Lecture 9- 1page

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Vector Mechanics for Engineers: StaticsE i gh t h

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3 - 1

MOMENT ABOUT AN AXIS





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3 - 2

APPLICATIONS

With the force F

, a person iscreating the moment M A

.

What portion of M A

is used in

turning the socket?

The force F is creating themoment M

O. How much of

M O

acts to unscrew the

pipe?





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3 - 3

Moment of a Force About a Given Axis

• Moment M O

of a force F applied at the point A

about a point O,

F r M Or

r

r

×=

• Scalar moment M OL about an axis OL is the

projection of the moment vector M O

onto the

axis,

)F r M M OOL

r

r

rrr

×•=•= λ λ

• Moments of F about the coordinate axes,

x y z

z x y

y z x

yF xF M

xF zF M

zF yF M

−=

−=

−=





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3 - 4

Moment of a Force About a Given Axis

• Moment of a force about an arbitrary axis,

( ) B A B A

B A

B BL

r r r

F r

M M

rrr

r

r

r

rr

−=

×•=

•=

λ

λ

• The result is independent of the point Balong the given axis.





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3 - 5

Sample Problem 3.5

a) about A

b) about the edge AB and

c) about the diagonal AG of the cube.

d) Determine the perpendicular distance between AG and FC .

A cube is acted on by a force P as

shown. Determine the moment of P





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3 - 6

Sample Problem 3.5

• Moment of P about A,

( )( ) ( )

( ) ( ) jiP jia M

jiP jiPP

jia jaiar

Pr M

A

AF

AF A

rrrrr

rrrrr

rrrrr

r

r

r

+×−=

+=+=

−=−=

×=

2

222

( ) k jiaP M A

rrrr

++=

2

• Moment of P about AB,

( )( )k jiaPi

M i M A ABr

rrr

rr

++•=

•=

2

2aP M AB =





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3 - 7

Sample Problem 3.5

• Moment of P about the diagonal AG,

( )

( )

( ) ( )

( )1116

231

2

3

1

3

−−=

++•−−=

++=

−−=

−−

==

•=

aP

k jiaPk ji M

k jiaP

M

k jia

k a jaia

r

r

M M

AG

A

G A

G A

A AG

rrrrrr

rrrr

rrr

rrrr

r

rr

λ

λ

6

aP M AG −=





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3 - 8

Sample Problem 3.5

• Perpendicular distance between AG and FC,

( ) ( ) ( )

0

11063

1

2

=

+−=−−•−=•P

k jik jP

Prrrrrrr

λ

Therefore, P is perpendicular to AG.

Pd

aP

M AG == 6

6

ad =







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3 - 10

Moment of a Couple

Two couples will have equal moments if

•2211 d F d F =

• the two couples lie in parallel planes, and

• the two couples have the same sense or

the tendency to cause rotation in the samedirection.





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3 - 11

Addition of Couples

• Consider two intersecting planes P1 and

P2 with each containing a couple

222

111

planein

planein

PF r M

PF r M r

r

r

r

r

r

×=

×=

• Resultants of the vectors also form a

couple

21 F F r Rr M rr

r

r

r

r

+×=×=

• By Varigon’s theorem

21

21

M M

F r F r M rr

r

r

r

r

r

+=

×+×=

• Sum of two couples is also a couple that is equal

to the vector sum of the two couples





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3 - 12

Couples Can Be Represented by Vectors

• A couple can be represented by a vector with magnitude

and direction equal to the moment of the couple.

• Couple vectors obey the law of addition of vectors.

• Couple vectors are free vectors, i.e., the point of applicationis not significant.

• Couple vectors may be resolved into component vectors.

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Lecture 9- 1page