CH2 Force Vectors

8/16/2019 CH2 Force Vectors

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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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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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= ()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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CH2 Force Vectors