[email protected] engr-36_lec-05_force_resultants-2.ppt 1 bruce mayer, pe engineering-36:...

[email protected] • ENGR-36_Lec-05_Force_Resultants-2.ppt1

Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engineering 36

Chp 4: ForceResultants

(2)



Scalar (Dot) Product of 2 Vectors

The SCALAR Product or DOT Product Between TwoVectors P and Q Is Defined As

resultscalarcosPQQP

PQQP

2121 QPQPQQP

undefined SQP

Scalar Product Math Properties• ARE Commutative• ARE Distributive• Are NOT Associative

– Undefined as (P•Q) is NO LONGER a Vector



Scalar Product – Cartesian Comps

Scalar Products With Cartesian Unit Components

Thus

000111 ikkjjikkjjii

kQjQiQkPjPiPQP zyxzyx

2222 PPPPPP

QPQPQPQP

zyx

zzyyxx



Scalar Product - Applications Angle Between Two Vectors

PQ

QPQPQP

QPQPQPPQQP

zzyyxx

zzyyxx

cos

cos

OL

OL

PPQ

QP

PQQP

OLPPP

cos

cos

along of projection cos

zzyyxx

OL

PPP

PP

coscoscos

ˆ

OL alongr unit vecto theis

Projection Of A Vector OnA Given Line

For Any Axis Defined By A Unit Vector



Vector Magnitude by DOT

A vector DOTed with itself reveals the Square of the Phythagorean Length

Thus the Vector Magnitude

This is IDEAL forMATLAB

2222 PPPPPP zyx

2222 PPPP zyx PPP

>> Pv = [-7 3 11] % [Px*i Py*j Pz*k]Pv = -7 3 11

>> Pm = sqrt(dot(Pv,Pv))Pm = 13.3791



DOT-Prod Application Summary

Given Two intersecting Vectors or Lines

Parallel & PerpendicularComponents• Given Vector VAB, and line

AC find the || & ┴ Components of VAB, VAD & VDB, relative to line AC

1800arccos BA

BA



DOT-Prod Application Summary

First Calc θ by method ofthe previous slide

Then Simply Use Trigon Right-Triangle ADB

ACABACAB arccos

sin

cos

ABDB

ABAD

VV

VV



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

• Fpar

• Fperp

cos|| FF

sinFF



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 = @(z) sqrt(dot(z,z))%% Find unit vector along EB, the Force DirectionEBv = [-4 -3 2] % in m => [delX*i delY*j delZ*k]EVm = MagV(EBv)uEB = EBv/EVm% % Find unit Vector along Pipe Segment DEDEv = [0 3 0]DEm = MagV(DEv)uDE = DEv/DEm%% Angle between the unit vectorsQ = acosd(dot(uEB,uDE))% in °%Fm = 600 % in Newtons%% the PARALLEL projection of F on DEFpar = Fm*cosd(Q)% the PERPENDICULAR projection of F on DEFperp = Fm*sind(Q)%disp(' ')disp('======================================')disp('Chk by finding F against ED (the opposite of DE)')% Find unit Vector along Pipe Segment DEEDv = [0 -3 0]EDm = MagV(EDv)uED = EDv/EDm%Qchk = acosd(dot(uEB,uED))% in °FparChk = Fm*cosd(Qchk)FperpChk = Fm*sind(Qchk)

Q = 123.8545

Fpar = -334.2516

Fperp = 498.2729

====================================Chk by finding F against ED (the opposite of DE)

Qchk = 56.1455

FparChk = 334.2516

FperpChk = 498.2729



WhiteBoard Work

Let’s WorkSome “Angle”

Problems

1

2

3

4



1050

1800

2400

1200

TBC = 5.3 kN



1050

1800

2400

1200

1050

1800

2400

1200

[email protected] engr-36_lec-05_force_resultants-2.ppt 1 bruce mayer, pe engineering-36:...

Documents

pe engineering

unit vector slide

previous slide

vector v ab

components of f

line ac slide

vectors projection

perpendicular projection