Section 3.4 Three-Dimensional Force Systems

Engineering Mechanics: Statics Fifteenth Edition Chapter 3 Equilibrium of a Particle Section 3.4 Three-Dimensional Force Systems Copyright © 2022 Pearson Education, Inc. All Rights Reserved The Equations of 3-D Equilibrium (1 of 2) When a particle is in equilibrium, the vector sum of all the forces acting on it must be zero ( F = 0 ). This equation can be written in terms of its x, y and z components. This form is written as follows. ( Fx ) i + ( Fy ) j + ( Fz ) k = 0 Copyright © 2022 Pearson Education, Inc. All Rights Reserved The Equations of 3-D Equilibrium (2 of 2) This vector equation will be satisfied only when Fx = 0 Fy = 0 Fz = 0 These equations are the three scalar equations of equilibrium. They are valid for any point in equilibrium and allow you to solve for up to three unknowns. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Example I (1 of 3) Given: The four forces and geometry shown. Find: The tension developed in cables A B, A C, and A D. Plan: 1. Draw a FBD of particle A. 2. Write the unknown cable forces TB, TC , and TD in Cartesian vector form. 3. Apply the three equilibrium equations to solve for the tension in cables. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Example I (2 of 3) Solution: TB = TB i TC = −(TC cos 60°) sin30°i + (TC cos 60°) cos30° j +TC sin60°k TC = TC(−0.25i +0.433 j + 0.866k ) TD = TD cos 120° i + TD cos120° j + TD cos45° k TD = TD(−0.5 i − 0.5 j + 0.7071 k ) W = − 300 k Copyright © 2022 Pearson Education, Inc. All Rights Reserved Example Ⅰ (3 of 3) Applying equilibrium equations: FR = 0 = TB i + TC(− 0.25 i +0.433 j + 0.866 k ) + TD(−0.5 i − 0.5 j + 0.7071k ) − 300k Equating the respective i, j, k components to zero, we have Fx = TB – 0.25TC –0.5TD =0 Fy = 0.433 TC –0.5TD =0 Fz = 0.866TC + 0.7071TD – 300 =0 (1 ) ( 2) (3) Using (2) and (3), we can determine TC = 203 lb , TD = 176 lb Substituting TC and TD into (1), we can find TB = 139 lb Copyright © 2022 Pearson Education, Inc. All Rights Reserved Example Ⅱ (1 of 3) Given: A 600 N load is supported by three cords with the geometry as shown. Find: The tension in cords AB, A and AD. Plan: 1. Draw a free body diagram of Point A. Let the unknown force magnitudes be FB, FC, FD. 2. Represent each force in its Cartesian vector form. 3. Apply equilibrium equations to solve for the three unknowns. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Example Ⅱ (2 of 3) FB = FB(sin 30 i + cos30 j ) N F = F  rAD  D D   rAD  = {0.5 FB i +0.866 FB j } N  FC = –FC i N = FD  (1 i – 2  j + 2k) N 1   (12 + 2 2 + 2) 2    = { 0.333 FD i – 0.667 FD j + 0.667 FD k } N Copyright © 2022 Pearson Education, Inc. All Rights Reserved Example Ⅱ (3 of 3) Now equate the respective i, j, and k components to zero.  Fx = 0.5 FB – FC + 0.333 FD = 0  Fy = 0.866 FB – 0.667 FD = 0  Fz = 0.667 FD – 600 = 0 Solving the three simultaneous equations yields FC = 646 N (since it is positive, it is as assumed, e.g., in tension) FD = 900 N FB = 693 N Copyright © 2022 Pearson Education, Inc. All Rights Reserved Concept Quiz (1 of 2) 1. In 3-D, when you know the direction of a force but not its magnitude, how many unknowns corresponding to that force remain? A. One B. Two C. Three D. Four Copyright © 2022 Pearson Education, Inc. All Rights Reserved Concept Quiz (2 of 2) 2. If a particle has 3-D forces acting on it and is in static equilibrium, the components of the resultant force ( Fx, Fy, and  Fz ) ________. A. have to sum to zero, e.g., −5 i + 3 j + 2 k B. have to equal zero, e.g., 0 i + 0 j + 0 k C. have to be positive, e.g., 5 i + 5 j + 5 k D. have to be negative, e.g., −5 i −5 j − 5 k Copyright © 2022 Pearson Education, Inc. All Rights Reserved Attention Quiz (1 of 2) 1. ​Four forces act at point A and point A is in equilibrium. Select the correct force vector P. A. {−20 i + 10 j –10 k } lb B. {−10 i − 20 j −10 k } lb C. {+ 20 i −10 j −10 k } lb D. None of the above Copyright © 2022 Pearson Education, Inc. All Rights Reserved Attention Quiz (2 of 2) 2. In 3-D, when you don’t know the direction or the magnitude of a force, how many unknowns do you have corresponding to that force? A. One B. Two C. Three D. Four Copyright © 2022 Pearson Education, Inc. All Rights Reserved Three-dimensional Force Systems Today’s Objectives: • Drawing a 3-D free body diagram, and, • Applying the three scalar equations (based on one vector equation) of equilibrium. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Reading Quiz (1 of 2) 1. Particle P is in equilibrium with five (5) forces acting on it in 3-D space. How many scalar equations of equilibrium can be written for point P? A. 2 B. 3 C. 4 D. 5 E. 6 Copyright © 2022 Pearson Education, Inc. All Rights Reserved Reading Quiz (2 of 2) 2. If a block is stationary, then the friction force acting on it is _______ A. ( ∑Fx ) i + ( ∑Fy ) j + ( ∑Fz ) k = 0 B.  F = 0 C.  Fx =  Fy =  Fz = 0 D. All of the above. E. None of the above. Copyright © 2022 Pearson Education, Inc. All Rights Reserved