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ENGINEERING STATICSCHAPTER 2: FORCE VECTORS
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Scalars and Vectors
• A scalar is any positive or negative physical quantity that can
completely specified by its magnitude. Examples: length, mtime.
• A vector is any physical quantity that requires both a magnidirection for its complete description. Examples: force, posi
moment.
Fig. 2.1 (Hib
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Vector Operations
• Multiplication/Division by a scalar
Multiplication by a positive scalar increase its magnitude by tof the scalar. Multiplication by a negative scalar will also chandirectional sense of the vector.
Figs. 2-2 and 2-3 (Hibbeler)
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Vector Operations
• Addition of Vectors (Parallelogram law )
The parallelogram law of addition can be used. To add two veand B:
1. Join the tails of the two vectors at a point. (This makes thconcurrent )
2. From the head of B, draw a line that is parallel to A. Draw
line from the head of A that is parallel to B. These line intepoint P to form the adjacent sides of a parallelogram.
3. The diagonal of this parallelogram, from the tails of the veis the resultant vector R.
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Vector Operations
• Addition of Vectors (Triangle rule)
The two vectors A and B can also be added using the triangleadd B to A:
1. Connect the head of A to the tail of B.
2. The resultant R extends from the tail of A to the head of B
In a similar manner, R can also be obtained by adding A to B.
Vector addition is commutative: R = A + B = B + A
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Vector Addition
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Vector Addition of Forces
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Resultant of two components forces F1 and F2 acting on a
Fig. 2-7 (Hibbeler)
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Math Review
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Finding the Components of a Force in Two Specific Dir
1. Set up an axes system, u and v , in the two directions of intere
2. Draw a line from the tip of F parallel to the u-axis until it interaxis.
3. Draw another line from the tip of F parallel to the v -axis until
the u-axes.
4. The force components Fu is obtained by joining the tail of F to
intersection with the u-axes.
5. The force component Fv is obtained by joining the tail of F to t
intersection with the v -axis.
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Finding the Components of a Forc
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Addition of More Than Two Force
FR = (F1 + F2) + F3
Fig. 2-9 (Hibbeler)
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Addition of a System of Coplanar For
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Rectangular Components: The two components of a force that is resolved
components along the x and y axes of a Cartesian coordinate system.
Two ways to represent these components: Scalar notation or Cartesian ve
Scalar Notation
F = F x + Fy
F x = F cos θ F y = F si
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Scalar Notation (Contd.)
=
or =
=
or =
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Cartesian Vector Notation:
Define Unit Vectors i and j, which have dimensionless magnitude of 1. Th
vectors are used to designate the directions of the x and y axes, respectiv
The Cartesian vector is:
F = F x i + F y j
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Resultant of a System of Coplanar Fo
Using Cartesian vector notati
F1 = F 1 x i + F 1y j
F2 = -F 2 x i + F 2y j
F3 = F 3 x i - F 3y j
FR = F1 + F2 + F3
= F 1 x i + F 1y j - F 2 x i + F 2y j + F 3 x
= (F 1 x – F 2 x + F 3 x )i + (F 1y + F 2
= (F Rx )i + (F Ry ) j
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Resultant of a System of Coplanar Fo
Using scalar notation:
In general, the components of the reany number of coplanar forces, can b
as the algebraic sum of the x and y c
the forces:
(F Rx ) = ΣF x (F Ry ) = ΣF y
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Resultant of a System of Coplanar Fo
= ()2 +
= tan− (
(
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THREE-DIMENSIONAL VECTOR ANAL
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Coordinate System and Unit Vectors
• A right-handed rectangular coordinate system is used. The system is right-handed if the thumb of the right hand points
direction of the positive z-axis when the right-hand fingers aabout this axis from the positive x -axis towards the positive
• The vectors are first represented in Cartesian vector form. Tvectors, i, j, k are used to designate the directions of the x , y
respectively.
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Components of a Vector, A
Using the parallelogram law t
A = Aʹ + Az
Aʹ = A x + Ay
Combining the above equatio
A = A x + Ay + Az
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Cartesian Vector Representation
A = A x i + Ay j + Azk
Note: Separating the magnand direction of each compovector , as done above, willsimplify the operations of vealgebra, particularly in threedimensions.
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Magnitude of a Cartesian Vector
= ʹ2 + 2
ʹ = 2 +
2
Combining the above equat
= 2 +
2 +
Note: The magnitude of A isthe positive square root of th
the squares of its componen
f
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Direction of a Cartesian Vector
Define the direction of A by coordinate direction angle αmeasured between the tail othe positive x , y and z axes, pthey are located at the tail o
Direction Cosines of A
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Direction Cosines of A
Considering the projection of A unto the x , y , z axes, give:
cos =
cos =
cos γ =
These numbers are called the direction cosines of A.
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Easy way to obtain the direction cosi
1. Define a unit vector u A in the dire
2. Since A = A x i + Ay j + Azk, then u A wmagnitude of one and be dimensdivided by its magnitude.
3. The above i, j, k components of udirection cosines of A, hence:
4 Si h i d f i l h i i
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4. Since the magnitude of a vector is equal to the positive sqof the squares of the magnitude of its component, and u Amagnitude of one, then:
5. If the magnitude and coordinate angles of A are known, thbe expressed in Cartesian vector form as:
Additi f T M C t i V t
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Addition of Two or More Cartesian Vect
In general:
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POSITION VECTORS
Position Vectors
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Position Vectors
• Important in formulating Cartvector directed between two space.
• Right-handed coordinate systewith the z axis directed upwarzenith direction).
• Points in space are located relorigins of coordinates, O, by smeasurements along the x , y ,Example, position of B is (6 m
Position Vectors
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Position Vectors
A position vector r is a fixed vector which locates a point in spaceanother point.
If r extends from the origins, O, to point P( x , y , z) then r can be gCartesian vector form as:
r = x i + y j + zk
Position Vectors
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Position Vectors
In general, r can be directed fro( x A, y A, z A) to a point B( x B, y B, zB)are the position vectors of poinrespectively, from the origin of then:
Position Vectors
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Position Vectors
Force Vector Directed Along a Line
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Force Vector Directed Along a Line
In three-dimensional static problems,of a force F is often specified by two pand B), through which its line of actioFig.).
Note that F has the same direction anthe position vector r directed from poB. The common vector is specified byvector, u = r/r. Hence, we can formulaCartesian vector, as follows:
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Dot Product of Two Vectors
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Dot Product
The dot product between two vectors can be used to find the anbetween two line or the components of a force parallel to or peto a line.
The dot product of two vectors A and B, is
written as A·B, read as “A dot B”, and defined as:
A·B = AB cos θ
The dot product is also referred to as the scalar
product since the results is a scalar quantity.
0° ≤ ≤ 180°
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Dot Product Laws of Operations
Dot Product of Any Two Cartesian Unit Ve
i · i = (1)(1) cos 0° = 1
i · j = (1)(1) cos 90° = 0
Etc.
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Cartesian Vector Formulation of Dot Prod
The dot product of two general vectors A and B in Cartesian vform is:
This reduces to:
So, to determine the dot product of two Cartesian vectors, mucorresponding x, y, z components and sum these products alg
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Applications of Dot Product
1. To find the angle θ between the tails of two vectors or intelines:
= cos− ·
= cos−
+ +
Note that if A · B = 0, then θ = 90° so that A is perpendicular t
(Since then, θ = cos-1 0 = 90°)
0° ≤ ≤ 180°
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Applications of Dot Product
2. To find the components of a vector parallel and perpendicline:
The scalar projection of A along a line aa
is determined from the product of A and
the unit vector ua which defines the
direction of the line.
The component of A that is perpendicular to the line aa is giv
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