Introduction to Force Vectors

1
Introduction to Force Vectors
2
Objective

• To show how to add forces and resolve them into
  components using graphical and/or mathematical
  solutions.

3
Definitions

• Scalar - A quantity characterized by a positive
  or negative number is called a scalar. Examples
  of scalars used in Statics are mass, volume or
  length.
• Vector - A quantity that has a magnitude, a
  direction and a point of application. Examples of
  vectors used in Statics are force and moment.

4
Line of Action
Direction
A
P
1
Head
?
Tail
O
Point of Application
Vector Definitions
5
A
RAB
A
B
B
RAB
A
B
B
A
RAB
Vector Addition
6
b
R
a
Resolution of a vector
7
b
R
B
a
A
Resolution of a vector
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Important Points

• A scalar is a positive or negative number.
• A vector is a quantity that has magnitude,
  direction and a point of application.
• Multiplication or division of a vector by a
  scalar will change the magnitude. The direction
  will change if the scalar is negative.
• If the vectors are collinear, the resultant is
  formed by algebraic or scalar addition.

9
F2 150 N
100
F1 100 N
150
Find the magnitude and direction of the resultant
force.
10
Trigonometry
Law of Sines
c
B
A
a
b
C
Law of Cosines
11
Parallelogram Law

• Make a sketch showing vector addition using the
  parallelogram law.
• Determine the interior angles of the
  parallelogram from the geometry of the problem.
• Label all known and unknown angles and forces in
  the sketch.
• Redraw one half of the parallelogram to show the
  triangular head-to-tail addition of the
  components and apply laws of sines and cosines or
  use ruler and protractor for graphical solution.

12
150 N
Y
FR
100
?
?
100 N
150
X
13
B
C
150 N
FR
Calculate angles
100
Angle COA 900 -150-100 650 Angle OAB 1800
-650 1150
?
A
?
100 N
150
O
14
Find FR from law of cosines. Find ? from law of
sines. Angle ? ? 150
FR
150 N
1150
Find FR from law of cosines.
?
100 N
15
Find ? from law of sines.
FR
150 N
1150
?
100 N
16
F2 150 N
100
F1 100 N
150
The resultant force has a magnitude of 213 N and
is directed 54.8o from the horizontal.
17
Mathematical Solution
B
Angle COA 900 -150-100 650
C
150 N
FR
100
?
A
?
100 N
150
O
The resultant force has a magnitude of 213 N and
is directed 54.8o from the horizontal.
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y
Resolve the 200 N force into components in the x
and y directions
F200 N
400
x
19
F Fx Fy Fx 200 cos 400 Fy 200 sin 400
y
F 200 N
200 N
Fy
400
400
Fx
x
20
y
F2
F1
x
F3
Find the resultant force FR F1 F2 F3
21
y
F2y
F1y
F1x
x
F2x
F3x
F3y
22
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