ENGI 1313 Mechanics I Lecture 18: Moment of a Force About a Specified Axis Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial.

Transcript ENGI 1313 Mechanics I Lecture 18: Moment of a Force About a Specified Axis Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial.

Slide 1

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 2

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 3

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 4

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 5

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 6

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 7

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 8

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 9

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 10

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 11

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 12

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 13

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 14

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 15

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 16

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 17

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 18

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 19

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 20

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 21

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Slide 22

ENGI 1313 Mechanics I

Lecture 18:

Moment of a Force About a
Specified Axis

Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]

Mid-Term


For This Class







October 18
830am-945am
to be determined
Up to and including problem set #4

Alternate Exam







2

Date:
Time:
Location:
Material:

Only if there is a compelling reason to miss October 18th and I
must be informed by October 15th with reason provided
Date:
October 22
Time:
700pm-815pm
Location:
EN 2007
Material:
Up to and including problem set #4
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Mid-Term (cont.)


General Comments




All students will be required to take the
examination

Resources
Tutorial problem sets
 Quiz problems
 Student societies


• http://www.engr.mun.ca/~societya/
• http://www.socb.ca/sample_finals.html
3

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Lecture 18 Objective


4

to provide scalar and vector methods for
determining the moment of a force about a
specified axis

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis






5

The axis of the moment
(Mo) about point O is
perpendicular to force
(F) and position vector
(r) plane
What is the component
moment about the x-, yor z-axes?
Why is this information
important?
© 2007 S. Kenny, Ph.D., P.Eng.

Mz

My
Mx
ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About Point O?
Mo  Fd  20N 0.5m  10N  m

6

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the X-Axis?


Tending to bend the pipe about O

4
M x  ( 4 / 5 )Mo   10N  m  8N  m
5 

Mx  Fd  20N 0.4m  8N  m
Mx

7

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


What is the Moment About the Y-Axis?


Tending to unscrew the pipe about O

3
M y  ( 3 / 5 )Mo   10N  m  6N  m
5 

My  Fd  20N 0.3m  6N  m
My

8

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)

L

Mz  Fd

9

© 2007 S. Kenny, Ph.D., P.Eng.

MZ  MA cos  FLcos  Fd'

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)


Moment Vector About Point O


Use cross product

 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m
 
F  F uˆ F  20N 0 î  0 ˆj  1kˆ

F   20kˆ N





10









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment About a Specified Axis (cont.)



 
MO  rA  F

rA  0.3î  0.4 ˆj  0kˆ m

F   20kˆ N

Moment Vector
About Point O
i

j

M O  0 .3 0 .4
0
0




k



0
 20

 0.4  20   0 0 î 


MO   0.3  20   0 0  ˆj  N  m
  0.3 0   0.4 0 kˆ 








MO   8î  6 ˆj N  m

11

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18



Moment About a Specified Axis (cont.)


Moment Component Projected on Y-axis


Recall dot product


My

My

My

My

12



 A  uˆ  MO  uˆ y

î  î  ˆj  ˆj  kˆ  kˆ  11cos0  1
î  ˆj  î  kˆ  ˆj  kˆ  11cos90  0


 
  8 î  6 ˆj  1 ˆj N  m
  8 î  6 ˆj  1 ˆj N  m

6 N  m

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product


Generalized Statement


Determine the component
moment magnitude (|Ma|) on
a specified axis due to the
moment (Mo)

MO

Ma

Ma

Ma

13

 
 r F


 MO cos  MO  uˆ a
 
 r  F  uˆ a
 
 uˆ a  r  F









© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)


Determine the component
moment magnitude (|Ma|)
on a specified axis due to
the moment (Mo)




 
Ma  uˆ a  r  F

14



uax

Ma  rx

uay

uaz

ry

rz

Fx

Fy

Fz

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uay

uaz

ry

rz

Fy

Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
15

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uax
rx
Fx

uay
ry
Fy

uay

uaz

ry

rz

Fy

Fz

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
16

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
17

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Triple Scalar Product (cont.)




 
Ma  uˆ a  r  F



uax


 M a  rx

Fx
uax

uay

uaz

rx

ry

rz

Fx

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz

uay

uaz

ry

rz

Fy

Fz

uax
rx
Fx

uay
ry
Fy

uaz
rz
Fz


Ma  uax ry Fz   rz Fy   uay rx Fz   rz Fx   uaz rx Fy   ry Fx 
18

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

Moment Vector on Specified Axis


Determine the moment
vector (Ma) on a specified
axis due to the moment
(Mo)

 

r





 
Ma  Ma uˆ a  uˆ a  r  F  uˆ a

19

© 2007 S. Kenny, Ph.D., P.Eng.

r

ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-01


The force F is acting along DC. Using the triple
product to determine the moment of F about the
bar BA, you could use any of the following
position vectors except ______.







20

A) rBC
A) rAD
C) rAC
D) rDB
A) rBD

Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.

Force Line of Action

Moment Axis
ENGI 1313 Statics I – Lecture 18

Comprehension Quiz 18-02


For finding the moment of the force F about the
x-axis, the position vector (r) in the triple scalar
product should be ___ .








21

A) rAC
A) rBA
C) rAB
D) rBC

Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1


22

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 18

ENGI 1313 Mechanics I Lecture 18: Moment of a Force About a Specified Axis Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial.

Transcript ENGI 1313 Mechanics I Lecture 18: Moment of a Force About a Specified Axis Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial.

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