Moments

1
Moments

• ENGR 221
• January 29, 2003

2
Lecture Goals

• 4.2 Moments and Their Characteristics
• 4.3 Vector Representation of a Moment
• 4.4 Couples

3
Vectors Operations

There are two components of vector operations,
which are used in the computation of forces and
moments in the equilibrium equations.

• Dot products
• Cross products

4
Vectors Operations

Dot product is a scalar product of two vectors,
where The dot product is
5
Vectors Operations

The dot product is The angle a is the angle
between the two vectors.
6
Vectors Operations

The dot product of two vectors is commutative
The magnitude of a vector is
7
Vectors Operations

Cross product is a vector quantity of two
vectors, where the resulting vector is
perpendicular to the plane of the two vectors.
The magnitude of the vector is where the angle
a is the between the two vectors
8
Vectors Operations

The cross product of two vectors is
9
Vectors Operations

The cross product of two vectors is
10
Vectors Operations

The cross product of two vectors is
11
Vectors Operations

The cross product of two vectors is not
commutative The cross product is
12
Vectors Operations

If the results of the cross product is zero, then
the two vectors are parallel and if the dot
product is zero, then the vectors are
perpendicular
13
Example Problem

• For given two vectors
• Determine the angle between the vectors.
• Determine the vector perpendicular to the two
  vectors.

14
Example Problem
The dot product is The magnitudes of A and B
15
Example Problem
The angle between the vectors
16
Example Problem
The cross product is
17
Example Problem
The magnitude is The unit vector is
18
Example Problem
Check the magnitude The two values agree
(Possible check!)
19
Class Problem

• For given two vectors
• Determine the angle between the vectors.
• Determine the vector perpendicular to the two
  vectors.

20
Moment Characteristics
A moment of a force about a point or axis
action. The magnitude is defined as the magnitude
of the force F and perpendicular distance d from
the line of action of force to the axis.
21
Moment Characteristics
The magnitude is defined as
22
Moment Characteristics
The magnitude is defined as Point 0 is the
moment center Distance d is the perpendicular
moment arm q is the angle between the two vectors
23
Moment Characteristics
The magnitude is defined as Counter clockwise
moment is positive Clockwise moment is negative
24
Moment Characteristics
The line of action follows the right hand rule to
compute the direction of the vector Use the
index finger as the direction of r vector and
middle finger as the direction of the F vector
and the thumb is the resulting direction.
25
Varigons Theorem
As with the summation of force combining to get
resultant force Similar resultant comes from
the addition of moments
26
Moment Characteristics
The force, F, is a sliding vector therefore it
produces the same moment about point O when it is
applied at a point along its line of action on or
off the body.
27
Characteristics of Moments
The angle a between the two vectors and e is
perpendicular to plane containing r and F
28
Example - Moment
A 100-lb vertical force is applied to the end of
a lever which is attached to a shaft at 0.
Determine (a) the moment of 100-lb force about
0.(b) the magnitude of the horizontal force
applied at A which will create the same moment
about 0.(c ) the smallest force applied at A
which will create same moment (d) how far from
the shaft a 240 lb vertical force must act to
create the same moment about 0. (e) whether any
one of the forces obtain in parts (a), (b), (c )
and (d) is equivalent to the original force.
29
Example - Moment

• The perpendicular distance from ) to the line of
  action of 100-lb force is

The magnitude of the moment about 0
The force rotates the lever in clockwise about 0
and M0 is perpendicular to the plane.
30
Example - Moment

• Horizontal force

Since the moment about 0 is 1200 lb-in the
resulting F
31
Example - Moment

• Smallest Force, since MFd, the smallest value of
  F occurs when d is a maximum. It will be
  perpendicular to 0A

32
Example - Moment

• A 240-lb vertical force In this case the force is
  given determine the distance

33
Example - Moment

• None of the force in parts b, c, and d is
  equivalent of original 100-lb force. Although
  they have the same moment about 0, they have
  different x and y components .

34
Example - Moment
A pole AB, 6 m long is held by three guy wires.
Determine the moment about C of the force exerted
by wire BE on point B. The tension in wire BE is
known to be 840 N.
35
Example - Moment
The moment at C is You will need to compute CB
vector B(3m, 6m, 0m) and C(0m, 2m, 3m)
36
Example - Moment
Determine the force vector You will need to
compute BE vector B(3m, 6m, 0m) and E(6m, 0m, 2m)
37
Example - Moment
The force vector is along BE with a magnitude of
840 N
38
Example - Moment
The moment at C is
39
Example - Moment

The cross product of two vectors is
40
Example - Moment

The cross product of two vectors is
41
Class - Example
A 450-N force is applied at A. Determine the
moment of 450-N about D, (b) the smallest force
applied at B which creates the same moment about
D.
42
Class - Example
A force Q of magnitude 450 N is applied at C.
Determine the moment of Q about the origin of the
coordinate system and (b) point D.
43
Problem
The jib crane is oriented so that the boom DA is
parallel to the x axis. At the instant shown the
tension in the cable AB is 13 kN. Determine the
moment about each of the coordinate axes of the
force exerted on D and moment along DA by the
cable at A.
44
Moment along an Axis?
Compute the value of the moment along an
axis? To find component of a vector we take a
dot product with the magnitude and use the vector
to compute the components
45
Moment Magnitude of line
Using the dot product of the line AB Using
the lAB to find the components of the moment
46
Example - Problem
The jib crane is oriented so that the boom DA is
parallel to the x axis. At the instant shown the
tension in the cable AB is 13 kN. Determine the
moment about each of the coordinate axes of the
force exerted on D and moment along DA by the
cable at A.
47
Example -Problem
Find the moment about D so that the vector DA
48
Example -Problem
Find the tension force in AB is A(3.2m, 4.8m, 0
m) and B(3.2m, 0m, 2m)
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Example - Moment
The force vector is along AB with a magnitude of
13 kN
50
Example - Moment
The moment at D is
51
Example - Moment

The cross product of two vectors is
52
Example - Moment

The cross product of two vectors is The dot
product of DA
53
Example - Moment

So there would be no moment on bar AB. What
would it have been if there was a bar DE going
off at 45 degrees The dot product of DE
54
Example - Moment

Moment would be
55
Class - Example
A vertical force P of magnitude 60 lb is applied
to the crank at A. Knowing that q 75o,
determine the moment P alone each of the
coordinate axes.
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Homework (Due 2/5/03)
Problems
4-6, 4-9, 4-11, 4-12, 4-14, 4-20, 4-29