Forces Classification of Forces Force System 1 – Concurrent : all forces pass through a point 2 – Coplanar : in the same plane 3 – Parallel : parallel line of action 4 – Collinear : common line of action Three Types 1 – Free (direction, magnitude and sense) 2 – Sliding 3 – Fixed A O Origin

Trigonometric Relations of a triangle Phythagorean theorem is valid only for a right angled triangle. For any triangle (not necessarily right angled) Sine Law Cosine Law A B C

Cartesian Coordinate Systems A much more logical way to add/subtract/manipulate vectors is to represent the vector in Cartesian coordinate system. Here we need to find the components of the vector in x,y and z directions. Simplification of Vector Analysis x y A z n en i j k x y z

Vector Representation in terms unit vector Suppose we know the magnitude of a vector in any arbitrary orientation, how do we represent the vector? Unit Vector : a vector with a unit magnitude A en ^

Right-Handed Coordinate System x y z Right-Handed System If the thumb of the right hand points in the direction of the positive z-axis when the fingers are pointed in the x-direction & curled from the x-axis to the y-axis.

Components of a Vector y A Ay Ax Az z x In 3-D y A Ay Ax x In 2-D Cartesian (Rectangular) Components of a Vector y A Ay Ax Az z x In 3-D y A Ay Ax x In 2-D

Cartesian Vectors z Az = Azk A = Aen en y Ax = Axi Ay = Ayj x Cartesian Unit Vectors y A = Aen Ay = Ayj Ax = Axi Az = Azk z x n i j k en

Cartesian Vectors Magnitude of a Cartesian Vector y A Ay Ax Az z x

Addition and Subtraction of vectors y F Fy Fx Fz z x

Force Analysis y F Fy Fx Fz z x

Force Analysis Ex: y z x 10ft 6ft 8ft F = 600

Summary of the Force Analysis z F y F x q = cos - 1 y y F

Addition/Subtraction of Cartesian vectors Since any vector in 3-D can be expressed as components in x,y,z directions, we just need to add the corresponding components since the components are scalars. Then the addition Then the subtraction

ΑΣΚΗΣΗ-36 Solution:

ΑΣΚΗΣΗ-36 (2) Unit vector and direction cosines From these components, we can determine the angles

Position Vectors z (xb, yb, zb) r ( xa, ya, za) y x Position vectors can be determined using the coordinates of the end and beginning of the vector

Force Vector Along a Line A force may be represented by a magnitude & a position Force is oriented along the vector AB (line AB) B z F Unit vector along the line AB A y x

Dot Product y x A Dot Product (Scalar Product)

Application of dot product Angle between two vectors A B