2

Force Vectors Part 2

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Chapter Objectives

• Cartesian vector form • Dot product and angle between 2 vectors

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Chapter Outline

1. 2. 3. 4. 5.

Cartesian Vectors Addition and Subtraction of Cartesian Vectors Position Vectors Force Vector Directed along a Line Dot Product

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2.5 Cartesian Vectors

• Right-Handed Coordinate System

A rectangular or Cartesian coordinate system is said to be right-handed provided: – Thumb of right hand points in the direction of the positive z axis – z-axis for the 2D problem would be perpendicular, directed out of the page.

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2.5 Cartesian Vectors

• Rectangular Components of a Vector

–

A vector A may have one, two or three rectangular components along the x, y and z axes, depending on orientation – By two successive application of the parallelogram law A = A’ + Az A’ = Ax + Ay – Combing the equations, A can be expressed as A = Ax + Ay + Az

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2.5 Cartesian Vectors

• Unit Vector

– Direction of A can be specified using a unit vector – Unit vector has a magnitude of 1 – If A is a vector having a magnitude of A ≠ 0, unit vector having the same direction as A is expressed by uA = A / A. So that A = A uA

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2.5 Cartesian Vectors

• Cartesian Vector Representations

– 3 components of A act in the positive i, j and k directions A = Axi + Ayj + AZk *Note the magnitude and direction of each components are separated, easing vector algebraic operations.

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2.5 Cartesian Vectors

• Magnitude of a Cartesian Vector

A'2 + Az2

– From the colored triangle, A =

2 2 A' = Ax + Ay – From the shaded triangle,

– Combining the equations gives magnitude of A

2 2 A = Ax + Ay + Az2

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2.5 Cartesian Vectors

• Direction of a Cartesian Vector

– Orientation of A is defined as the coordinate direction angles α, β and γ measured between the tail of A and the positive x, y and z axes – 0 ≤ α, β and γ ≤ 180 ° 0°

– The direction cosines of A are

Ax cos α = A

cos β = Ay A

Az cos γ = A

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2.5 Cartesian Vectors

• Direction of a Cartesian Vector

– Angles α, β and γ can be determined by the inverse cosines Given A = Axi + Ayj + AZk

then, uA = A /A = (Ax/A)i + (Ay/A)j + (AZ/A)k

2 where A = Ax2 + Ay + Az2

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2.5 Cartesian Vectors

• Direction of a Cartesian Vector

– uA can also be expressed as uA = cosαi + cosβj + cosγk 2 2 Ax + Ay + Az2

– Since A =

and uA = 1, we have

cos 2 α + cos 2 β + cos 2 γ = 1

– A as expressed in Cartesian vector form is A = AuA = Acosαi + Acosβj + Acosγk = Axi + Ayj + AZk Copyright © 2010 Pearson Education South Asia Pte Ltd

2.6 Addition and Subtraction of Cartesian Vectors

• Concurrent Force Systems

– Force resultant is the vector sum of all the forces in the system

FR = ∑F = ∑Fxi + ∑Fyj + ∑Fzk

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Example 2.8

Express the force F as a Cartesian vector.

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Solution

Since two angles are specified, the third angle is found by

cos 2 α + cos 2 β + cos 2 γ = 1

cos 2 α + cos 2 60 o + cos 2 45o = 1

2 2 cos α = ± 1 − (0.5) − (0.707 ) =±0. 5

Two possibilities exit, namely

α = cos −1 (0.5)= 60 o

α = cos −1 (− 0.5) = 120o

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Solution

By inspection, α = 60º since Fx is in the +x direction Given F = 200N F =...