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CHAPTER 2 FORCE VECTORS

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2.1 Scalars and Vector 1.Scalar A quantity characterized by a positive or negative number. LengthVolumeMass symbolLVM EX: 2.Vector A quantity having both a magnitude and a direction. EX: position, force, moment Symbol: or A line of action magnitude:A= direction: degree c.c.w sense: arrowhead

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2.2 Vector operation 1.Scalar Multiplication and Division 2.Vector Addition Parallelogram Law

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3.Vector Subtraction 4.Resolution of Vector (1) Known (2) Two line of action are known.

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2.3 Vector Addition of Force 1. Engineering problems (a) Find the resultant force, knowing its components. resultant force = ?? (b) Resolve a known force into two components.

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2. Analysis method a> Parallelogram law b> Trigonometry i> sine law ii> cosine law

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2.4 Addition of a System of Coplanar Forces 1. Rectangular components 2. Notation (1) scalar notation Algebraic scalars are used to express the magnitude and directional sense of the rectangular components of a force.

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(2) Cartesian Vector Notation Cartesian unit vector i and j are used to express the directions of x and y axes respectively. The magnitude of each component of force F is always a positive quantity.

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3. Coplanar Force Resultants (1) Resolve each force into it x and y components using Cartesian vector notation.

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(2) Add the respective components using scalar algebra (3) Form the resultant force by adding the resultant of the x and y components using the parallelogram law. magnitude orientation

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2.5 Cartesian Vectors 1. Right-Hand Coordinates system 2. Rectangular components of a vector

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3. Unit vector A vector having a magnitude of 1. :a vector, unit vector of =?? Unit vector (dimensionless) 4. Cartesian unit vectors i, j,k. Cartesian unit vector i,j,k are used to designate the direction of the x, y, z axes respectively.

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5. Cartesian vector representation unit vector of

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(1) Magnitude of Cartesian vector A (2) Direction of Cartesian vector A The orientation of A is defined by the coordinate direction angles α β γ Direction consines cosα=Ax/A cosβ=Ay/A cosγ=Az/A Direction of A is defined by its unit of vector u A

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2. Subtraction A-B R`=A-B=(Ax-By)i+(Ax-By)j+ (Az-Bz)k 2-6 Operations of cartesian vectors Additon:A+B A=Axi+Ayj+Azk B=Bxi+Byj+Bzk R=A+B=(Ax+Bx)i+(Ay+By)j+ (Az+Bz)k 3. Concurrent Force System are algebraic sums of the respective x,y,z components of each force in the system.

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2-7 Position Vectors 1. Definition A position vector is a fixed vector which locates a point in space relative to another point. 2. Position vector from origin 0 to point P(x,y,z) Cartesian vector form of r 3. Position vector from point A to point B

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2-8 Force Vector Directed Along A Line u: Unit vector defines the direction and sense of position vector r and force vector F.

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2-9 Dot Product (Scalar Product) (1) A. B=B. A (2) α (A. B)=( α A. B)=(A. α B)= α (A. B) (3) A(B+D)=A. B+A. D 1. Definition θ A. B=ABcos θ,0< θ <180° 2. Law of operation

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3. Cartesian vector formulation (1) Dot product of Cartesian unit vector i. i=1*1*cos0 。 =1 j. j=1, k. k=1 i. j=1*1cos90 。 =0, i. k=k. j=0 (2) Dot product of vectors A & B in Cartesian vector form A=Axi+Ayj+Ajk B=Bxi+Byi+Bjk A. B=(Axi+Ayj+Ajk). (Bxi+Byi+Bjk) =AxBx+AyBy+AzBz

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4. Applications (1) Angle between two vectors θ (2) Components of a vector parallel and perpendicular to a line. A: known vector θ: angle b/w A and u. u: unit vector of a line (known) θ A

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