1. Force Vector

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Engineering Mechanics:

Statics


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Course Outline

X.Moment of inertia

IX.Center of gravity and centroidVII. Friction

VI. Internal forces

V.Equilibrium of a rigid body

II.Force vectors

IV.Force system resultants

III.Equilibrium of a particle

I.Introduction


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Mechanics can be defined as that branch of the physical sciences

concerned with the state of rest or motion of bodies that are subjected to

the action of forces. In general, this subject is subdivided into two

branches: rigid-body mechanics and fluid mechanics. Rigid-body

mechanics will be the main target for this course of Mechanics for Civil

Engineering.

Introduction

Newton's Three Laws of Motion: The entire subject of rigid-body

mechanics is formulated on the basis of Newton three laws of motion,

the validity of which is based on experimental observation. They apply tothe motion of a particle as measured from a non-accelerating reference

frame. with reference to Fig. 1, they may be briefly stated as follows:


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First Law: A particle original at rest, or

moving in a straight line with constant

velocity, will remain in this state provided the

particle is not, subjected to an unbalanced

force.

Second Law: A particle acted upon by anunbalanced force F experiences an

accelerationa that has the same direction as

the force and a magnitude that is directly

proportional to the force. IfFis applied to a

particle of mass m, this law may be expressed

mathematically as F=ma.

Third Law: The mutual forces of action and reaction between two

particles are equal, opposite, and collinear.


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2 2

11 3 2

24

2411

23

2

9.81 m/s 10 m/sNewton's law of universal gravitation:

6.673 10 m / kg.s

5.976 10 kg

6371 km

5.976 106.673 10 9.81

6371 10

where

Unit:

similar to

2

9.81m/s

g

G

M

R

mF m mg

g

W mg F ma

mMF GR

3 2 222

kg kgN m / kg.s kg.m/s

m

(N) (kg) (m/s )F m g


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Example 1:


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Basic conceptsThe following concepts and definitions are basic to the study of mechanics, and they

should be understood at the outset.

Spaceis the geometric region occupied by bodies whose positions are described by

linear and angular measurements relative to a coordinate system. For three-

dimensional problems, three independent coordinates are needed. For two-dimensional problems, only two coordinates are required.

Timeis the measure of the succession of events and is a basic quantity in dynamics.

Time is not directly involved in the analysis of statics problems.

Mass is a measure of the inertia of a body. which is its resistance to a change of

velocity. Mass can also be thought of us the quantity of matter in a body. The mass of

a body affects the gravitational attraction force between it and other bodies. This force

appears in many applications in statics.

Force is the action of one body on another. A force tends to move a body in the

direction of its action. The action of a force is characterized by its magnitude, by the

direction of its action, and by its point of application. Thus force is a vector quantity,and its properties are discussed.


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A particle is a body of negligible dimensions. In the mathematical

sense, a particle is a body whose dimensions are considered to be near

zero so that we may analyze it as a mass concentrated at a point. Weoften choose a particle as a differential element of a body. We may treat

a body as a particle when its dimensions are irrelevant to the description

of its position or the action of forces applied to it.

Rigid body. A body is considered rigid when the change in distance

between any two of its points is negligible for the purpose at hand. For

instance, the calculation of the tension in the cable which supports theboom of a mobile crane under load is essentially unaffected by the

small internal deformations in the structural members of the boom. For

the purpose, then, of determining the external forces which act on the

boom, we may treat it as a rigid body. Statics deals primarily with thecalculation of external forces which act on rigid bodies in equilibrium.

Determination of the internal deformations belongs to the study of the

mechanics of deformable bodies, which normally follows statics in the

curriculum.


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Most of the physical quantities in mechanics can be expressed

mathematically by means of scalars and vectors:

Scalar: A quantity characterized by a positive or negative number is

called a scalar. For example, mass, volume, and length are scalar

quantities often used in statics. In this course, scalars are indicated byletters in italic type, such as the scalarA.

Vector:A vector is a quantity that has both a magnitude and a direction.

In statics the vector quantities frequently encountered are position, force,

and moment.


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For handwritten work, a vector is generally represented by a letter with arrow

written over it, such as . The magnitude is designated as simply .

In this

A

c

or A

ou

or

rse vectors will be

I. Vec

A

tor :

symbolized in type; for example, is used

to designate the vector.

boldface A


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Example 2:The screw is subjected to two forces,F1andF2. Determine

the magnitude and direction of the resultant force?


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Example 3:Resolve the 1000-N force acting on the pipe in Fig. below

into components in the

(a) xandydirections?(b)x'andydirections?


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Example 4: The force F acting on the frame shown in Fig. below a

magnitude of 500 N and is to be resolved into two components acting

along members ABand AC. Determine the angle , measured below thehorizontal, so that the componentFACis directed from Atoward Cand

has a magnitude of 400 N.


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III. Cartesian vectors:

Right-hand rule

Unit vector:The direction ofAcan be specified using a unit vector. This vector is so

named since it has a magnitude of 1. IfAis a vector having a magnitudeA# 0, then the

unit vector having the same direction asAis represented by:

uA=A/A so that

A=AuA


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Unit Cartesian vectors:

SinceuAhas a magnitude of 1, then from equation above an importantrelation between the direction cosines can be formulated as:

Finally, if the magnitude and coordinate direction angles ofAare given.

Amay be expressed in Cartesian vector form as:


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Example 8:ExpressFas a Cartesian vector?


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Force Vector Directed along a Line


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Force Vector Directed along a Line


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1. Force Vector