MMET 275 502 MECHANICS FOR TECHNOLOGISTS Department of Engineering Technology and Industrial Distribution Texas A M University Spring 2023 MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 1 REVIEW 1 Problem 1 Conceptual Questions 10 points 10 points 10 points Problem 2 Force Vectors 35 points Problem 3 Equilibrium of a Particle 35 points MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 2 2 VECTOR NOTATION USED IN THIS CLASS An overbar arrow will be used to designate a general vector For example a force vector will be written as A unit vector a dimensionless vector having a magnitude of 1 will be denoted with an overbar hat For example the unit vectors used for a set of x y and z coordinate axes will be denoted as respectively The magnitude of a vector will be denoted as the scalar or simply MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 3 F and ijkFFF VECTOR OPERATIONS Vector Addition MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 4 VECTOR ADDITION OF FORCES Procedure for Analysis Trigonometry Law of cosines Law of sines MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 5 ADDITION OF A SYSTEM OF COPLANAR FORCES Coplanar Force Resultants Vector resultant is therefore If scalar notation is used MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 6 123 RRxRyFFFFFiFj yyyRyxxxRxFFFFFFFF321321 CARTESIAN VECTORS Direction of a Cartesian Vector Orientation of A is defined as the coordinate direction angles and measured between the tail of A and the positive x y and z axes 0 180 The direction cosines of A are MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 7 AAx cosAAy cosAAz cos CARTESIAN VECTORS Direction of a Cartesian Vector uA can also be expressed as uA cos i cos j cos k Since and uA 1 we have A as expressed in Cartesian vector form is A AuA Acos i Acos j Acos k Axi Ayj AZk MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 8 222zyxAAAA 1coscoscos222 EXAMPLE 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 MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 9 THE DOT PRODUCT OF TWO VECTORS MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 10 EQUILIBRIUM OF PARTICLES ANALYSIS STRATEGY The following is a set of formal steps that will enable us to solve many types of problems in equilibrium analysis Draw your FBDs 1 Draw outlined shape 2 Show all the forces applied forces weight forces 3 Identify each force forces of reaction from supports and or other bodies in the system Known forces with proper magnitude and direction Letters used to represent magnitude and directions MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 11 EQUILIBRIUM OF PARTICLES ANALYSIS STRATEGY Writing down your equilibrium equations 1 Choose a set of coordinate axes that best fits the constraints of the problem 2 Write down the components of the forces in your FBD that are aligned with your choice of coordinate axes These projections will likely involve the usage of direction angles or spherical angles is setting up the unit vectors for the forces particularly in three dimensional problems Verifying the solvability of your equilibrium equations Count the number of scalar equilibrium equations and the number of unknowns in these equations If the number of unknowns exceeds the number of equations you need to look for other bodies in the system for which FBDs are needed Solve MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 12 PRACTICE MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 13 PRACTICE MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 14