### Chapter Four

Force System ResultantsAsst.Lecturer :

M ohanad N ihad

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Introduction

➢ 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 .

This tendency to rotate is sometimes called a torque , but most often it

is called the moment of a force or simply the moment .

➢ The magnitude of the moment is directly proportional to the magnitude

of F and the perpendicular distance or moment arm d .

➢ The larger the force or the longer the moment arm, the greater the

moment or turning effect . Note that if the force F is applied at an angle

Ɵ ≠ 90 ° , then it will be more difficult to turn the bolt since the moment

arm d'= d sin Ɵ will be smaller than d .

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where d is the moment

arm or perpendicular

distance from the axis

at point O to the line of

action of the force .

�� �� = F (d ����� Ɵ )

where d �� ���� Ɵ is the moment

arm or perpendicular

distance from the axis at point

O to the line of action of the

force .

�� �� = 0

If F is applied along the

wrench, its moment arm

will be zero since the line

of action of F will intersect

point O (the z axis) . Where

Ɵ = 0 .

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Moment of Force

1 - Scalar Formulation

The moment is a vector quantity since it has a specified magnitude

and direction .

➢ Magnitude:

➢ Direction:

The direction of �� �� is defined by its moment axis, which is

perpendicular to the plane that contains the force F and its moment

arm d . The right -hand rule is used to establish the sense of

direction of �� �� . According to this rule, the natural curl of the

fingers of the right hand, as they are drawn towards the palm,

represent the rotation, or if no movement is possible, there is a

tendency for rotation caused by the moment .

Units of moment is N.m or lb .ft.

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➢ Resultant Moment

For two -dimensional problems, where all the

forces lie within the x – y plane, 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 system .

As a convention,

1 - If ( �� �� ) �� positive, its direction will be

counterclockwise .

2 - If ( �� �� ) �� negative, its direction will be

Clockwise .

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Cross Product

The moment of a force will be formulated using Cartesian vectors in the next section .

Before doing this, however, it is first necessary to expand our knowledge of vector

algebra and introduce the cross -product method of vector multiplication .

➢ The cross product of two vectors A and B yields the vector C , which is written :

➢ Magnitude :

The magnitude of C is defined as the product of the magnitudes of A and B and the sine

of the angle Ɵ between their tails (0 ≤ Ɵ ≤ 180 °). Thus, C = AB sin Ɵ .

➢ Direction :

Vector C has a direction that is perpendicular to the plane containing A and B such that

C is specified by the right -hand rule .

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Curling the fingers of the right hand from vector A (cross) to vector

B , the thumb points in the direction of C , as shown in Fig.

Knowing both the magnitude and direction of C , we can write:

where the scalar AB sin Ɵ defines the magnitude of C and the unit

vector �� �� defines the direction of C .

Laws of Operation:

• The commutative law is not valid; i.e., A × B ≠ B × A Rather,

A × B= - B × A

by using the right -hand rule. The cross product B × A yields a vector

that has the same magnitude but acts in the opposite direction to C ;

i.e., B × A = -C.

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• If the cross product is multiplied by a scalar a , it obeys the

associative law;

• The vector cross product also obeys the distributive law of

addition,

Cartesian Vector Formulation:

Before we start, we have to mention that:

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Let us now consider the cross product of two general vectors A and B which are expressed

in Cartesian vector form. We have

Carrying out the cross -product operations and combining terms yields:

There is another way to solve Cartesian Vector form. We can use the matrix then find a

determinant.

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This equation may also be written in a more compact determinant form as:

It is necessary to expand a determinant whose first 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 .

➢ A determinant having three rows and three columns can be expanded using three minors, each of which is

multiplied by one of the three terms in the first row. There are four elements in each minor, for example,

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By definition , this determinant notation represents the terms (�� 11 �� 22 - �� 12 �� 21 ), which is simply

the product of the two elements intersected by the arrow slanting downward to the right (�� 11 �� 22 )

minus the product of the two elements intersected by the arrow slanting downward to the left

(�� 12 �� 21 ).

Adding the results and noting that the j element must include the minus sign yields the expanded form of A ×

B .

For correct signs

+ − +

− + −

+ − +

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2 - Vector Formulation

The moment of a force F about point O , or actually about the

moment axis passing through O and perpendicular to the plane

containing O and F , �� �� = r × ��

r represents a position vector directed from O to any point on

the line of action of F .

➢ Magnitude :

The magnitude of the cross product is defined from

�� �� = �� �� ������ �� , where the angle �� is measured between

The tails of r and F .

To establish this angle, r must be treated as a sliding vector so that �� can be

constructed properly .Since the moment arm d = �� ������ �� ,then

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➢ Direction:

The direction and sense of �� �� in are determined by the right -hand

rule . Thus, sliding r to the dashed position and curling the right -

hand fingers from r toward F , “r cross F ,” the thumb is directed

upward or perpendicular to the plane containing r and F and this is

in the same direction as MO, the moment of the force about point

O . Note that the “curl ” of the fingers, like the curl around the

moment vector, indicates the sense of rotation caused by the force .

Principle of Transmissibility

We can use any position vector r measured from point O to any

point on the line of action of the force F because of the cross

product operation is often since the perpendicular distance or

moment arm from point O to the line of action of the force is not

needed .

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Cartesian Vector Formulation

1. Establish x, y, z coordinate axes, then we can express r (position

vector) in i, j, k , so, r= �� � � + �� � � + �� � � as in fig b .

2. Represent F force as a Cartesian vector in terms of i, j, k.

F= �� � � + �� � � + �� �� as in fig b .

3. Then apply Cross product to find the moment and sort the vectors in

a determinate matrix:

4. Solve the determinate to get:

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Resultant Moment of a System of Forces:

If a body is acted upon by a system of forces, the resultant

moment of the forces about point O can be determined by vector

addition of the moment of each force . This resultant can be

written symbolically as :

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The moment of a force about a point is equal to the sum of

the moments of the components of the force about the point.

Because the resultant of forces is zero.

For two -dimensional problems as in Fig, we can use the

principle of moments by resolving the force into its

rectangular components and then determine the moment

using a scalar analysis . Thus,

This method is generally easier than finding the same

moment using �� �� = Fd .

Principle of Moments

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Moment of a Couple

A couple is defined as two parallel forces that have the same magnitude, but opposite directions,

and are separated by a perpendicular distance d as shown in Fig .

➢ Since the resultant force is zero, the only effect of a couple is to produce an

actual rotation.

➢ The moment produced by a couple is called a couple moment . We can

determine its value by finding the sum of the moments of both couple

forces about any arbitrary point .

➢ However �� � = �� � + r or r = �� � - �� � ,so that

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This result indicates that a couple moment is a free vector , i.e ., it can act at any point since M

depends only upon the position vector r directed between the forces and not the position vectors

�� and �� .

Moment of a Couple Formulation

Scalar Formulation Vector Formulation

We can utilize the cross

product to find the

moment of couple

Magnitude Direction

F is the magnitude of

one of the forces .

d is the perpendicular

distance or moment arm

between the forces .

Using the right -hand rule,

where fingers are curled with

the sense of rotation caused by

the couple forces . In all cases,

M will act perpendicular to

the plane containing these

forces .

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Equivalent Couples:

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

couples are equivalent . For example, the two couples shown in Fig . below are equivalent

because each couple moment has a magnitude of M = 30 N( 0 .4 m) = 40 N( 0 .3 m) = 12 N .

m , and each is directed into the plane of the page .

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Resultant Couple Moment:

Since couple moments are vectors, their resultant can be

determined by vector addition.

For example, consider the couple moments �� 1 and �� 2 acting

on the pipe in Fig .a .Since each couple moment is a free

vector, we can join their tails at any arbitrary point and find the

resultant couple moment, �� �� = �� 1 + �� 2 as shown in Fig .b .

If more than two couple moments act on the body, we may

generalize this concept and write the vector resultant as

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Sometimes it is convenient to reduce a system of forces and couple moments acting on a

body to a simpler form by replacing it with an equivalent system , consisting of a single

resultant force acting at a specific point and a resultant couple moment . force acting on a

body (stick) is a sliding vector since it can be applied at any point along its line of action .

Simplification of a Force and Couple System

Consider holding the stick in

which is subjected to the

force F at point A .

If we attach a pair of equal

but opposite forces F and -F

at point B , which is on the

line of action of F .

We observe that -F at B and

F at A will cancel each other,

leaving only F at B . Force F

has now been moved from A

to B without modifying its

external effects on the stick .

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We can also use the above procedure to move a force to a point that is not on the line of action of

the force.

Consider holding the stick in

which is subjected to the

force F at point A .

If we attach a pair of equal

but opposite forces F and -F

at point B , which is on the

line of action of F .

Force F is now applied at B ,

and the other two forces, F at

A and -F at B , form a couple

that produces the couple

moment M = Fd . Therefore,

the force F can be moved

from A to B provided a

couple moment M is added

to maintain an equivalent

system .

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A system of several forces and couple moments acting on a body can be reduced

to an equivalent single resultant force acting at a point O and a resultant couple

moment .

System of Forces and Couple Moments