Engineering Mechanics: Statics in SI Units, 12e 2 Force Vectors 1 Chapter Objectives Parallelogram Law Cartesian vector form Dot product and an angle between two vectors 2
Chapter Outline 1. Scalars and Vectors 2. Vector Operations 3. Vector Addition of Forces 4. Addition of a System of Coplanar Forces 5. Cartesian Vectors 6. Addition and Subtraction of Cartesian Vectors 7. Position Vectors 8. Force Vector Directed along a Line 9. Dot Product 3 2.1 Scalars and Vectors Scalar A quantity characterized by a positive or negative number Indicated by letters in italic such as A (italics) e.g. 4
2.1 Scalars and Vectors Vector A quantity that has magnitude and direction e.g. Arrow above a letter A! A magnitude is represented by A! In this course, we sometimes use a bold font, A, for a vector and the magnitude is represented by A 5 2.2 Vector Operations Multiplication and Division of a Vector by a Scalar - Product of vector A and scalar a - Magnitude = - Law of multiplication applies e.g. A/a = ( 1/a ) A, a 0 6
2.2 Vector Operations Vector Addition - Two vectors addition, A and B, gives a vector R, can be done by parallelogram law - R can be obtained by triangle - R = A + B = B + A - For collinear vectors, i.e. A and B are on the same straight line. 7 2.2 Vector Operations Vector Subtraction - Special case of addition, similar to vector addition e.g. R = A B = A + ( - B ) 8
2.3 Vector Addition of Forces Resultant Force Parallelogram law Resultant, F R = ( F 1 + F 2 ) 9 2.3 Vector Addition of Forces Analysis Procedure Parallelogram Law Draw a parallelogram, parallel to the two forces The resultant force is a diagonal of the parallelogram 10
2.3 Vector Addition of Forces - - law of cosines law of sines 11 Example 2.1 The screw eye is subjected to two forces, F 1 and F 2. Determine the magnitude and direction of the resultant force. 12
Solution Parallelogram Law Unknown: magnitude of F R and angle θ 13 Solution 14
Solution Trigonometry Direction Φ of F R measured from the horizontal! φ = 39.8 + 15! φ = 54.8! 15 2.4 Addition of a System of Coplanar Forces Scalar Notation Use scalar values and use axis x and y, consider positive and negative values accordingly F y 16
2.4 Addition of a System of Coplanar Forces Cartesian Vector Notation Caretesian vectors i and j for x and y, respectively i and j is a unit vector Examples 17 2.4 Addition of a System of Coplanar Forces Coplanar Force Resultants Find components in x and y for all forces Adding all the forces for each direction The resultant force found by parallelogram or pythagorus Cartesian vector notation: 18
2.4 Addition of a System of Coplanar Forces Coplanar Force Resultants 19 2.4 Addition of a System of Coplanar Forces Coplanar Force Resultant Find F R by pythagorus 20
Example 2.5 Determine x and y components of F 1 and F 2 acting on the boom. Express each force as a Cartesian vector. 21 Solution Scalar Notation Cartesian Vector Notation 22
Solution By similar triangles we have Scalar Notation: Cartesian Vector Notation: 23 Example 2.6 The link is subjected to two forces F 1 and F 2. Determine the magnitude and orientation of the resultant force. 24
Solution I Scalar Notation: 25 Solution I Resultant Force From vector addition, direction angle θ is 26
Solution II Cartesian Vector Notation Thus, 27 2.5 Cartesian Vectors (3D) Right- Handed Coordinate System Right thumb z Other fingers sweeping from x to y z axis in 2D is pointing outwards from the paper 28
2.5 Cartesian Vectors Rectangular Components of a Vector Vector A can be found in x, y and z Use parallelogram twice A = A + A z A = A x + A y A = A x + A y + A z 29 2.5 Cartesian Vectors Unit Vector Direction A find by unit vector Magnitude 1 u A = A / A ด งน A = A u A 30
2.5 Cartesian Vectors Cartesian Vector Representations 3 componenets A in i, j and k directions 31 2.5 Cartesian Vectors Magnitude of a Cartesian Vector 32
2.5 Cartesian Vectors Direction of a Cartesian Vector Direction of A defined by α, β and γ 0 α, β and γ 180 The direction cosines of A is cosα = A x A cos β = A y A cosγ = A z A 33 2.5 Cartesian Vectors Direction of a Cartesian Vector A = A x i + A y j + A Z k u A = A /A = (A x /A)i + (A y /A)j + (A Z /A)k A = A + A + 2 x 2 y A 2 z 34
2.5 Cartesian Vectors Direction of a Cartesian Vector u A = cosαi + cosβj + cosγk A = A + A + A cos 2 2 x 2 y 2 z 2 2 α + cos β + cos γ = 1 = Acosαi + Acosβj + Acosγk = A x i + A y j + A Z k 35 2.6 Addition and Subtraction of Cartesian Vectors Concurrent Force Systems Resultant forces F R = F = F x i + F y j + F z k 36
Example 2.8 Express the force F as Cartesian vector. 37 Solution 38
Solution 39 2.7 Position Vectors x,y,z Coordinates Right- handed coordinate system All other points are relative to O 40
2.7 Position Vectors Position Vector Position vector r used to indicate a location from a reference position E.g. r = xi + yj + zk 41 2.7 Position Vectors Position Vector (between 2 points) Vector addition r A + r = r B Solving r = r B r A = (x B x A )i + (y B y A )j + (z B z A )k or r = (x B x A )i + (y B y A )j + (z B z A )k 42
2.7 Position Vectors Direction and magnitude of a cable can be found by position vectors A and B Position vector r Angles α, β and γ Unit vector, u = r/r 43 Example 2.12 An elastic rubber band is attached to points A and B. Determine its length and its direction measured from A towards B. A (1, 0, -3) m B (-2, 2, 3) m 44
Solution Position vector Magnitude = length of the rubber band Unit vector in the director of r 45 Solution 46
2.8 Force Vector Directed along a Line For 3D, direction of F F = F u = F (r/r) F with unit (N) r with unit (m) (r/r) 47 2.8 Force Vector Directed along a Line F is force along the cable - need x, y, z - find position vector r along a cable Find unit vector u = r/r F = Fu 48
Example 2.13 The man pulls on the cord with a force of 350N. Represent this force acting on the support A, as a Cartesian vector and determine its direction. 49 Solution End points of the cord are A (0m, 0m, 7.5m) and B (3m, - 2m, 1.5m) Magnitude = length of cord AB Unit vector, 50
Solution Force F has a magnitude of 350N, direction specified by u. 51 2.9 Dot Product Dot product of A and B written as A B (A dot B) A B = AB cosθ where 0 θ 180 The result is scalar 52
2.9 Dot Product Laws of Operation 1. Commutative law A B = B A 2. Multiplication by a scalar a(a B) = (aa) B = A (ab) = (A B)a 3. Distribution law A (B + D) = (A B) + (A D) 53 2.9 Dot Product Cartesian Vector Formulation - Dot product of Cartesian unit vectors i i = (1)(1)cos0 = 1 i j = (1)(1)cos90 = 0 - Similarly i i = 1 j j = 1 k k = 1 i j = 0 i k = 0 j k = 0 54
2.9 Dot Product Cartesian Vector Formulation Dot product of 2 vectors A and B A B = A x B x + A y B y + A z B z Application Find angle between two vectors θ = cos - 1 [(A B)/(AB)] 0 θ 180 Find vector parallel and perpendicular components A a = A cos θ = A u 55 Example 2.17 The frame is subjected to a horizontal force F = {300j} N. Determine the components of this force parallel and perpendicular to the member AB. A (0, 0, 0) B (2, 6, 3) 56
Solution Since Thus 57 Solution Since result is a positive scalar, F AB has the same sense of direction as u B. Express in Cartesian form Perpendicular component 58
Solution Magnitude can be determined from F or from Pythagorean Theorem, or 59