplemoment.


Example 4.10

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Example 4.10

IfF = 200 lb, determine the resultant couple

moment.

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If M1 = 180 ld.ft, M2 = 90

lb.ft, and M3 = 120 lb.ft,

determine the magnitude

and coordinate direction

angles of the resultant

couple moment.


Example 4.11

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Simplification of a Force and Couple System

Sometimes it is convenient to reduce a system of forces and couple momentsacting on a body to a simpler form by replacing it with an Equiv alent System

consisting of a Single Resultant Force acting at a specific point and a Resultant

Couple Moment

Equiv alent System:A system is Equivalent if the External Effects it produces on

a body are the same as those caused by the original force and couple moment

system

External Effects o f a sys temrefer to the Translating and Rotating motion of the

body if the body is free to move, or it refers to the Reactive Forces at the

supports if the body is held fixed

Force F is moved from A to B without

modifying its external effects on the stick; i.e.,

reaction at the grip remains same

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Force can be moved to a point that is not on the line of action of the force

oFis applied perpendicular to the stick,

o then we can attach a pair of equal but opposite

forces FandF

oForce Fis now applied at B, and the other two

forces, Fat A and F at B, form a Couple thatproduces the Couple Moment

force Fcan be moved from A toBprov ided a

couple moment M is added to maintain an

Equivalent Systemo In both cases the systems are equivalent which

causes a Downward Force F and Clockwise

Couple MomentM= Fdto be felt at the grip

B

B

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System of Forces and Couple Moments

Ois not on the line of action ofF1, and so this force

can be moved to point Oprovided a couple moment

M1= r1 F1is added to the body

=

Similarly, M2= r

2 F

2couple moment should be added to

the body when we move F2to pointO

Since the Couple Moment M is a free

vector, it can just be moved to point O

Equivalent System is now obtained,

which produces the same external

effects on the body as that of the

original Force and Couple System

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By summing the forces and couple moments:

Resultant Force = FR= F1+ F2

Resultant Couple Moment = (MR)O= M + M1+ M2

In general, the method of reducing a Force and

Couple System to an Equivalent System isrepresented using following Equations:

Resultant Force FR of the system isequivalent to the sum of all the forces

Resultant Couple Moment (MR)oof the

system is equivalent to the sum of all the

couple moments Mplus the moments ofall the forces MO

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Weights of these traffic lights can be replaced by their equivalent resultant force:WR= W1+ W2, and a couple moment (MR)O= W1d1+ W2d2at the support, O

In both cases the Support must provide the same resistance to Translation and

Rotation in order to keep the member in the Horizontal Position

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Replace the force and couple system acting on the member in Fig. by anequivalent resultant force and couple moment acting at point O.


Example 4.12

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Example 4.13

The structural member is subjected to a couple moment Mand forces F1and F2

and shown in Fig. Replace this system by an equivalent resultant force andcouple moment acting at its base, point O.

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Example 4.13

The slab is to be hoisted using the three slings shown. Replace the system of

forces acting on slings by an equivalent force and couple moment at point O. The

forceF1is vertical.

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