Vector Representation of a Moment

1
Lecture 10

• Vector Representation of a Moment

2
Moments
y

• The moment of a force F about point O is
• Where r is a position vector.
• Mo is the moment about the axis out of the page.

Line of action
a
d
F
r
O
x
3
Moments

• By definition, the cross-product of two
  intersecting vectors r and F is

Where e is the unit vector perpendicular to the
plane containing the vectors r and F.
Note
Changing the sequence will produce a moment with
the opposite sense.
4
Moments

• The position vector from point B (the point about
  which the moment is to be determined) to any
  point on the line of action of the force F (point
  A), can be expressed in terms of unit vectors
  i,j,k, and the coordinates (xA,yA,zA) and
  (xB,yB,zB).

5
Moments

• For the two dimensional case,

and
6
Moments

• For the three dimensional case,

and
7
Moments
Where
8
Moments

• This can be extended to any force system.

Where
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Example 1

• Problem
• Determine the moment of F about point O.

10
Example 1

• Solution
• (1) Determine the components Fx and Fy.

(2) Determine the components rx and ry of the
position vector r.
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Example 1

• (3) Find Mo

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Example 2

• Problem
• Determine the moment of F about points B and C.

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Example 2

• Solution
• (1) Determine the components Fx and Fy.

(2) Determine rB and rC.
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Example 2

• (3) Find MB.

Or
Ans.
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Example 2

• (4) Find MC.

Or
Ans.
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Example 3

• Problem
• Determine the moment of F about point B.

17
Example 3

• Solution
• (1) Determine the Cartesian vector for F.

18
Example 3

• (2) Determine rB.

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Example 3

• (3) Find MB.

Ans.
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Example 3

• (4) Find the magnitude of the moment MB.

(5) Find the direction angles.
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Example 3

• (6) Calculate the value of d.

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Additional Material

• Read Sections 4.3-4.4
• Problems 4-35, 37, 41, 49