PE2113-Chapter 2 - Force Vectors_082715

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Chapter 2 – Force Vectors

Scalars and Vectors

Procedures for Analysis(Fall 2015)

Instructor: Dr. Ben Shiau

Associate ProfessorMewbourne School of Petroleum & Geological

Engineering

(Tel: 5-6817, SEC 1346, [email protected])

mailto:[email protected]

mailto:[email protected]



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Objectives

• To show how to add forces and resolve them intocomponents using the Parallelogram Law

• To express force and position in Cartesian Vector

form and explain how to determine the vector’s

magnitude and direction

• To introduce the DOT PRODUCT in order to

determine the angle between vectors or theprojection of one vector onto another.



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2.1 Scalars and Vectors

• Scalar is a quantity characterized by a positiveor negative number. That is, it has magnitude

but no direction. Generally written in italic ( not

boldface ). – Examples are mass, volume, length, speed, and

time

• Vector has magnitude as well as direction.Written with an arrow on top (or boldface ).

– Examples are displacement, acceleration,

weight, force, and velocity



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2.2 Vector Operations• Multiplication and Division of a Vector by a Scalar

– The product or division of a vector (by a scalar) is also a vector. However,

the magnitude will change and the sense will change if the scalar isnegative.

– A special case: if the vectors are collinear (that is, on the same straight

line), the resultant is formed by algebraic or scalar addition.

• Vector Addition

– The addition of vectors A and B results in a resultant R = A + B. This can

be achieved using the Parallelogram Law.

• Vector Subtraction

– The resultant difference between vectors A and B results in a resultant R’

= A – B.• Resolution of Vector

– A vector may be resolved into two components having known lines of

action by using the parallelogram law .



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2.3 Vector Addition of Forces

• Since force is a vector , it adds according to the

parallelogram law.• Two common problems in statics involve either finding

resultant of force, knowing its components or resolving a

known force into two components.

• If more than two forces are to be added, successive

application of the parallelogram law is carried out in order

to obtain the resultant force.

– Using the parallelogram law requires extensive geometricand trigonometric calculations (basically sine, cosine, and

Pythagoras laws). Instead problems of this type are easily

(best) solved by using the “rectangular -component

method.”



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Procedure for Analysis of Vector Addition of Two Forces

Problems that involve the addition of two forces can be solved as follows:

Parallelogram Law

• Make a sketch placing the vectors together at their tails and then showing thevector addition using the parallelogram law.

• Two “component” forces add according to the parallelogram law, yielding a result

force that forms the diagonal of the parallelogram

• If a force is to be resolved into components along 2 axes directed from the tail of

the force, then start at the head of the force and construct lines parallel to theaxes, thereby forming the parallelogram. The sides of the parallelogram represent

the components.

• Label all the known and unknown force magnitudes and the angles on the sketch

and identify the two unknowns.

• Trigonometry• Redraw a half portion of the // to illustrate the – head-to-tail addition of the

components

• The magnitude of the resultant force can be determined from the law of cosines,

and its direction is determined from the sine law.

• The magnitude of the two force components are determined from sine law.



Ch 2.3 EXAMPLE 2.2



EXAMPLE 2.2 (continued)




Class Exercise: Fundamental Problems* F2-4. (p. 27)



Ch. 2.4 Addition of a System of Coplanar Forces

EXAMPLE 2.6




Ch 2 4 Addi i f S f C l F



Ch. 2.4 Addition of a System of Coplanar Forces

(Cont.)

EXAMPLE 2.7




Class Exercise: Problem 2-50



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2.5 Cartesian Vector Notation

• The operation of vector algebra, when applied to

solving problems in 3-D, are greatly simplified if thevectors are first represented in Cartesian vector form.

• Right-Handed Coordinate System: The RHR is used

to develop the theory of vector algebra as follows: – A rectangular or Cartesian coordinate system is said to be

R-H provided the thumb of the right hand points in the

direction of the positive (+) z axis when the right handfingers are curled about this axis and directed from the +ve

x towards the +ve y axis. According to this rule, the z-axis

for a 2-D problem would be directed outward ,

perpendicular to the page.



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Cartesian Vector Notation

• Rectangular Components of a Vector• A vector A may have 1, 2, or 3 rectangular

components along the x, y, and z coordinate

axes, depending on how the vector is orientedrelative to the axes. In general, though, when A

is directed within an octant of the x, y, z frames,

it can be represented by the sum of its 3rectangular components as follows:

A = A x i + Ay j + Az k



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Cartesian Vector Notation • Cartesian Unit Vectors: In 3-D, the set of Cartesian unit vectors, i, j, and

k is used to designate the directions of the x, y, and z axes respectively.• Cartesian Vector Representation: A = A x i + Ay j + Az k

Magnitude of a Cartesian Vector

Direction of Cartesian Vector

(see Fig. 2-26, Hibbeler for α, β, & γ)

• Since the magnitude of a vector is equal to the +ve square root of the

sum of the squares of the magnitudes of its components, and u A has amagnitude of 1, then an important relation between the direction

cosines can be derived as follows:

cos2 + cos2 +cos2 = 1

Question: Show the relationship cos2 + cos2 +cos2 = 1

222

Z Y X A A A A

A

A

A

A

A

A Z Y X cos;cos;cos



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Cartesian Vector Notation • An easy way of obtaining the direction cosines is to form a unit vector

u A in the direction of A. If A is expressed in Cartesian vector form,A = A x i + Ay j + Az k , then u A will have a magnitude of one and be

dimensionless provided A is divided by its magnitude, i.e.

u A = A/A = (Ax/A)i + (Ay/A)j + (Az/A)k

• Thus, u A = cos i + cos j +cos k

• If the magnitude and coordinate direction angles of A are known, then

A may be expressed in Cartesian vector form as

A = Au A

A = Acos i + Acos j +Acos k

A = Axi + Ay j + Azk A

A

A

A

A

A Z Y X cos;cos;cos



Ch. 2.6 Addition of Cartesian Vectors

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EXAMPLE 2.8



EXAMPLE 2.9




Class Exercise: F2-17



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**Ch. 2.7 & 2.8 Position and Force Vectors

• A position vector r is defined as a fixed vector which locates

a point in space relative to another point.• The easiest way to formulate the components of a position

vector is to determine the distance and direction that one

must travel along the x, y, and z directions – going from the

tail to the head of the vector. That is

– r = (xB – x A)i + (yB – y A) j + (zB – z A)k

• Thus, the i, j, k components of the position vector r may be

formed by taking the coordinates of the tail of the vector A

(x A, y A, z A) and subtracting them from the corresponding

coordinates of the head B (xB, yB, zB)



EXAMPLE 2.12



EXAMPLE 2.12

(continued)



29Exercise: Problem 2-99



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Ch 2.9 Dot Product• In two dimensions, it is easy to find the angle between two

lines or the components of a force parallel or perpendicularto a line. This is by using trigonometry.

• In 3-D, however, this is often difficult, and hence vector

methods is employed to solve the problem. The dot product

defines a particular method for “multiplying” two vectors andis used to solve the 3-D problems.

• The DOT product of vector A and B, written as A•B, and

reads “A dot B” is defined as the product of the magnitudesof A and B and the cosine of the angle θ between their tails.

• A•B = ABcosθ (0o θ 180o), and it is a scalar . Hence, the

dot product is also called a scalar product.



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Laws of Operation of a DOT Product

1. Commutative law: A•B = B•A 2. Multiplication law: a(A•B) = (a A)•B = A•(aB) = (A•B)a

3. Distributive law: A•(B + D) = (A•B) + (A•D)

Cartesian Vector Formulation

• The dot product can be used to find the dot product for

each of the Cartesian unit vectors

i•i = 1 j•j = 1 k•k = 1

i•j = 0 i•k = 0 k•j = 0



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Dot Product of Two General Vectors A and B

• If A = A x i + Ay j + Az k and B = B x i + By j + Bz k ,then

A•B = A x B x + Ay By + Az Bz

• Thus, to determine the dot product of twoCartesian vectors, multiply their corresponding x,

y, z components and sum their products

algebraically.• Note: Since the result is a scalar, we must be

careful not to include any unit vectors in the final

result.



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Applications of Dot Product

• The dot product has two important applications inmechanics:

To determine the angle formed between two vectors

or intersecting lines (i.e. the angle between the tails of

vector A and B)

θ = cos-1(A•B/AB) 0o ≤ θ ≤ 180o

To determine the components of a vector parallel and

perpendicular to a line



EXAMPLE 2.16



EXAMPLE 2.16

(continued)



EXAMPLE 2.17



EXAMPLE 2.17

(continued)

PE2113-Chapter 2 - Force Vectors_082715

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