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ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
1
ME 2560 STATICS
Chapter IV. Resultant of a System of Forces
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
2
1. Moment of a Force – Scalar FormulationThe moment of a force about a point or axis provides a measure of thetendency of the force to produce rotation on a body about the point oraxis. In the figure, Fx acts perpendicular to the handle of the wrench andis located at a distance dy from the point O.
Moment created by force Fx about axis z.
dyFx
x
y
z
(Mo)z
Fx tends to cause the pipe tospin around the z axis. Thelarger Fx or dy the greaterthe spinning effect. Thistendency of rotation causedby Fx is called torque,moment of a force ormoment. The moment axis(z) is perpendicular to theplane formed by the vectorof the force and the vectorof the distance dy.
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
3
Applying Fz to the wrench as shown below, Fz will result in a rotationeffect on the pipe around x. It may not be possible to turn the pipearound x but still, Fz creates a tendency for rotation and thus, themoment (Mo)x is produced. The force vector, Fz, and the distance(position vector), dy, are in the same plane and the moment (Mo)x isperpendicular to this plane.
xdy Fz
Moment created by force Fz about axis x.
y
z
(Mo)x
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
4
Finally, if a force Fy is applied to the wrench, as shown below, nomoment is produced about point O. This is because the line of action ofthe force passes exactly through z–axis and therefore the distance thatcreates the moment is zero.
Fy
Moment created by force Fy is zero.
y
z
x
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
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Thus, the moment Mo about the point O or about the axis passingthrough O is equal to the magnitude of the force F times theperpendicular distance d comprised between the point where the force isapplied and the point O (or axis O).
Fd
MoF
Mo O
d
O
Schematic of the concept for the moment of a force.
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
6
The Moment of a force has both magnitude and direction and thus it isalso a vector quantity.
The magnitude of the moment of a force F that has a moment arm of dunits is
The moment arm is the perpendicular distance from the axis at point Oto the line of action of the force.
The units for the moment are Nm in the SI system and lbft in theEnglish system.
The direction of the Moment MO is specified using the right hand rule.
dFM O
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
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Resultant Moments of a System of Coplanar ForcesIf a system of forces lies in the plane x–y, then, the moment produced byeach force about point O will be directed along axis z.
x
y
System of coplanar forces in the x – y plane producing a vector moment directed in the z direction.
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
8
The resultant moment MRO of the system can be determined by simplyadding the moments of all the forces algebraically since all themoment vectors are collinear.
dFM RO
The counterclockwise curl shown at the left of the equation expressesthat by the scalar sign convention, the moment of any force will bepositive if it is directed along the z axis (counterclockwise), andnegative if it is directed along the –z axis (clockwise).In summary:•A moment is a turning effect.• A moment always acts about a given point and is either clockwise orcounterclockwise in nature.• The moment about a point O caused by F is defined as the F times theperpendicular distance from the force’s line of action to the point.
Examples
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
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2. Cross ProductThe cross product of two vectors A and B yields a third vector C that isperpendicular to A and B. This operation is expressed as
The magnitude of C is defined as the product of the magnitudes of Aand B and the sine of the angle θ between their tails (0o θ 180o).
C = AB sinθ
Direction. Vector C has a direction that is perpendicular to the planecontaining A and B such that C is specified by the right hand rule.
BAC
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
10
If the magnitude and direction of C are know then we can express
The scalar ABsinθ defines the magnitude of C and the unit vector ucdefines the direction of C. The following figure illustrates the concept ofthe cross product defined above.
CuABBAC ˆ)sin(
C=AB sinθuC
AB
C
θ
The cross product of two vectors yields a third vector perpendicular to a plane containing the two original vectors.
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
11
Laws of Operation1. Commutative law is not valid
2. Multiplication by a scalar
3. The distributive law
ABBA
ABBA
aBABaABAaBAa )()()()(
)()()( DABADBA
In cross product operations is very important to maintain the proper order of the vectors since they are not commutative.
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
12
To determine the cross product of Cartesian unit vectors the right handrule can be used if it is kept in mind that the magnitude of a unit vector is“1” and that the cross product results in a third vector perpendicular tothe two original vectors, thus
Cartesian Vector Formulation
0ˆˆˆˆˆˆˆˆ0ˆˆˆˆˆˆˆˆ
0ˆˆˆˆˆˆˆˆ
kkjkijik
jjijkikj
iikijkji
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
13
Thus the cross product of two vectors A and B is obtained as
kbjbibkajaiaBA
kbjbibBkajaiaA
zyxzyx
zyxzyx
ˆˆˆˆˆˆ
ˆˆˆˆˆˆ
kbabajbabaibabaBA xyyxxzzxyzzyˆˆˆ
A more convenient and conventional form to express the previous expression is:
zyx
zyx
bbbaaakji
BA
ˆˆˆ
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
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3. Moment of a Force–Vector FormulationThe moment of a force F about an axis that passes through point O isperpendicular to the plane containing r and F. This moment can bedetermined using vector cross product:
FrMO
Moment axis
Mo
OrF
A
Moment of force F about axis O.
In this equation, r represents aposition vector going from Oto any point lying on the lineof action of F
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
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The magnitude of the moment Mo is given by the definition of the crossproduct of two vectors
Mo = Fr sin θ
θ is measured between the tails of the two vectors.
Magnitude of a Moment
θθ
Mo
O
d
rr
FCross product of vector force and position vector results
in the moment vector.
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
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The direction and the sense of Mo are determined by the right hand rule.Direction of a Moment
Principle of TransmissibilityThe principle of transmissibility states that a force may be applied at any point on its given line of action without altering the resultant effects of the force external to the rigid body on which acts.
When only the resultant external effects of a force are to be investigated the force may be treated as a sliding vector, and it is necessary and sufficient to specify the magnitude, direction and line of action of the force.
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
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If a force and a vector are expressed in Cartesian coordinates themoment vector produced by the force can be found as
rx, ry, and rz are the components of the position vector drawn from thepoint O to any point on the line of action of the force F whosecomponents are Fx, Fy, and Fz. Expansion of the determinant results in
Cartesian Vector Formulation
zyx
zyxO
FFFrrrkji
FrM
ˆˆˆ
kFrFrjFrFriFrFrM xyyxxzzxyzzyOˆ)(ˆ)(ˆ)(
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
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If a system of forces is acting on a body as shown below, the resultantmoment of the forces about the point O can be determined by vectoraddition of the moment produced by each force:
Resultant Moment of a System of Forces
FrM RO
Resultant Moment of a system of forces acting on a rigid body.
x
y
z
F1
F2 F3
r1
r2r3 MRO
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
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4. Principle of MomentsThe principle of moments often named also Varignon’s theorem since it was developed by the French mathematician Varignon (1654–1722) states that:
The moment of a force about a point is equal to the sum of the moments of the force’s components about the point.
Examples
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
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5. Moment of a Force about a Specified Axis
When the moment of a force is computed about a point, the moment andits axis are always perpendicular to the plane containing the force andthe moment arm.
To determine the component of this moment along a specified axis thatpasses through the point, we can use two approaches:
•Scalar analysis
•Vector analysis.
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
21
Scalar AnalysisThe force of 20 N that is applied to the pipe on the x–y plane produces a totalmoment Mo.The force is directed in the negative z–direction and it is applied at a location of0.3 m in the x–direction and 0.4 m the y–direction. This point of applicationproduces a double effect.1) Due to the location in the x–direction the force tends to unscrew the pipe.2) Due to the moment arm in the y–direction, the force generates a tendency
for the pipe to rotate around the x–axis.F = 20 N. Force produces a moment with two components:1) Moment in the y–direction: My = F dx = (20 N) (0.3 m) = 6 N–m (+)2) Moment in the x–direction: Mx = F dy = (20 N) (0.4 m) = 8 N–m (–)
Thus, the moment that produces tendency to unscrew the pipe is My
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
22
Vector Analysis
This same problem can be analyzed using vector algebra directly. Firstwe need to determine the position vector form the origin to point Aand also notice that the force is expressed in vector form as = –20k.
= 0.3 i + 0.4 jFr
r
m]-[Nˆ8ˆ6
)ˆ20()ˆ4.0ˆ3.0(
ijM
kjiFrM
Examples
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
23
6. Moment of a CoupleA couple is defined as two parallel forces with •Same magnitude•Opposite directions •Separated by a perpendicular distance d
The resultant force is zero, thus the only effect of the couple is toproduce rotation or tendency of rotation in a specific direction.
–F
F
d
Schematic of the definition for a couple.
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
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The moment produced by a couple is named couple moment and its value can be determined by finding the sum of the moments of both couple forces at any arbitrary point:
FrFrM BA
)(
rB rA
F –F
ODetermination of the moment of a couple in space.
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
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The computation of a couple moment can be done by simply finding the moment about a point lying on the line of action of one of the forces produced by the remaining force.
F
–F
O
A
BrA
rB
rA-B
FrM
FrM
FrFrM
ABo
BAo
BAo
)(
)(
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
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Equivalent Couples
Two couples are said to be equivalent if they produce the same moment.
Since the moment produced by a couple is always perpendicular to theplane containing the couple forces, it is therefore necessary that theforces of equal couples lie in the same plane or in parallel planes.
Resultant Couple Moment
Since couple moments are free vectors, they may be applied to any point P on a body and added vectorially.
Examples
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
27
7. Equivalent SystemA force applied on a body has two effects, it can produce:•Translation•RotationThe degree of these effects depends on where and how the force isapplied.1) Line of Action of Force Passes through Point O
= =
O
AF
O
AF
F
–F
O
A
F
Reduction of a force acting on a point without moment arm with respect to O.
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
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2) Line of Action of Force Produces a Moment about Point O.
=
O
A
F
A
F
F
–F O O
AF
M=rF
Reduction of a force acting on a point with moment arm with respect to O.
=
ME2560 – Statics
Chapter IV. Resultant of a System of Forces
Spring 2020
29
8. Resultant of a Force and a Couple SystemA system of forces and couple moments acting on a rigid body can bereplaced by an equivalent single resultant force acting at a specifiedpoint O and a resultant couple moment.
O
F1F2
MC
r1r2
A system of forces and moments applied to a rigid body can be replaced by one resultant force and
one resultant moment.
OCRo
R
MMM
FF
Examples
