CHAPTER 2 FORCE VECTORS 2.1 Scalars and Vector 1.Scalar A quantity characterized by a positive or negative number. LengthVolumeMass symbolLVM EX: 2.Vector.
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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
2.2 Vector operation 1.Scalar Multiplication and Division 2.Vector Addition Parallelogram Law
3.Vector Subtraction 4.Resolution of Vector (1) Known (2) Two line of action are known.
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.
2. Analysis method a> Parallelogram law b> Trigonometry i> sine law ii> cosine law
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.
(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.
3. Coplanar Force Resultants (1) Resolve each force into it x and y components using Cartesian vector notation.
(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
2.5 Cartesian Vectors 1. Right-Hand Coordinates system 2. Rectangular components of a vector
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.
5. Cartesian vector representation unit vector of
(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
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.
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
2-8 Force Vector Directed Along A Line u: Unit vector defines the direction and sense of position vector r and force vector F.
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
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
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