Force System Resultants4

STATICSAssist. Prof. Dr. Cenk Üstündağ

Chapter Objectives

• Concept of moment of a force in two and three dimensions

Chapter Outline

1. Moment of a Force – Scalar Formation2. Cross Product3. Moment of Force – Vector Formulation4. Principle of Moments

4.1 Moment of a Force – Scalar Formation

When a force is applied to a body it will produce a tendencyfor the body to rotate about a point that is not on the line ofaction of the force. This tendency to rotate is sometimescalled a torque, but most often it is called the moment of aforce or simply the moment.

If a force is applied to the handle ofthe wrench it will tend to turn thebolt about point O (or the z axis).The magnitude of the moment isdirectly proportional to themagnitude of F and theperpendicular distance or momentarm d.

4.1 Moment of a Force – Scalar Formation

As shown, d is the perpendicular distance from point O to the line of action of the force.

In 2-D, the direction of MO is either clockwise (CW) orcounter-clockwise (CCW), depending on the tendency forrotation.

In a 2-D case, the magnitude of the moment is MO = F d

4.1 Moment of a Force – Scalar Formation

Often it is easier to determineMO by using the components ofF as shown.Then MO = (FY a) – (FX b). Note the different signs on theterms! The typical sign convention for a moment in 2-D isthat counter-clockwise is considered positive. We candetermine the direction of rotation by imagining the bodypinned at O and deciding which way the body would rotatebecause of the force.

For example, MO = F d and the direction is counter-clockwise.

Fa

b

dO

abO

F

F x

F y

4.1 Moment of a Force – Scalar Formation

Resultant Moment • Resultant moment, MRO = moments of all the forces

MRO = ∑Fd

Example

For each case, determine the moment of the force about point O.

Solution

Line of action is extended as a dashed line to establish moment arm d.Tendency to rotate is indicated and the orbit is shown as a colored curl.

)(.200)2)(100()( CWmNmNMa o

Solution

)(.5.37)75.0)(50()( CWmNmNMb o

)(.229)30cos24)(40()( CWmNmmNMc o

Solution

)(.4.42)45sin1)(60()( CCWmNmNMd o

)(.0.21)14)(7()( CCWmkNmmkNMe o

4.2 Cross Product

While finding the moment of a force in 2-D isstraightforward when you know the perpendiculardistance d, finding the perpendicular distances can behard—especially when you are working with forces inthree dimensions.

So a more general approach to finding the moment of aforce exists. This more general approach is usually usedwhen dealing with three dimensional forces but can beused in the two dimensional case as well.

This more general method of finding the moment of aforce uses a vector operation called the cross product oftwo vectors.

4.2 Cross Product

Cross product of two vectors A and B yields C, which is written as

C = A X Band is read “C equals A cross B”

Magnitude • Magnitude of C is the product of

the magnitudes of A and B• For angle θ, 0º≤ θ ≤ 180º

C = AB sinθ

4.2 Cross Product

Direction • Vector C has a direction that is

perpendicular to the plane containingA and B such that C is specified bythe right hand rule.

• Expressing vector C whenmagnitude and direction are knownC = A X B = (AB sinθ)uC

As shown, uC is the unit vectorperpendicular to both A and Bvectors (or to the plane containingthe A and B vectors).

4.2 Cross Product

Laws of Operations1. Commutative law is not valid

A X B ≠ B X ARather,

A X B = - B X ACross product A X B yields avector opposite in direction to C

B X A = -C

4.2 Cross Product

Laws of Operations2. Multiplication by a Scalar

a( A X B ) = (aA) X B = A X (aB) = ( A X B )a

3. Distributive LawA X ( B + D ) = ( A X B ) + ( A X D )

Proper order of the cross product must be maintained since they are not commutative

4.2 Cross Product

Cartesian Vector FormulationThe right-hand rule is a useful tool for determining the direction of the vector resulting from a cross product.

For example: i j = kNote that a vector crossed into itself is zero, e.g., i i = 0

4.2 Cross Product

Cartesian Vector FormulationSimilarly

i j = k i k = - j i i = 0j k = i j i = - k j j = 0k i = j k j = - i k k = 0

4.2 Cross Product

Cartesian Vector FormulationCross product of 2 vectors A and B

A B = (Axi + Ayj + Azk) (Bxi + Byj + Bzk)= AxBx(i i) + AxBy(i j) + AxBz(i k)+ AyBx(j i) + AyBy(j j) + AyBz(j k)+ AzBx(k i) + AzBy(k j) + AzBz(k k)

A B = (AyBz - AzBy)i - (AxBz - AzBx)j + (AxBy - AyBx)k

4.2 Cross Product

Cartesian Vector FormulationA B = (AyBz - AzBy)i - (AxBz - AzBx)j + (AxBy - AyBx)kThis equation may also be written in a more compact determinant form as

4.2 Cross Product

Cartesian Vector FormulationA determinant having three rows and three columns can beexpanded using three minors, each of which is multiplied byone of the three terms in the first row. There are fourelements in each minor, for example,

By definition, this determinant notation represents the terms(A11A22 - A12A21), which is simply the product of the twoelements intersected by the arrow slanting downward to theright (A11A22) minus the product of the two elementsintersected by the arrow slanting downward to the left(A12A21).

4.2 Cross Product

Cartesian Vector FormulationFor a 3 3 determinant, the three minors can be generatedin accordance with the following scheme:

4.3 Moment of Force - Vector Formulation

Moment of force F about point O can be expressed using cross product

MO = r FHere r represents a position vector directed from O to any point on the line of action of F.Magnitude• For magnitude of cross product,

MO = rF sinθ• Treat r as a sliding vector. Since d = r sinθ,

MO = rF sinθ = F (rsinθ) = Fd

4.3 Moment of Force - Vector Formulation

Direction• Direction and sense of MO are determined

by right-hand rule

Note that the “curl” of the fingers, like thecurl around the moment vector, indicates thesense of rotation caused by the force. Sincethe cross product does not obey thecommutative law, the order of r F must bemaintained to produce the correct sense ofdirection for MO.

4.3 Moment of Force - Vector Formulation

Principle of Transmissibility• For force F applied at any point A, moment created about

O is MO = rA F • F has the properties of a sliding vector, thus

MO = r1 F = r2 F = r3 F

4.3 Moment of Force - Vector Formulation

Moments in 3-D can be calculated using scalar (2-D)approach, but it can be difficult and time consuming. Thus, itis often easier to use a mathematical approach called thevector cross product.Using the vector cross product, MO = r F.

Here r is the position vector from point O to any point on the line of action of F.

4.3 Moment of Force - Vector Formulation

So, using the cross product, a moment can be expressed as

By expanding the above equation using 2 2 determinants, we get (units are N - m )

MO = (ryFZ - rZFy)i - (rxFz - rzFx)j + (rxFy - ryFx)k

The physical meaning of the above equation becomesevident by considering the force components separately andusing a 2-D formulation.

4.3 Moment of Force - Vector Formulation

Resultant Moment of a System of Forces• Resultant moment of forces about point O can be

determined by vector addition

MRO = ∑(r F)

Example

Two forces act on the rod. Determine the resultantmoment they create about the flange at O. Express theresult as a Cartesian vector.

Solution

Position vectors are directed from point O to each force as shown.These vectors are

The resultant moment about O is

m 254

m 5kjir

jr

B

A

mkN 604030 304080254

204060050

kji

kjikji

FrFrFrM BAO

4.4 Principles of Moments

• Also known as Varignon’s Theorem“Moment of a force about a point is equal to the sum ofthe moments of the forces’ components about thepoint”

• Since F = F1 + F2,

MO = r F= r (F1 + F2) = r F1 + r F2

Pierre Varignon (1654 – 1722) a French mathematician.

Example

Determine the moment of the force about point O.

Solution

The moment arm d can be found from trigonometry,

Thus,

Since the force tends to rotate or orbit clockwise about point O, the moment is directed into the page.

m 898.275sin3 d

mkN 5.14898.25 FdMO

Example

Determine the moment produced by the force F aboutpoint O. Express the result as a Cartesian vector.

Solution

Solution on whiteboard

Example

Force F acts at the end of the angle bracket in figure. Determine the moment of the force about point O .

Solution

Solution on whiteboard