CHAPTER TWO Force Vectors

2.1 Scalars and Vectors Force Vectors Scalars : A quantity represented be a number (positive or negative) Ex: Mass, Volume, Length (in the book scalars are represented by italics) Vectors : A quantity which has both A – magnitude (scalar) B – direction (sense) Ex: position, force, moment Line of action Magnitude Sense

Forces Classification of Forces Contact 1 – Contacting or surface forces (mechanical) 2 – Non-Contacting or body forces (gravitational, weight) Area 1 – Distributed Force, uniform and non-uniform 2 – Concentrated Force

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

2.2 Basic Vector Operations Properties of Vectors 1 – Vector Addition 2 – Vector Subtraction 3 – Vector Multiplication

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

Example Ex: If the angle between F1 and F2 =60o and F1 = 54N Find: the Resultant force and the angle b. R F1 F2 120o 60o

Example… contd R F1 F2 120o 60o

Properties of Parallelogram Parallelogram Law Properties of Parallelogram A – sum of the three angles in a triangle is 180o B – sum of the interior angles is 360o C- opposite sides are equal

Vector Addition R A B B C A + B A R

Vector Subtraction Vector Subtraction A A - B -B B A

2.3 Vector addition of forces If we consider, only two forces at a time then the result can be obtained using parallelogram law, and by using law of sines and cosines of triangles. Even if we have multiple (say 5) forces, we take two at a time to resolve the resultant one by one. Consider another example in Example 2.4 (a) page 24 Given To find: Resultant Procedure: Use law of cosines to find the magnitude, law of sines find the angles

2.3 Vector addition of forces (example 2.4) Given To find: Solution: Draw the vector diagram Use the sine law to find

2.4 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 ^

2.5 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

2.6. 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

2.6 Addition/Subtraction of Cartesion 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

2.6 ---Example 2-9, pg 41 Determine the magnitude and the coordinate direction angles of the resultant force on the ring Solution:

2.6 ---Example 2-9…2 Unit vector and direction cosines From these components, we can determine the angles

2.4---Problem 2-27 (pg 35) Solution: Four concurrent forces act on the plate. Determine the magnitude of the resultant force and its orientation measured counterclockwise from the positive x axis. Solution: Draw the free body diagram. Then resolve forces in x and y directions.

2.4---Problem 2-27…..2 Find the magnitude and direction.

2.7. 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

2.8---Problem 2-51, page 56 Solution Determine the length of the connecting rod AB by first formulating a Cartesian position vector form A to B and then determining its magnitude. Solution

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

2.6---Problem 2-36 (pg 47) Solution:

2.6---Problem 2-36…..2

2.6---Problem 2-36…..3

2.9. Dot Product y x A Dot Product (Scalar Product)

2.9---Problem 2-76, pg 65 A force of F = 80 N is applied to the handle of the wrench. Determine the magnitudes of the components of the force acting along the axis AB of the wrench handle and perpendicular to it. Hand first rotates 45 in the xy plane and 30 in that plane

2.9---Problem 2-76…..2

Application of Dot Product Component of a Vector along a line A n A

Application of dot product Angle between two vectors A B