Statics
Chapter 2 Resultant of Coplannar Force Systems
2-1 Intro
• Two systems of forces = equivalent – if produce the same
mechanical effect on rigid body
• Single force that is equivalent to a force system is called the
resultant of the force system
2-2 Vector Representation
• Figure 2-1 represented graphically by a line segment AB with
an arrowhead at one end – length of the line segment AB
represents the magnitude of the force - direction is indicated
by the angle from the reference axis.
• Equal Vectors – two vectors having same magnitude and
same direction are said to be equal – figure 2-2
• Negative Vector – two vectors having the same magnitude and
opposite direction.
2-3 Resultant of Concurrent
Forces
• Parallelogram Law – Figure 2-4 shows two vectors are added
according to this law
• Triangle Rule – the sum of two vectors can also be determined
by constructing one half of the parallelogram or triangle
• To find the vector sum , we first lay out p at A then lay out Q from
the tip of P in a tip to tail fashion, the closing side of the triangle
represents the sum of the two vectors
• Examples 2-1
• Example 2-2
• Example 2-3
2-4 Rectangular Components
• Any two or more forces whose resultant is equal to a force F
are call the components of the force
• Two mutually perpendicular components are called the
rectangular components Fx and Fy are the rectangular
components of F in the x and y direction
• If the magnitude F and the direction angle of a force are
known then from the right triangle
• Fx=Fcos @ and Fy = Fsin@
• Counterclockwise measurement is regarded as positive, clockwise
measurement as negative
2-4 Rectangular Components
• Magnitude and Direction – when the scalar components Fx and Fy of
the force F are give , the magnitude of F may be determined from
formula 2-3 page 47
• And the reference angle from 2-4 page 47
• Depending on which quadrant the force vector is in the direction
angle in the standard position is
•
•
•
•
•
•
•
•
First quadrant 0=@
Second quadrant 0= 180 -@
Third quadrant 0=180 + @
Fourth quadrant 0=360-@
Example 2-4 page 47
Example 2-5 page 48
Example 2-6 page 49
Example 2-7 page 50
2-5 Resultants by Rectangular
Components
• The resultant of any number of concurrent coplanar forces can
be determined by using their rectangular components
•
•
•
•
•
Figure 2-9 R=F1 +F2+F3
Rx = F1x +F2x +f3x
Rx = F1cos @1 + F2cos @2 + F3cos @3
Ry = F1y +F2y +f3y
Rx = F1sin @1 + F2sin @2 + F3sin @3
• Example 2-8 page 52
• Example 2-9 page 53
2-6 Moment of Force
• Effects of a force – force tends to move a body along its line of
action – it also tends to rotate a body about an axis
• The ability of a force to cause a body to rotate is measured by
a quantity called the moment of the force.
• Wrench example – rotating moment also called the torque is
produced by the applied force depends not only on the
magnitude of the force but als on the perpendicular distance d
from the center 0 of the bolt to the lne of action of the force
• Definition of Moment – two dimensional case – moment of a
force about a point ie equal to the magnitude of the force
multiplied by the perpendicular distance from 0 to the line of
action of the force
• Formula 2-7 page 55
• Units of moment are Ib* ft or Ib * in or N* m or kN * m
2-6 Moment of a Force
• Direction of Moments – example 2-13 P is CCW and Q is CW
• Summation of Moments – CCW will be considered positive
CW will be considered negative.
• Example 2-10 page 56
2-7 Varignon’s Theorem
• States that the moment of a force about any point is equal to the
sum of the moments produced by the components of the forces
abut the same point
• The sum of the moments of the components must be the same as
the moment of the force itself
• Formula 2-8 page 59
• Principle of transmissibility
• The point of application of a force acting on a rigid body may be
place anywhere along its line of action – moment arm is clearly
independent of the point of application - therefore as long as the
magnitude the direction and the line of actin of a force are defined
the moment of a force about a given point may b determined by
placing the force at any point along its line of actin.
• Example 2-11
• Example 2-12
2-8 Couple
• Effect of a couple – two equal and opposite forces having parallel lines
of actin form a couple – the sum of the moment of the two forces
however is not zero – the effect of a couple acting on a rigid body
therefore is to cause the rigid body to rotate about an axis
perpendicular to the plane of the forces
• Moment of a couple – denoting the perpendicular distance between the
two forces by d , the moment of a couple about an arbitrary point 0 is
equal to formula 2-9 page 63
• Since 0 is an arbitrary point – the moment of a couple about any point is
equal to the magnitude of the forces times the perpendicular distance
between the forces.
• Equivalent couples – two couples acting on the same plane or parallel planes
are equivalent if they have the same moment acting in the same direction
• Addition of Couples – addition of two or more couples in a plane or parallel
planes is the algebraic sum of their moments
• Example 2-13 page 63
• Example 2-14 page 64
2-9 Replacing a Force with a
Force couple system
• Two systems of forces are said to be equivalent if they
produce the same mechanical effect on a rigid body –
• Equivalent force systems – are said to be equivalent if they
have the same resultant force and the same resultant moment
about the same point
• Force couple system – move a force to another point using the
principle of transmissibility – we may add two equal and
opposite forces without altering the mechanical effect of the
original force – figure 2-17 page 66
2-10 Resultant of a nonconcurrent
coplanar force system
• In this system there are no point of concurrency – so the
locating of the line of action of the resultant is not
immediately known
• Choose convenient x and y coordinate axes and then resolve
each force into rectangular components – formula 2-11
• Example 2-16 page 68
• Example 2-17 page 69
• Example 2-18 page 70
2-11 Resultant of Distributed
Line Loads
• Distributed Load – occurs whenever the load applied to a body is not
concentrated at a point – could be exerted along a line , over a area,
or throughout an entire solid body
• Load intensity – distributed load along a line is characterized by a
load intensity expressed as force per unit length.
• Examples 1000 Ib/ft , 1 kip/ft, 1 N/m or 1kn/m
• Uniform load – distributed load with a constant intensity is called
uniform load – represented with a load diagram – shape like
rectangular block
• Triangular load – distributed load whose intensity varies linearly
from zero to a maximum intensity – represented as triangle
• Equivalent concentrated force – to determine the resultant of a
force system – each distributed load may be replaced by its
equivalent concentrated force
• Example 2-19 page 74
• Example 2-20 page 75
• Example 2-21 page 76