Engineering Mechanics Statics
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1 Mechanical Systems Engineering Engineering Mechanics Statics 2. Force Vectors; Operations on Vectors Dr. Rami Zakaria
2 MECHANICS, UNITS, NUMERICAL CALCULATIONS & GENERAL PROCEDURE FOR ANALYSIS Today s we will learn how to: a) Resolve a 2-D vector into components. b) Add 2-D vectors using Cartesian vector notations. c) Represent a 3-D vector in a Cartesian coordinate system. d) Find the magnitude and coordinate angles of a 3-D vector e) Add vectors (forces) in 3-D space
3 Introduction 1. Which one of the following is a scalar quantity? A) Force B) Position C) Mass D) Velocity 2. The result of adding two vectors together is a/an. A) scalar quantity B) vector C) circle D) angle FR APPLICATION OF VECTOR ADDITION There are three concurrent forces acting on the hook due to the chains. We need to decide if the hook will fail (bend or break)? To do this, we need to know the resultant force acting on the hook.
4 Scalars and Vectors Scalar: A quantity characterized by a positive or negative number. Vector: A quantity characterized by a magnitude (number) and direction. Scalars Vectors Examples Mass, Volume Force, Velocity Characteristics It has a magnitude (+ -) It has a magnitude and a direction Addition rule (+) Simple arithmetic Parallelogram law Notation A (in the book A) A
5 Vector Operations 1- Multiplication and Division of a Vector by a Scalar If (a) is a scalar then aa has a magnitude aa
6 2- Vector Adding We can add two vectors using either the parallelogram law or the triangle method. Parallelogram Law: Triangle method (connect tail to head): -How can we add two collinear vectors? (see page 19) -How do you subtract a vector? (see page 19) -How can you add more than two concurrent vectors graphically? (see page 21)
7 3- Resolution of a Vector Resolution of a vector is breaking up a vector into components -It is like using the parallelogram law in reverse. -In 2D system, a vector can be resolved into two components Remember: in a triangular ABC Sine law: A B C sin a sin b sin c A c B Cosine law: C A 2 B 2 2AB cos c b C a
8 ADDITION OF A SYSTEM OF COPLANAR FORCES We resolve vectors into components using the x and y axis system (cartesian). The directions are based on the x and y axes. We use the unit vectors i and j to designate the x and y axes. For example, F = Fx i + Fy j or F' = F'x i + ( F'y ) j The x and y axis are always perpendicular to each other. Together, they can be directed at any inclination.
9 Addition of Several Vectors Step 1 is to resolve each force into its components. Step 2 is to add all the x-components together, followed by adding all the y components together. These two totals are the x and y components of the resultant vector. Step 3 is to find the magnitude and angle of the resultant vector. (see page 33) You can also represent a 2-D vector with a magnitude and angle. F Rx F R cos F F sin Ry R
10 Example Given: Three concurrent forces acting on a tent post. Find: The magnitude and angle of the resultant force. Plan: a) Resolve the forces into their x-y components. b) Add the respective components to get the resultant vector. c) Find magnitude and angle from the resultant components.
11 Solution F1 = {0 i j } N F2 = { 450 cos (45 ) i sin (45 ) j } N = { i j } N F3 = { (3/5) 600 i + (4/5) 600 j } N = { 360 i j } N
12 Summing up all the i and j components respectively, we get, FR = { ( ) i + ( ) j } N = { i j } N Using magnitude and direction: FR = ((41.80) 2 + (1098) 2 ) 1/2 = 1099 N = tan -1 (1098/41.80) = 87.8 y FR x
13 Example group work Given: Three concurrent forces acting on a bracket Find: The magnitude and angle of the resultant force. Plan: a) Resolve the forces into their x and y components. b) Add the respective components to get the resultant vector. c) Find magnitude and angle from the resultant components.
14 Solution:
15 y FR x
16 Example 1. Resolve F along x and y axes and write it in vector form. F = { } N y A) 80 cos (30 ) i 80 sin (30 ) j B) 80 sin (30 ) i + 80 cos (30 ) j C) 80 sin (30 ) i 80 cos (30 ) j D) 80 cos (30 ) i + 80 sin (30 ) j 30 x 2. Determine the magnitude of the resultant (F1 + F2) force in N when F1 = { 10 i + 20 j } N and F2 = { 20 i + 20 j } N. A) 30 N B) 40 N C) 50 N D) 60 N E) 70 N
17 CARTESIAN VECTORS Application In the case of this radio tower, if you know the forces in the three cables, how would you determine the resultant force acting at D, the top of the tower?
18 CARTESIAN UNIT VECTORS For a vector A, with a magnitude of A, an unit vector is defined as u A = A / A. Characteristics of a unit vector : a) Its magnitude is 1. b) It is dimensionless (has no units). c) It points in the same direction as the original vector (A). The unit vectors in the Cartesian axis system are i, j, and k. They are unit vectors along the positive x, y, and z axes respectively.
19 CARTESIAN VECTOR REPRESENTATION Consider a box with sides A X, A Y, and A Z meters long. The vector A can be defined as A = (A X i + A Y j + A Z k) The projection of vector A in the x-y plane is A. The magnitude of A is found by using the same approach as a 2-D vector: A = (A X 2 + A Y2 ) 1/2. The magnitude of the position vector A can now be obtained as A = ((A ) 2 + A Z2 ) ½ = (A X 2 + A Y 2 + A Z2 ) ½
20 DIRECTION OF A CARTESIAN VECTOR The direction or orientation of vector A is defined by the angles ά, β, and γ. These angles are measured between the vector and the positive X, Y and Z axes, respectively. Their range of values are from 0 to 180 Using trigonometry, direction cosines are found using These angles are not independent. They must satisfy the following equation. cos ² + cos ² + cos ² = 1 This result can be derived from the definition of a coordinate direction angles and the unit vector. Recall, the formula for finding the unit vector of any position vector: or written another way, u A = cos i + cos j + cos k.
21 GROUP PROBLEM SOLVING Given: The screw eye is subjected to two forces, F 1 and F 2. Find: Plan: The magnitude and the coordinate direction angles of the resultant force. 1) Using the geometry and trigonometry, resolve and write F 1 and F 2 in the Cartesian vector form. 2) Add F 1 and F 2 to get F R. 3) Determine the magnitude and angles,,.
22
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25 Attention Questions 1. What is not true about an unit vector, u A? A) It is dimensionless. B) Its magnitude is one. C) It always points in the direction of positive X- axis. D) It always points in the direction of vector A. 2. If F = {10 i + 10 j + 10 k} N and G = {20 i + 20 j + 20 k } N, then F + G = { } N A) 10 i + 10 j + 10 k B) 30 i + 20 j + 30 k C) 10 i 10 j 10 k D) 30 i + 30 j + 30 k
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