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**Force, Moment, Couple and Resultants**

2 Force Systems Force, Moment, Couple and Resultants

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**Objectives Students must be able to #1**

Course Objective Describe the characteristics and properties of forces and moments, analyse the force system, obtain the resultant and equivalent force systems. Chapter Objectives Use mathematical formulae to manipulate physical quantities Obtain position vectors with appropriate representation. Use and manipulate force vectors Use and manipulate moment vectors Analyse the force system resultants Describe and obtain equivalent systems

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**Force Definition Force is a vector quantity (why?)**

Force is the action of one body on another. [Statics] Force is an action that tends to cause acceleration of an object. [Dynamics] The SI unit of force magnitude is the newton (N). One Newton is equivalent to one kilogram-meter per second squared (kg·m/s2 or kg·m · s – 2) Examples of mechanical force include the thrust of a rocket engine, the impetus that causes a car to speed up when you step on the accelerator, and the pull of gravity on your body. Force can result from the action of electric fields, magnetic fields, and various other phenomena. 3 3

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**FORCE SYSTEMS Force is a vector**

Line of action is a straight line colinear with the force Force System: concurrent if the lines of action intersect at a point parallel if the lines of action are parallel y coplanar if the lines of action lie on the same plane x

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**Writing Convention Hand Print Scalar Vector Unit Vector**

Magnitude of Vector same symbol In this course, you have to write in this convention. Recommended Style

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**FORCE SYSTEMS 2-D Force Systems 3-D Force Systems Moment Moment Couple**

Vector (2D&3D) Basic Concept 2-D Force Systems 3-D Force Systems Moment Couple Resultants Moment Couple Resultants

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**Free Vectors: associated with “Magnitude” and “Direction”**

Representation parallelogram or V Magnitude: or V Vector : : Direction triangle +

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Vector Operation Addition #5 Commutative

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Vector Operation Addition #6 Associative

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**Operation Scalar Multiplication #2**

wrt = with respect to associative distributive wrt scalar addition distributive wrt vector addition

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**Component Resolution of a Vector**

A vector may be resolved into two components.

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**Basic relations of Triangle (C/6, law of cosine, sine)**

Law of sine b

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**a c b Hint Given V, and , find Law of cosine Law of sine 2 1 b b**

q b c b Given V, and , find Law of cosine (Law of sine) Law of sine

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**Vector Component and Projection**

b : vector components of (along axis a and b) a : projections of (onto axis a and b) = b special case: projection vectors are orthogonal to each other a : orthogonal projections & vector components

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**Rectangular Components**

vector component = vector projection Most commonly used y q x

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Fx=? Fy=? y F x p-b minus (b>90) b x y b-q y x b y x

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**EXAMPLE 2-1 y T ANS x kN Given the magnitude of the tension**

in the cable, T = 9 kN, express T in terms of unit vector i and j T x y 3 S.F. Correct? kN ANS

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**(a) parallel and perpendicular to arm AB**

We are using robot arm to put the cylindrical part into a hole. Determine the components of the force which the cylindrical part exerts on the robot along axes (a) parallel and perpendicular to arm AB (b) parallel and perpendicular to arm BC P = 90 N par per Defining direction P = 90 N per par arm AB ANS arm BC ANS

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**Vector Component (Algebraic)**

2/2 Combine the two forces P and T, which act on the fixed structure at B, into a single equivalent force R Graphics P=800 N (8cm) R T=600 N (6cm) Geometric P T R Vector Component (Algebraic) Correct? Point of application is B

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**Two forces is not acting**

Example Hibbeler Ex 2-1 #1 Determine the magnitude and direction of the resultant force. Two forces is not acting at the same point. Geometric

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**Vector Component (Algebraic)**

Geometric Good? (get full score?) - more explanation - mark answer - 5S.F. Then 3S.F.

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Good Answer Sheet Geometric O a

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Point of Application

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Example Hibbeler Ex 2-6 #1 Vector

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Example Hibbeler Ex 2-6 #2 Vector

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**Reference axis (very very important)**

Many problems do not come with ref. axis. Assignment based on convenience/experience Originally pass through O Vector summation (addition) Three ways to be mastered y x o F1y F1x F2y F2x 1. Graphically Ry Rx 2. Geometrically 3. Vector component (algebraically) The calculations do not reveal the point of application of the resultant force. In case where forces do not apply at the same point of application, you have to find it too!

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Recommended Problem 2/9, H2-17, 2/12, 2/26, H2-28

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**Three Dimensional Coordinate System**

Real-life Coordinate System is 3D. Introduce rule for defining the 3rd axis - “right-hand rule”: x-y-z - for consistency in math calculation (cross vector) z x How does 2D differs from 3D? y 2D z x

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**Rectangular Components (3D)**

projection & component z y x - cos(x), cos(y), cos(z) : “directional cosines” of (maybe +/-) - cos2(x)+cos2(y)+cos2(z) = 1 is a unit vector in the direction of - If you known the magnitude and all directional cosines, you can write force in the form of directional cosine Method

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Example Hibbeler Ex 2-8 Find Cartesian components of F z x y

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**Given the cable tension T = 2 kN. Write the vector expression of**

1) directional cosine method x y z A B Real directional cosine B directionl cosine = -0.92 A

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x y z A B A B x y z A B B A Thus, ANS

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**Directional Cosines by Graphics**

cos2(x)+cos2(y)+cos2(z) = 1

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**(a) Two points on the line of action of force is given (F also given).**

- Usually, the direction of force is not given using the directional cosines. Need some calculation. - Two examples (a) Two points on the line of action of force is given (F also given). z B (x2, y2, z2) Position vector Two-Point Method A (x1, y1, z1) y x

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z 0.5 2) 2-point construction 0.4 y B A 0.3 1.2 x kN Ans

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**ANS Write vector expression of . Also determine angle x, y, z, of T**

with respect to positive x, y and z axes Consider: T as force of tension acting on the bar where = unit vector from B to A Thus ANS

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Example Hibbeler Ex 2-9 #1 Vector Determine the magnitude and the coordinate direction angles of the resultant force acting on the ring

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Example Hibbeler Ex 2-9 #2 Vector

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Example Hibbeler Ex 2-11 #1 Vector Specify the coordinate direction angles of F2 so that the resultant FR acts along the positive y axis and has a magnitude of 800 N.

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Example Hibbeler Ex 2-11 #2 Vector

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Example Hibbeler Ex 2-11 #3 Vector

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Example Hibbeler Ex 2-15 #1 Force The roof is supported by cables as shown. If the cables exert forces FAB = 100 N and FAC = 120 N on the wall hook at A as shown, determine the magnitude of the resultant force acting at A.

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Example Hibbeler Ex 2-15 #2 Force

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Example Hibbeler Ex 2-15 #3 Force

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Example Hibbeler Ex 2-15 #4 Force

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**Fx = Fxy cos() = F cos() cos() Fy = Fxy sin() = F cos() sin()**

(b) Two Angles orienting the line of action of force are given (, ) Othorgonal projection Method Resolve into two components at a time z y Fz = F sin() Fxy = F cos() Fx = Fxy cos() = F cos() cos() Fy = Fxy sin() = F cos() sin() x

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y x z F Fxy Fx Fy Fz 65o 50o Ans

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y x z T A C B 15o x TAB TZ Ans

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**2/110 A force F is applied to the surface of the sphere as shown.**

The 2 angles (zeta, phi) locate Point P, and point M is the midpoint of ON. Express F in vector form, using the given x-,y- z-coordinates.

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Recommended Problems 3D Rectangular Component: 2/99 2/ / /110

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**Operation Products Dot Products Cross Products Mixed Triple Products**

Vector Dot Products Cross Products Mixed Triple Products

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scalar product (unit vector) ( three orthogonal vector )

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**Application of Dot Operation**

Angle between two vectors Component’izing Vector line which direction?

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y x z F Fxy Fx Fy Fz 65o 50o Ans

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y x z T A C B 15o x TAB TZ which direction?? Ans

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**“Cross Product” of Vectors**

right-hand rule (A then B) line which are perpendicular with both vectors

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**Operation Cross Product**

Laws of Operations Commutative Law is not valid Associative wrt scalar multiplication Distributive wrt vector addition

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**x-y-z complies with right-hand rule**

+ z x

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**How to calculate cross product**

This term can be written in a determinant form

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Cross Product - - - + + +

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**Why cross product? Mathematical Representation of Moments, Torque**

Perpendicular Direction Area Calculation z A C y B Area = ? O x

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Mixed Triple Product

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**Why mixed triple product?**

Why mixed triple product? Mathematical Representation of Moments along the axis. O Volume Calculation Volume must always + Volume = ? Area * Height

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**Operation Product Summary**

Vector Dot Product Scalar Cross Product Vector Mixed Triple Product Scalar

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Homepage URLs Statics official HP (User: Prince Password: Caspian) Session 1 HP (after the end of registration period)

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**FORCE SYSTEMS 2-D Force Systems 3-D Force Systems Moment Moment Couple**

Vector Basic Concept 2-D Force Systems 3-D Force Systems Moment Couple Resultants Moment Couple Resultants

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**Force Definition Force is a vector quantity (why?)**

Force is the action of one body on another. [Statics] Force is an action that tends to cause acceleration of an object. [Dynamics] The SI unit of force magnitude is the newton (N). One newton is equivalent to one kilogram-meter per second squared (kg·m/s2 or kg·m · s – 2) Examples of mechanical force include the thrust of a rocket engine, the impetus that causes a car to speed up when you step on the accelerator, and the pull of gravity on your body. Force can result from the action of electric fields, magnetic fields, and various other phenomena.

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**Force Representation Use different colours in diagrams**

Body outline blue Load red Miscellaneous black (dimension, angle, etc.) Vector quantity Magnitude Direction Point of application 10 N

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Type of Forces Applied force External force Reactive force Force Stress Internal force Strain Concentrated Contact force Force Force Distributed Body force

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Force Cables & Springs Cable in tension

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2/2 Combine the two forces P and T, which act on the fixed structure at B, into a single equivalent force R P=800 N (8cm) Graphical R T=600 N (6cm) Geometric P T R Algebraic Correct? Point of application is B

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**How to add sliding vectors (forces)?**

Principle of Transmissibility is applied at point A A Point of application Not OK. ! Still OK. Point of Application is wrong A A

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**Special case: Addition of Parallel Sliding Force**

Point of application line of action R2 R R1 R2 R1 R R This graphical method can be used to find Line of action The better and efficient way will be discussed later, when we learn the concept of “moment”, “couple”, and “resultant force” 80

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**T VD1 Ty Tx x y 60 Ans Move all forces to that concurrent point**

Point of application, But no physical meaning Ans Application Point

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**How to add sliding vectors (forces)?**

is applied at point A Point of application There is better way to find the point of application (or line of action), but you have to learn the concept of moment and couples first.

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Moment In addition to the tendency to move a body, force may also tend to rotate a body about an axis (magnitude) summation From experience (experiment) magnitude depends only on “F” and “d” moment axis Direction Moment is a vector

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**Moment Definition Moment is a vector quantity. Magnitude Direction**

x y z O Moment is a vector quantity. Magnitude Direction Axis of Rotation The unit of moment is N·m The moment-arm d (perpendicular distance) The right-hand rule determined by vector cross product Sign convention: 2D +k or CCW is positive. Moment of a force or torque

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**Mathematical Definition (3D)**

from A to point of application of the force Moment about point A : -Magnitude: a -Direction: right-hand rule X r -Point of application: point A A d (Unit: newton-meters, N-m) 2D - 2D, need sign convention and be consistent; e.g. + for counter- clockwise and – for clockwise M=Fd d +

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**sum of moment (of each force) = moment of sum (of all force)**

Varignon’s Theorem (Principle of Moment) can be used with more than 2 components Same? The moment of a force about any point is equal to the sum of the moments of the components of the force about that point sum of moment (of each force) = moment of sum (of all force) Useful with rectangular components d2 Mo = -Fxd2+Fyd1 y O d1 + x

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**Principle of Transmissibility & Moment**

Principle of Transmissibility is based on the fact that “moving force along the line of action causes no effect in changing moment” position vector: from A to any point on line of action of the force. O convenient a X r A d Y Z - direction: same - magnitude: M = Fr sin a = Fd Sliding force has the same moment O

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Sample 2/5 Calculate the magnitude of the moment about the base point O of 600N force in five different ways. 2m A 400 2m 4m 600N A d 400 y x O 600N Solution II: 3D Vector Approach 4m Solution I: 2D Scalar Approach O CW or CCW? CW Correct? CW

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**Solution III: Varignon’s theorem**

F1 B 2m F1 F2 A 400 F2 600N 2m A 400 4m d1 600N 4m F1 O C O d2 F2 Solution III: Varignon’s theorem + Solution V: Transmissibility Solution IV: Transmissibility

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EXAMPLE 2.8 In raising the flagpole, the tension T in the cable must supply a Moment about O of 72 kN-m. Determine T. o 12 m 30 m d 15 m ANS

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**Example Hibbeler Ex 4-7 #1 Correct?**

Moment Determine the moment of the force about point O. Correct?

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**Scalar Approach (Varignon’s theorem)**

Example Hibbeler Ex 4-7 #2 Moment 3D Vector Approach Scalar Approach (Varignon’s theorem)

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Couple - Couple is a summed moment produced by two force of equal magnitude but opposite in direction. a d M = F(a+d) – Fa = Fd magnitude does not depend on distance a (point O), i.e. any point on the body has the same magnitude. Effect of Pure Rotation + O - tendency to rotate the “whole” object. - no effect on moving object as translation. 2D representations: (Couples) C C C couple is a free vector

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**Moment Couple Definition #2**

Moment of a Couple B A O A couple moment is a free vector It can act at any point since M depends only upon the position vector r directed between the forces.

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**Force-couple systems **

- Line of action of a force on a body may be changed if a couple is added to compensated for the change in the tendency to rotate of that body. No changes in net external effect A B A B B d Principle of transmissibility A Force-couple system The direction and magnitude of Force can not be changed, only line of action (i.e. only change to other pararell line) Procedure may be reversed to combine a force with a couple

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**F A B B F C A C A B F B A F F F A B A B from new location (B)**

to old location (A) C A C A B F B A F F No Moment: Principle of Transmissibility F A B A Principle of Transmissibility is based on the fact that moving force along the line of action causes no effect in changing moment B

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**Why using equivalent system?**

B B Principle of transmissibility A Force-couple system All force systems are equal. real (physical) system In the viewpoint of Mechanics, Result of force to these systems are equal equivalent system equivalent system

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**Understanding Force-Couple system**

Moment about point B of force F = tendency of force F to rotate the object at point B couple occurs when moving Force F from A to B ( couple occurs when moving Force F parallel to its line of action to the point B) Equivalent System A B B D D A

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**Be careful of the direction of moment**

P Vector Diagram F 12m P Ans

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**2/11 Replace the force F by an equivalent force-couple system at point O.**

x y 50 kN 0.25 m 0.1m 50 kN M Couple occurred when moving F to O = Moment of F about O CCW Correct? Ans

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**Engine number 3 fails. Determine the force-couple system on the body about point o.**

Moving all 3 forces to point O (direction: left) couples occuring when moving forces. x y sum of moments? + (CW) ANS Sum of couples Got the meaning?

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Example Hibbeler Ex 4-14 #1 Resultant Replace the current system by an equivalent resultant force and couple moment acting A.

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Example Hibbeler Ex 4-14 #2 Resultant

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b cos20 exactly cancelled b 300N 20o 60 N-m D Ans

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2/6 Simplest Resultant Resultant of many forces-couple is the simplest force-couple combination which can replace the original forces/couples without changing the external effects on the body they act on y x q Point of application -Add two at a time get line of action of -Add many do not get line of action of

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**Easier way to get a resultant + its location**

F1d1 2) Replace each force with a force at point O + a couple F2d2 F3d3 3) Add forces and moments O Mo=(Fidi) O arbitrary d1 d2 d3 1) Pick a point (easy to find moment arms) (forces + couples : same procedures) resultant d=Mo/R 2D any forces + couples system O single-force system (no-couple) Mo=Rd or single-couple system 4) Replace force-couple system with a single force 3D any forces + couples system single-force + special single-couple (wrench)

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**2/87 Determine the resultant and its line of action of the following three loads.**

why? Move 3 forces to point O, Sums their force and couples M (force-couple system) O R Note: M depends on the location where we move the force to + M = -2.4*0.2cos *0.12cos20 -3.6*0.3cos20 kN-m Note: R is the same regardless with the location point we move the force to R = ( 2.4cos sin cos20 ) i +( -2.4sin cos cos20 ) j kN move the R to point X where Resultant Moment is zero find the point X where the Resultant Moment is zero R O X M + N = 0 But we want to find the line of action of the “pure” resultant force (the one which has no additional couple)

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**M R O (0,0) P (x,y) M O N R P R M = -1.635 kN-m + Sys 1 couples**

At point O (0,0) Sys 1 M O R M = kN-m P At point X (x,y) O (0,0) P (x,y) couples cancelled Correct? Sys 2 M Two equivalent systems Moment at any point must be the same on both system O Pick Point O N R P R ( line of action )

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**Manually Canceling Couples**

At point O (0,0) + M O R P M = kN-m d Manually Canceling Couples How to locate Point P How to find line of action ? O (0,0) d O P d or P

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**Equivalent System Definition**

R P R Two force-couple systems are equivalent

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A car stuck in the snow. Three students attempt to free the car by exert forces on the car at point A, B and C while the driver’s actions result in a forward thrust of 200 N as shown in picture. Determine 1) the equivalent force-couple system at the car center of mass G 2) locate the point on x-axis where the resultant passes. x y G

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**ANS For line of action of resultant y x y b x G G**

Sys II Sys I Couple Cancellation At y = 0; x = m. ANS Two equivalent systems Moment at point G must be the same on both system If you want to find only b (not line of action itself) Two equivalent systems (2D) + or - , you have to find out manually

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**Determine the resultant (vector) and the point on x and y axes which must pass.**

G

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**ANS y y x x For line of action of resultant O O If y = 0; x = 7.42 m.**

x = 0; y = m. ANS

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