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Transcript of 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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