Chapter 2 Force Vectors

Engineering Mechanics: Statics Fifteenth Edition Chapter 2 Force Vectors Section 2.5 Cartesian Vectors Section 2.6 Addition of Cartesian Vectors Copyright © 2022 Pearson Education, Inc. All Rights Reserved Cartesian Vectors and their Addition & Subtraction Today’s Objectives: a. Represent a 3-D vector in a Cartesian coordinate system. b. Find the magnitude and coordinate angles of a 3-D vector c. Add vectors (forces) in 3-D space Copyright © 2022 Pearson Education, Inc. All Rights Reserved Cartesian Unit Vectors For a vector A, with a magnitude of A, an unit vector is defined as uA = 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. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Cartesian Unit Vectors Consider a box with sides A X , A Y , and A Z meters long. The vector A can be defined as A = (A Xi + A Y j + A Zk ) m Copyright © 2022 Pearson Education, Inc. All Rights Reserved Direction of a Cartesian Vector (1 of 2) The direction or orientation of vector A is defined by the angles ,, and . 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 Ay Ax A cos = cos = cos = z A A A These angles are not independent. They must satisfy the following equation. cos² + cos² + cos²  = 1 Copyright © 2022 Pearson Education, Inc. All Rights Reserved Direction of a Cartesian Vector (2 of 2) 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: uA = A A A Ax = i+ y j+ z k A A A A or written another way, u A = cos  i + cos  j + cos  k . Copyright © 2022 Pearson Education, Inc. All Rights Reserved Section 2.6: Addition of Cartesian Vectors Once individual vectors are written in Cartesian form, it is easy to add or subtract them. The process is essentially the same as when 2-D vectors are added. FR = F = Fx i + Fy j + Fzk For example, if A = A X i + A Y j + A Z k and B = B X i + B Y j + BZ k , then A + B = (A X + B X ) i + (A Y + B Y ) j + (A Z + B Z ) k or A − B = (A X − B X ) i + (A Y − B Y ) j + (A Z − BZ ) k . Copyright © 2022 Pearson Education, Inc. All Rights Reserved Important Notes Sometimes 3-D vector information is given as: a. Magnitude and the coordinate direction angles, or, b. Magnitude and projection angles. You should be able to use both these sets of information to change the representation of the vector into the Cartesian form, i.e., F = 10 i − 20 j + 30 k  N . Copyright © 2022 Pearson Education, Inc. All Rights Reserved Example (1 of 4) Given: Two forces F1 and F2 are applied to a hook. Find: The resultant force in Cartesian vector form. Plan: 1. Using geometry and trigonometry, write F1 and F2 in Cartesian vector form. 2. Then add the two forces (by adding x and y-components). Copyright © 2022 Pearson Education, Inc. All Rights Reserved Example (2 of 4) Solution: First, resolve force F1. Fx = 0 = 0 lb 4 Fy = 500   = 400 lb 5 3 Fz = 500   = 300 lb 5 Now, write F1 in Cartesian vector form (don’t forget the units!). F1 = 0i + 400 j + 300k  lb Copyright © 2022 Pearson Education, Inc. All Rights Reserved Example (3 of 4) Now, resolve force F2. F2z = − 800 sin 45o = −565.7 lb F2 ’ = 800 cos 45o = 565.7 lb F2′ can be further resolved as, F2x = 565.7 cos 30o = 489.9 lb F2y = 565.7 sin 30o = 282.8 lb Thus, we can write: F2 = {489.9 i + 282.8 j − 565.7 k } lb Copyright © 2022 Pearson Education, Inc. All Rights Reserved Example (4 of 4) So FR = F1 + F2 and F1 = 0i + 400 j + 300k  lb F2 = 489.9i + 282.8 j − 565.7k  lb FR = 490i + 683 j − 266k  lb Copyright © 2022 Pearson Education, Inc. All Rights Reserved Concept Quiz (1 of 2) 1. If you know only uA, you can determine the __________ of A uniquely. A. Magnitude B. angles ( , and  ) C. components A X , A Y , and A Z D. All of the above. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Concept Quiz (2 of 2) 2. For a force vector, the following parameters are randomly generated. The magnitude is 0.9 N,  = 30o ,  = 70o ,  = 100o. What is wrong with this 3-D vector ? A. Magnitude is too small. B. Angles are too large. C. All three angles are arbitrarily picked. D. All three angles are between 0o to 180o. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Group Problem Solving (1 of 4) Given: The screw eye is subjected to two forces, F1 and F2. Find: The magnitude and the coordinate direction angles of the resultant force. Plan: 1. Using the geometry and trigonometry, resolve and write F1 and F2 in the Cartesian vector form. 2. Add F1 and F2 to get FR. 3. Determine the magnitude and angles , , . Copyright © 2022 Pearson Education, Inc. All Rights Reserved Group Problem Solving (2 of 4) First resolve the force F1. F1z = − 250 sin 35o = − 143.4N F´ = 250 cos 35o = 204.8 N F′ can be further resolved as, F1x = 204.8 sin 25o = 86.6 N F1y = 204.8 cos 25o = 185.6 N Now we can write: F1 = 86.6 i + 185.6 j − 143.4 k N Copyright © 2022 Pearson Education, Inc. All Rights Reserved Group Problem Solving (3 of 4) Now, resolve force F2. The force F2 can be represented in the Cartesian vector form as:   F2 = 400 cos 120o i + cos 45o j + cos 60o k N = −200 i + 282.8 j + 200 k  N F2 = −200 i + 282.8 j + 200 k  N Copyright © 2022 Pearson Education, Inc. All Rights Reserved Group Problem Solving (4 of 4) So FR = F1 + F2 and F1 = 86.6 i + 185.6 j − 143.4 k  N F2 = −200 i + 282.8 j + 200 k  N FR = −113.4 i + 468.4 j + 56.6 k  N Now find the magnitude and direction angles for the vector. FR = ( −113.4 ) 2 + 468.4 2 + 56.6  1 2 2 = 485.2 = 485 N F   −113.4   = cos−1  RX  = cos−1  = 104o   485.2   FR  F   468.4   = cos−1  RY  = cos−1  = 15.1o   485.2   FR  F   56.6   = cos−1  Rz  = cos−1  = 83.3o   485.2   FR  Copyright © 2022 Pearson Education, Inc. All Rights Reserved Attention Quiz (1 of 2) 1. What is not true about an unit vector, e.g., uA? A. B. C. D. It is dimensionless. Its magnitude is one. It always points in the direction of positive X- axis. It always points in the direction of vector A. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Attention Quiz (2 of 2) 2. If F = {10i + 10 j + 10k } N and G = {20i + 20 j + 20k } N, then F + G = A. B. C. D.  ____  N 10 i + 10 j + 10 k 30 i + 20 j + 30 k −10 i − 10 j − 10 k 30 i + 30 j + 30 k Copyright © 2022 Pearson Education, Inc. All Rights Reserved Reading Quiz (2 of 2) 2. The symbols , , and  designate the ___________ of a 3-D Cartesian vector. A. B. C. D. Unit vectors Coordinate direction angles Greek societies X, Y and Z components Copyright © 2022 Pearson Education, Inc. All Rights Reserved