Ishik University / Sulaimani Civil Engineering Department. Chapter -2-

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1 Ishik University / Sulaimani Civil Engineering Department Chapter -- 1 orce Vectors Contents : 1. Scalars and Vectors. Vector Operations 3. Vector Addition of orces 4. Addition of a System of Coplanar orce 1

2 Objectives To show how to add forces and resolve them into components using the Parallelogram Law and Trigonometry Analysis. To express force and position in Cartesian vector form and explain how to determine the vector s magnitude and direction Scalars and Vectors Scalar : A quantity characterized by a positive or negative number. Indicated by letters in italic such as A. Example: mass, volume and length Vector : A quantity that has both magnitude and direction. Example: position, force and moment Represent by a letter with an arrow over it such as A or A Magnitude is designated as A or simply A. In this subject, vector is presented as A and its magnitude (positive quantity) as A. 4

Vector Represented graphically as an arrow. Length of arrow = Magnitude of Vector. Angle between the reference axis and arrow s line of action = Direction of Vector. Arrowhead = Sense of Vector 5 A.

3 Vector Represented graphically as an arrow. Length of arrow = Magnitude of Vector. Angle between the reference axis and arrow s line of action = Direction of Vector. Arrowhead = Sense of Vector 5 A. Vector Addition. Vector Operations - Addition of two vectors A and B gives a resultant vector R by the parallelogram law. - Result R can be found by triangle construction. Example: R = A + B = B + A 6 3

4 B. Vector Subtraction The resultant of the difference between two vectors A and B of the same type may be expressed as: - Special case of addition Example: R = A B = A + ( - B ) - Rules of Vector Addition Applies 7 C. Multiplication and division of vector by a scalar: If a vector is multiplied by a positive scalar, its magnitude is increased by that amount. When multiplied by a negative scalar it will also change the directional sense of the vector 8 4

3. Vector Addition of orces Experimental evidence has shown that a force is a vector quantity since it has a specified magnitude, direction, and sense and it adds according to the parallelogram law.

5 3. Vector Addition of orces Experimental evidence has shown that a force is a vector quantity since it has a specified magnitude, direction, and sense and it adds according to the parallelogram law. When two or more forces are added, successive applications of the parallelogram law is carried out to find the resultant 9 inding a resultant force: The two component forces 1 and acting on the pin in figure, can be added together to form the resultant force. 10 5

inding the components of a force: o o Sometimes it is necessary to resolve a force into two components in order to study its pulling and pushing effect in two specific directions.

6 inding the components of a force: o o Sometimes it is necessary to resolve a force into two components in order to study its pulling and pushing effect in two specific directions. or example, in figure below, is to be resolved into two components along two members, defined by u and v. 11 Addition of several forces: o o If more than two forces are to be added successive applications of the parallelogram law can be carried out in order to obtain the resultant force. or example if the three forces 1,, 3 act at a point o, the resultant of any two of the forces is found (1 + ) and then this resultant is added to the third force yielding the resultant of all three forces. (R = (1 + ) + 3) 1 6

Then start at the head of force, and construct lines parallel to the axes, thereby forming the parallelogram. The sides of the parallelogram represent the components u and v.

7 Procedure of Analysis / Review a) Parallelogram Law b) Trigonometry 1. sine law. cosine law a) Parallelogram Law Two "component" forces 1 and in ig.(a) add according to the parallelogram law yielding, a resultant force R that forms the diagonal of the parallelogram. If a force is to be resolved into components along two axes u and v, ig.(b). Then start at the head of force, and construct lines parallel to the axes, thereby forming the parallelogram. The sides of the parallelogram represent the components u and v. 13 b) Trigonometry Analysis Redraw a half portion of the parallelogram to illustrate the triangular head to tail addition of the components. rom this triangle, the magnitude of the resultant force can be determined using the law of cosines, and its direction is determined from the law of sines. The magnitudes of two force components are determined from the law of sines. 14 7

8 Example -1- The screw eye is subjected to two forces 1 and. Determine the magnitude and the direction of a Resultant force. Ans: R = 13 N = Solution 1: Parallelogram Unknown: magnitude of R and angle θ 16 8

9 Solution : Trigonometry Law of Cosines: R 1.6N 13N 100N 150N 100N 150N cos Law of Sines 150N 1.6N sin sin N sin N sin 39.8 Direction Φ of R measured from the horizontal is, Note: The results seem reasonable since it shows R to have a magnitude larger than its components and a direction that is between them. 18 9

10 Example -- Determine the magnitude of the component force in figure below and the magnitude of the resultant force R if R is directed a long the positive y axis. Ans: = 45 lb R = 73 lb 19 Solution 1: Parallelogram Solution : Trigonometry The magnitudes of R and are the two unknowns. They can be determined by applying the law of sines. 0 10

11 Example -3- Determine the magnitude of the resultant force acting on the screw eye and its direction measured clockwise from the x axis. Ans: R = 6.80 kn θ = Example -4- Resolve the horizontal 600lb force in figure below, into components action along the u and v axes and determine the magnitudes of these components. Ans: u = 1039 lb v = 600 lb 11

12 4. Addition of a System of Coplanar orce or resultant of two or more forces: ind the components of the forces in the specified axes Add them algebraically orm the resultant In this subject, we resolve each force into rectangular forces along the x and y axes. x y 3 Scalar Notation - x and y axes are designated positive and negative - Components of forces expressed as algebraic scalars Example: Sense of direction along positive x and y axes Instead of using the angle θ, the direction of can also be defined using a small "slope" triangle, shown in figure. It is also possible to represent the x and y components of a force in terms of Cartesian unit vectors i and j 4 1

13 Cartesian Vector Notation We can express as a Cartesian vector. = x i + y j Coplanar orce Resultants In coplanar force resultant case, each force is resolved into its x and y components. Then the respective components are added using scalar algebra since they are collinear. or example, consider the three concurrent forces, 5 Each force is represented as a Cartesian vector. Cartesian vector notation; 1 = 1x i + 1y j = - x i + y j 3 = 3x i 3y j Vector resultant is therefore, R = = 1x i + 1y j - x i + y j + 3x i 3y j = ( 1x - x + 3x )i + ( 1y + y 3y )j = ( Rx )i + ( Ry )j 6 13

14 We can represent the components of the resultant force of any number of coplanar forces symbolically by the algebraic sum the x and y components of all the forces. In all cases, Rx = x Take note of sign conventions Magnitude of R can be found by Pythagorean Theorem, Direction angle θ (orientation of the force) can be found by trigonometry, Ry = y R tan 1 Rx Ry Rx Ry 7 Example -5- The link is subjected to two forces 1 and. Determine the magnitude and orientation of the resultant force. 8 14

15 Solution : Solution 1: Scalar Notation; 600cos N Rx Rx Ry Ry : x y : 600sin N N 400sin 45 N N 400cos 45 N 9 Resultant orce R 69N 36.8N 58.8N rom vector addition, Direction angle θ is 58.8 tan 1 N 36.8N 67.9 Solution : Cartesian Vector Notation Thus, 1 = { 600cos30 i + 600sin30 j } N = { -400sin45 i + 400cos45 j } N R = 1 + = (600cos30 N - 400sin45 N)i + (600sin30 N + 400cos45 N)j = {36.8i j}N *The magnitude and direction of R are determined in the S.1nle manner as before

16 Example -6- Determine the x and y components of 1 and acting on the boom shown in figure, express each force as a Cartesian vector. 31 Solution; Scalar Notation; 1x 1y 00sin 30 N 100N 100N 00cos30 N 173N 173N By similar triangles we have; Scalar Notation; Cartesian Vector Notation; x y x y N N 13 40N 100N 100N 100i 173 j N 1 40i 100 jn 3 16

17 33 Example -7- The link is subjected to two forces 1 and. Determine the magnitude and orientation of the resultant force

18 Scalar Notation: Rx Rx : 600cos 30 N 400sin 45 N 36.8N x Ry Ry : 600sin 30 N 400cos 45 N 58.8N Resultant orce; R y 36.8N 58.8N 69N rom vector addition, direction angle θ is; 58.8N tan N 35 Resultant orce; 36.8N 58.8N 69N R rom vector addition, direction angle θ is; 58.8N tan N Cartesian Vector Notation; 1 = { 600cos30 i + 600sin30 j } N = { -400sin45 i + 400cos45 j } N Thus, R = 1 + = (600cos30ºN - 400sin45ºN)i + (600sin30ºN + 400cos45ºN)j = {36.8i j}N The magnitude and direction of R are determined in the same manner as before

19 Example -8- Resolve each force acting on the post into its x and y components. Ans: 1x = O N 1y = 300 N x = 318 N y = 318 N 3x = 360 N 3y = 480 N 37 Example -9- Determine the magnitude and direction of the resultant force. Ans: R = 567 N θ =

68 θ = 180 + Φ = 59 39 Example -11- If the resultant force acting on the bracket is to be 750

20 Example -10- Determine the magnitude of the resultant force acting on the corbel and its direction θ measured counterclockwise from the x axis. Ans: R = 154 lb Φ = θ = Φ = Example -11- If the resultant force acting on the bracket is to be 750 N directed along the positive x axis, determine the magnitude of and its direction θ. Ans: θ = = 36 N 40 0

8 = 31. kn 41 References : 1. Engineering Mechanics-Statics by R.C.

21 Example -1- Determine the magnitude of the resultant force and its direction θ measured counterclockwise from the positive x axis. Ans: θ = 39.8 = 31. kn 41 References : 1. Engineering Mechanics-Statics by R.C.-Hibbeler, 1th Edition.. Lecture Notes and Exercises on Statics, by Dr. Abdulwahab Amrani. 4 1

Engineering Mechanics: Statics in SI Units, 12e Force Vectors

Engineering Mechanics: Statics in SI Units, 1e orce Vectors 1 Chapter Objectives Parallelogram Law Cartesian vector form Dot product and angle between vectors Chapter Outline 1. Scalars and Vectors. Vector