ENGR-36_Lec-05_Force_Resultants-2.ppt 1 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Engineering 36 Chp 4: Force Resultants (2)

ENGR-36_Lec-05_Force_Resultants-2.ppt 2 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Scalar (Dot) Product of 2 Vectors  The SCALAR Product or DOT Product Between Two Vectors P and Q Is Defined As  Scalar Product Math Properties ARE Commutative ARE Distributive Are NOT Associative –Undefined as (PQ) is NO LONGER a Vector

ENGR-36_Lec-05_Force_Resultants-2.ppt 3 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Scalar Product – Cartesian Comps  Scalar Products With Cartesian Unit Components  Thus

ENGR-36_Lec-05_Force_Resultants-2.ppt 4 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Scalar Product - Applications  Angle Between Two Vectors  Projection Of A Vector On A Given Line  For Any Axis Defined By A Unit Vector

ENGR-36_Lec-05_Force_Resultants-2.ppt 5 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Vector Magnitude by DOT  A vector DOTed with itself reveals the Square of the Phythagorean Length  Thus the Vector Magnitude  This is IDEAL forMATLAB >> Pv = [ ] % [Px*i Py*j Pz*k] Pv = >> Pm = sqrt(dot(Pv,Pv)) Pm =

ENGR-36_Lec-05_Force_Resultants-2.ppt 6 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics DOT-Prod Application Summary  Given Two intersecting Vectors or Lines  Parallel & Perpendicular Components Given Vector V AB, and line AC find the || & ┴ Components of V AB, V AD & V DB, relative to line AC

ENGR-36_Lec-05_Force_Resultants-2.ppt 7 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics DOT-Prod Application Summary  First Calc θ by method of the previous slide  Then Simply Use Trig on Right-Triangle ADB

ENGR-36_Lec-05_Force_Resultants-2.ppt 8 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example: P2-120 by MATLAB  Determine the magnitudes of the components of F = 600N acting along and perpendicular to segment DE of the pipe assembly  Notes The Angle θ between DE & EB (the direction of F) appears to be OBTUSE F par F perp

ENGR-36_Lec-05_Force_Resultants-2.ppt 9 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example: P2-120 by MATLAB % Bruce Mayer, PE % ENGR36 * 18Jul2 % ENGR36_parNperp_Projection_H13e_P2_120_1207.m % % Magnitude of a vector by ANON fcn MagV sqrt(dot(z,z)) % % Find unit vector along EB, the Force Direction EBv = [ ] % in m => [delX*i delY*j delZ*k] EVm = MagV(EBv) uEB = EBv/EVm % % Find unit Vector along Pipe Segment DE DEv = [0 3 0] DEm = MagV(DEv) uDE = DEv/DEm % % Angle between the unit vectors Q = acosd(dot(uEB,uDE))% in ° % Fm = 600 % in Newtons % % the PARALLEL projection of F on DE Fpar = Fm*cosd(Q) % the PERPENDICULAR projection of F on DE Fperp = Fm*sind(Q) % disp(' ') disp('======================================') disp('Chk by finding F against ED (the opposite of DE)') % Find unit Vector along Pipe Segment DE EDv = [0 -3 0] EDm = MagV(EDv) uED = EDv/EDm % Qchk = acosd(dot(uEB,uED))% in ° FparChk = Fm*cosd(Qchk) FperpChk = Fm*sind(Qchk) Q = Fpar = Fperp = ==================================== Chk by finding F against ED (the opposite of DE) Qchk = FparChk = FperpChk =

ENGR-36_Lec-05_Force_Resultants-2.ppt 10 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics WhiteBoard Work Let’s Work Some “Angle” Problems

ENGR-36_Lec-05_Force_Resultants-2.ppt 11 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics T BC = 5.3 kN

ENGR-36_Lec-05_Force_Resultants-2.ppt 12 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics