Tenth Edition
2
CHAPTER
VECTOR MECHANICS FOR ENGINEERS:
STATICS
Ferdinand P. Beer
E. Russell Johnston, Jr.
David F. Mazurek
Lecture Notes:
John Chen
Statics of Particles
California Polytechnic State University
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Tenth
Edition
Vector Mechanics for Engineers: Statics
Contents
Introduction
Resultant of Two Forces
Vectors
Addition of Vectors
Resultant of Several Concurrent
Forces
Sample Problem 2.1
Sample Problem 2.2
Rectangular Components of a
Force: Unit Vectors
Addition of Forces by Summing
Components
Sample Problem 2.3
Equilibrium of a Particle
Free-Body Diagrams
Sample Problem 2.4
Sample Problem 2.6
Expressing a Vector in 3-D Space
Sample Problem 2.7
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Vector Mechanics for Engineers: Statics
Application
The tension in the cable supporting
this person can be found using the
concepts in this chapter
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Vector Mechanics for Engineers: Statics
Introduction
• The objective for the current chapter is to investigate the effects of forces
on particles:
- replacing multiple forces acting on a particle with a single
equivalent or resultant force,
- relations between forces acting on a particle that is in a
state of equilibrium.
• The focus on particles does not imply a restriction to miniscule bodies.
Rather, the study is restricted to analyses in which the size and shape of
the bodies is not significant so that all forces may be assumed to be
applied at a single point.
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Vector Mechanics for Engineers: Statics
Resultant of Two Forces
• force: action of one body on another;
characterized by its point of application,
magnitude, line of action, and sense.
• Experimental evidence shows that the
combined effect of two forces may be
represented by a single resultant force.
• The resultant is equivalent to the diagonal of
a parallelogram which contains the two
forces in adjacent legs.
• Force is a vector quantity.
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Vector Mechanics for Engineers: Statics
Vectors
• Vector: parameters possessing magnitude and direction
which add according to the parallelogram law. Examples:
displacements, velocities, accelerations.
• Scalar: parameters possessing magnitude but not
direction. Examples: mass, volume, temperature
• Vector classifications:
- Fixed or bound vectors have well defined points of
application that cannot be changed without affecting
an analysis.
- Free vectors may be freely moved in space without
changing their effect on an analysis.
- Sliding vectors may be applied anywhere along their
line of action without affecting an analysis.
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Vector Mechanics for Engineers: Statics
Addition of Vectors
• Trapezoid rule for vector addition
• Triangle rule for vector addition
• Law of cosines,
C
B
C
B
R 2  P 2  Q 2  2 PQ cos B
  
R  PQ
• Law of sines,
sin A sin B sin C


P
R
Q
• Vector addition is commutative,
   
PQ  Q P
• Vector subtraction
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Vector Mechanics for Engineers: Statics
Resultant of Several Concurrent Forces
• Concurrent forces: set of forces which all
pass through the same point.
A set of concurrent forces applied to a
particle may be replaced by a single
resultant force which is the vector sum of the
applied forces.
• Vector force components: two or more force
vectors which, together, have the same effect
as a single force vector.
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Vector Mechanics for Engineers: Statics
Sample Problem 2.1
SOLUTION:
The two forces act on a bolt at
A. Determine their resultant.
• Graphical solution - construct a
parallelogram with sides in the same
direction as P and Q and lengths in
proportion. Graphically evaluate the
resultant which is equivalent in direction
and proportional in magnitude to the the
diagonal.
• Trigonometric solution - use the triangle
rule for vector addition in conjunction
with the law of cosines and law of sines
to find the resultant.
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Vector Mechanics for Engineers: Statics
Sample Problem 2.1
• Graphical solution - A parallelogram with sides
equal to P and Q is drawn to scale. The
magnitude and direction of the resultant or of
the diagonal to the parallelogram are measured,
R  98 N   35
• Graphical solution - A triangle is drawn with P
and Q head-to-tail and to scale. The magnitude
and direction of the resultant or of the third side
of the triangle are measured,
R  98 N   35
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Vector Mechanics for Engineers: Statics
Sample Problem 2.1
• Trigonometric solution - Apply the triangle rule.
From the Law of Cosines,
R 2  P 2  Q 2  2 PQ cos B
 40N 2  60N 2  240N 60N  cos155
R  97.73N
From the Law of Sines,
sin A sin B

Q
R
sin A  sin B
Q
R
 sin 155
A  15.04
  20  A
  35.04
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60 N
97.73N
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Vector Mechanics for Engineers: Statics
Sample Problem 2.2
SOLUTION:
A barge is pulled by two
tugboats. If the resultant of
the forces exerted by the
tugboats is 5000 lbf directed
along the axis of the barge,
determine the tension in each
of the ropes for  = 45o.
Discuss with a neighbor how
you would solve this problem.
• Find a graphical solution by applying
the Parallelogram Rule for vector
addition. The parallelogram has sides
in the directions of the two ropes and a
diagonal in the direction of the barge
axis and length proportional to 5000 lbf.
• Find a trigonometric solution by
applying the Triangle Rule for vector
addition. With the magnitude and
direction of the resultant known and
the directions of the other two sides
parallel to the ropes given, apply the
Law of Sines to find the rope tensions.
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Vector Mechanics for Engineers: Statics
Sample Problem 2.2
• Graphical solution - Parallelogram Rule
with known resultant direction and
magnitude, known directions for sides.
T1  3700 lbf
T2  2600 lbf
• Trigonometric solution - Triangle Rule
with Law of Sines
T1
T2
5000 lbf


sin 45 sin 30 sin 105
T1  3660 lbf
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T2  2590 lbf
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Vector Mechanics for Engineers: Statics
What if…?
• At what value of  would the tension in rope
2 be a minimum?
Hint: Use the triangle rule and think about
how changing  changes the magnitude of T2.
After considering this, discuss your ideas with
a neighbor.
• The minimum tension in rope 2 occurs when
T1 and T2 are perpendicular.
T2  5000 lbf sin 30
T2  2500 lbf
T1  5000 lbf  cos 30
T1  4330 lbf
  90  30
  60
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Vector Mechanics for Engineers: Statics
Rectangular Components of a Force: Unit Vectors
• It’s possible to resolve a force vector into perpendicular
components so that the resulting parallelogram is a
rectangle. Fx and Fy are referred to as rectangular
vector components and
 

F  Fx  Fy


• Define perpendicular unit vectors i and j which are
parallel to the x and y axes.
• Vector components may be expressed as products of
the unit vectors with the scalar magnitudes of the
vector components.



F  Fx i  Fy j

Fx and Fy are referred to as the scalar components of F
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Vector Mechanics for Engineers: Statics
Addition of Forces by Summing Components
• To find the resultant of 3 (or more) concurrent
forces,
   
R  PQ S
• Resolve each force into rectangular components,
then add the components in each direction:








Rx i  R y j  Px i  Py j  Qx i  Q y j  S x i  S y j


 Px  Qx  S x i  Py  Q y  S y  j
• The scalar components of the resultant vector are
equal to the sum of the corresponding scalar
components of the given forces.
R y  Py  Q y  S y
Rx  Px  Qx  S x
  Fx
  Fy
• To find the resultant magnitude and direction,
2
2
1 R y
R  Rx  R y
  tan
Rx
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Vector Mechanics for Engineers: Statics
Sample Problem 2.3
SOLUTION:
• Resolve each force into rectangular
components.
• Determine the components of the
resultant by adding the corresponding
force components in the x and y
directions.
Four forces act on bolt A as shown.
Determine the resultant of the force
on the bolt.
• Calculate the magnitude and direction
of the resultant.
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Vector Mechanics for Engineers: Statics
Sample Problem 2.3
SOLUTION:
• Resolve each force into rectangular components.
force mag
r
F1 150
r
F2
80
r
F3 110
r
F4 100
x  comp
129.9
y  comp
75.0
27.4
0
75.2
110.0
96.6
25.9
Rx  199.1 R y  14.3

• Determine the components of the resultant by
adding the corresponding force components.
• Calculate the magnitude and direction.
R  199.12  14.32
14.3 N
tan  
199.1 N
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R  199.6N
  4.1
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Vector Mechanics for Engineers: Statics
Equilibrium of a Particle
• When the resultant of all forces acting on a particle is zero, the particle is
in equilibrium.
• Newton’s First Law: If the resultant force on a particle is zero, the particle will
remain at rest or will continue at constant speed in a straight line.
• Particle acted upon by
two forces:
- equal magnitude
- same line of action
- opposite sense
• Particle acted upon by three or more forces:
- graphical solution yields a closed polygon
- algebraic solution


R  F  0
 Fx  0
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 Fy  0
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Vector Mechanics for Engineers: Statics
Free-Body Diagrams
Space Diagram: A sketch showing
the physical conditions of the
problem, usually provided with
the problem statement, or
represented by the actual
physical situation.
Free Body Diagram: A sketch showing
only the forces on the selected particle.
This must be created by you.
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Vector Mechanics for Engineers: Statics
Sample Problem 2.4
SOLUTION:
• Construct a free body diagram for the
particle at the junction of the rope and
cable.
• Apply the conditions for equilibrium by
creating a closed polygon from the
forces applied to the particle.
In a ship-unloading operation, a
3500-lb automobile is supported by
a cable. A rope is tied to the cable
and pulled to center the automobile
over its intended position. What is
the tension in the rope?
• Apply trigonometric relations to
determine the unknown force
magnitudes.
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Vector Mechanics for Engineers: Statics
Sample Problem 2.4
SOLUTION:
• Construct a free body diagram for the
particle at A, and the associated polygon.
• Apply the conditions for equilibrium and
solve for the unknown force magnitudes.
Law of Sines:
T
T AB
3500 lb
 AC 
sin 120 sin 2 sin 58
TAB  3570 lb
TAC  144 lb
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Vector Mechanics for Engineers: Statics
Sample Problem 2.6
SOLUTION:
• Decide what the appropriate “body” is
and draw a free body diagram
It is desired to determine the drag force
at a given speed on a prototype sailboat
hull. A model is placed in a test
channel and three cables are used to
align its bow on the channel centerline.
For a given speed, the tension is 40 lb
in cable AB and 60 lb in cable AE.
• The condition for equilibrium states
that the sum of forces equals 0, or:


R  F  0
 Fx  0
 Fy  0
• The two equations means we can solve
for, at most, two unknowns. Since
there are 4 forces involved (tensions in
3 cables and the drag force), it is easier
to resolve all forces into components
Determine the drag force exerted on the
and apply the equilibrium conditions
hull and the tension in cable AC.
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Vector Mechanics for Engineers: Statics
Sample Problem 2.6
SOLUTION:
• The correct free body diagram is shown
and the unknown angles are:
7 ft
 1.75
4 ft
  60.25
tan  
1.5 ft
 0.375
4 ft
  20.56
tan  
• In vector form, the equilibrium
condition requires that the resultant
force (or the sum of all forces) be zero:
 



R  TAB  TAC  TAE  FD  0
• Write each force vector above in
component form.
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Vector Mechanics for Engineers: Statics
Sample Problem 2.6
• Resolve the vector equilibrium equation into
two component equations. Solve for the two
unknown cable tensions.
r
r
r
TAB  40 lbsin60.26 i  40 lbcos 60.26 j
r
r
 34.73 lbi  19.84 lb j
r
r
r
TAC  TAC sin20.56 i  TAC cos 20.56 j
r
r
 0.3512TAC i  0.9363TAC j
r
r
TAE  60 lb j
r
r
FD  FD i
r
R0
r
 34.73 0.3512TAC  FD  i
r
 19.84  0.9363TAC  60 j
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Vector Mechanics for Engineers: Statics
Sample Problem 2.6

R0

  34.73  0.3512 T AC  FD  i

 19.84  0.9363T AC  60 j
This equation is satisfied only if each component
of the resultant is equal to zero
 Fx  0
0  34.73 0.3512TAC  FD
 Fy  0
0 19.84  0.9363TAC  60
T AC  42.9 lb

FD  19.66 lb
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Vector Mechanics for Engineers: Statics
Expressing a Vector in 3-D Space
If angles with some of the axes are given:

• The vector F is
contained in the
plane OBAC.

• Resolve F into
horizontal and vertical
components.
Fy  F cos y
Fh  F sin  y
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• Resolve Fh into
rectangular components
Fx  Fh cos 
 F sin  y cos 
Fy  Fh sin 
 F sin  y sin 
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Vector Mechanics for Engineers: Statics
Expressing a Vector in 3-D Space
If the direction cosines are given:

• With the angles between F and the axes,
Fx  F cos x Fy  F cos y Fz  F cos z




F  Fx i  Fy j  Fz k



 F cos x i  cos y j  cos z k

 F




  cos x i  cos y j  cos z k


•  is a unit vector along the line of action of F
and cos x , cos
  y , and cos z are the direction
cosines for F

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
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Vector Mechanics for Engineers: Statics
Expressing a Vector in 3-D Space
If two points on the line of action are given:
Direction of the force is defined by
the location of two points,
M  x1 , y1 , z1  and N  x2 , y2 , z 2 
r
d  vector joiningM and N
r
r
r
 dxi  dy j  dz k
d x  x 2  x1 d y  y 2  y1 d z  z 2  z1
r
r
F  F
r 1
r
r
r
  d x i  d y j  d z k
d
Fd y
Fd
Fd x
Fx 
Fy 
Fz  z
d
d
d

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
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Vector Mechanics for Engineers: Statics
Sample Problem 2.7
SOLUTION:
• Based on the relative locations of the
points A and B, determine the unit
vector pointing from A towards B.
• Apply the unit vector to determine the
components of the force acting on A.
The tension in the guy wire is 2500 N.
Determine:
• Noting that the components of the unit
vector are the direction cosines for the
vector, calculate the corresponding
angles.
a) components Fx, Fy, Fz of the force
acting on the bolt at A,
b) the angles x, y, z defining the
direction of the force (the direction
cosines)
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Vector Mechanics for Engineers: Statics
Sample Problem 2.7
SOLUTION:
• Determine the unit vector pointing from A
towards B.
r
r
r
AB  40mi  80m j  30mk
AB 
40m  80m  30m
2
2
2
 94.3 m


r 40 r  80 r  30 r
  
i  
j  
k
94.3  94.3  94.3 
r
r
r
 0.424 i  0.848 j  0.318k
• Determine the components of the force.
r
r
F  F
r
r
r
 2500 N 0.424 i  0.848 j  0.318k
r
r
r
 1060N i  2120 N j  795 Nk
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

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Vector Mechanics for Engineers: Statics
Sample Problem 2.7
• Noting that the components of the unit vector are
the direction cosines for the vector, calculate the
corresponding angles.




  cos x i  cos y j  cos z k



 0.424 i  0.848 j  0.318k
 x  115.1
 y  32.0 
 z  71.5
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Vector Mechanics for Engineers: Statics
What if…?
SOLUTION:
FBA
FAB
• Since the force in the guy wire must be
the same throughout its length, the force
at B (and acting toward A) must be the
same magnitude but opposite in
direction to the force at A.
r
r
FBA  FAB
What are the components of the
force in the wire at point B? Can
you find it without doing any
calculations?

Give this some thought and discuss
this with a neighbor.
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r
r
r
 1060Ni  2120 N j  795 Nk
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