ENGI 1313 Mechanics I

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ENGI 1313 Mechanics I

• Lecture 15 Force System Resultant and Cross
  Product

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Chapter 4 Objectives

• to discuss the concept of the moment of a force
  and show how to calculate it in two and three
  dimensions.
• to provide a method for finding the moment of a
  force about a specified axis.
• to define the moment of a couple.
• to present methods for determining the resultants
  of non-concurrent force systems.
• to indicate how to reduce a simple distributed
  loading to a resultant force having a specified
  location.

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Lecture 15 Objectives

• to examine concepts of the moment of a force in
  scalar notation
• Slides in Lecture 13 revised for consistent
  notation with the text and provide summary
  classification of textbook problems
• to examine concepts on the cross product with
  respect to the moment of a force

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Moment Force Line of Action (cont.)

• Find Line of Action

x
O
y
d
F
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Moment Force Line of Action (cont.)

• Find Components

x
O
y
Fx
Fy
F
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Comprehension Quiz 15-01

• If a force of magnitude F can be applied in four
  different 2-D configurations (P,Q,R, S), select
  the cases resulting in the maximum and minimum
  (Max, Min) moment (torque) values on the wrench
  about Point A.
• A) (Q, P)
• B) (R, S)
• C) (P, R)
• D) (Q, S)
• Answer D

P
S
A
R
Q
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Example 15-01

• For each case illustrated, determine the moment
  of the force about point O.

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Example 15-01 (cont.)

• For each case illustrated, determine the moment
  of the force about point O.

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Example 15-01 (cont.)

• For each case illustrated, determine the moment
  of the force about point O.

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Example 15-01 (cont.)

• For each case illustrated, determine the moment
  of the force about point O.

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Moment of Coplanar Force System

• System of Forces Acting on a Single Plane

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Moment of Coplanar Force System

• System of Forces Acting on a Single Plane

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Cross Product

• Vector Resultant
• Magnitude
• Direction

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Cross Product (cont.)

• Commutative Law

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Cross Product (cont.)

• Vector Direction

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Cross Product (cont.)

• Cartesian Vectors

Terms go to zero
More convenient determinant form
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Cross Product (cont.)

• Cartesian Components from Determinant

Element i
Element j
Element k
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Applications of Cross Product

• Vector Notation Moment of a Force

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Applications of Cross Product

• Vector Notation Moment of a Force

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Principle of Transmissibility

• The force vector F can act at any point along the
  force vector line of action (sliding vector)
  and still create the same moment about point O.

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Principle of Moments

• Varignons Theorem
• The moment of a force about a point is equal to
  the sum of the moments of the forces components
  about the point.

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Comprehension Quiz 15-01

• Using the CCW direction as positive, the net
  moment of the two forces about point P is
• A) 10 Nm
• B) 20 Nm
• C) -20 Nm
• D) 40 Nm
• E) -40 Nm
• Answer B

10 N
5 N
3 m P 2 m
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Comprehension Quiz 15-02

• If r 5 j m and F 10 k N, the moment r x
  F equals
• A) 50 i Nm
• B) 50 j Nm
• C) -50 i Nm
• D) -50 j Nm
• E) 0 Nm
• Answer A

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Example 15-01

• Given a 3 in, b 6 in and c 2 in. Find the
  moment of F on the block about point O.

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References

• Hibbeler (2007)
• http//wps.prenhall.com/esm_hibbeler_engmech_1