TM chapter 2 moment and couple

VIETNAM OIL AND GAS GROUP
PETROVIETNAM UNIVERSITY

THEORETICAL MECHANICS
CHAPTER 2: MOMENT OF A FORCE AND A COUPLE

Lecturer : Dr. Vo Quoc Thang
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Contents
• Scalar and Vector Products of Vectors
• Principle of Transmissibility

• Moment of a Force About a Point
• Varignon’s Theorem
• Rectangular Components of the Moment of a Force

• Moment of a Force About a Given Axis
• Moment of a Couple
• Addition of Couples
• Resolution of a Force Into a Force and a Couple
• Reduction of a System of Forces

Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

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Scalar Product of Two Vectors
• Scalar product between two vectors P and Q:

P  Q  PQ cos 
• Properties:
- Commutative:
- Distributive:
- NOT associative:

PQ  QP

P  Q1  Q 2   P  Q1  P  Q 2

P  Q   S  P  Q  S 

• Scalar products with Cartesian unit vectors:
i  i 1 j j 1 k  k 1 i  j  0 j k  0 k  i  0
P  Q  Px i  Py j  Pzk  Q x i  Q y j  Q zk   Px Q x  Py Q y  Pz Q z

P  P  Px2  Py2  Pz2  P 2
Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

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Scalar Product of Two Vectors

• Angle between two vectors:
P  Q Px Q x  Py Q y  Pz Q z
cos  

PQ
PQ

• Projection POL of a vector P on a given axis OL:
PQ
Q
 P  λ  P  cos  x i  cos  y j  cos zk 

POL  P cos  

 Px cos  x  Py cos  y  Pz cos z

Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

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Vector Product of Two Vectors
• Vector product V of two vectors P and Q:

V  P Q
1.Line of action: perpendicular to plane
containing P and Q.
2.Magnitude: V  PQ sin

3.Direction: right-hand rule.

or

• Properties:
- NOT commutative:

Q  P   P  Q 

- Distributive:

P  Q1  Q 2   P  Q1  P  Q 2

- NOT associative:

P  Q   S  P  Q  S 

Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

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Vector Product of Two Vectors
• Vector products of Cartesian unit vectors:
ii  0

j  i  k k  i  j

i j  k

j j  0

k  j  i

i  k  j j k  i

k k  0

• Vector products in Cartesian coordinates:
V  Px i  Py j  Pzk  Q x i  Q y j  Q zk 

 Py Q z  Pz Q y i  Px Q z  Pz Q x j  Px Q y  Py Q x k
Px

Qx i

 Py

Qy j

Pz

Qz k

Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

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Mixed Triple Product of Three Vectors
• Mixed triple product of three vectors:
S  P  Q   Scalar

• The six mixed triple products formed from S, P and
Q have equal magnitudes but not the same sign:
S  P  Q   P  Q  S   Q  S  P 

 S  Q  P    P  S  Q   Q  P  S 

• Mixed triple product in Cartesian coordinates:
Sx

S  P  Q   S y
Sz



Px

Qx

Py

Qy

Pz

Qz

 Sx Py Q z  Pz Q y   S y Px Q z  Pz Q x 
 Sz Px Q y  Py Q x 

Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

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Principle of Transmissibility

Conditions of equilibrium or motion of a rigid body are
not affected by transmitting a force along its line of action.

NOTE: F and F’ are equivalent forces.
Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

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Principle of Transmissibility

• NOT always applied. Ex: in determining internal forces and deformations.

Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

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Moment of a Force About a Point

The moment of F applied at the point A about
a point O is defined as:
M O  rA / O  F
1. Line of action: perpendicular to the plane
containing O and the force F.
2. Magnitude: M O  rF sin   Fd

N.m  or lb.ft 

3. Direction: right-hand rule.

Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

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Moment of a Force About a Point

M O  Fd

M O  Fd'  Fd sin 

MO  0

• If any force F’ has the same line of
action, same magnitude and same
direction as F, does it produce the
same moment about O as F ? YES

M O  r1  F  r2  F    ri  F
Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

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Varignon’s Theorem

The moment about a given point O of the resultant of several
concurrent forces is equal to the sum of the moments of the
various moments about the same point O.
r  F1  F2    Fi   r  F1  r  F2    r  Fi

Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

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Rectangular Components of the Moment of a Force
The moment of F applied at A about O in Cartesian coordinates:
M O  rA  F rA  xi  yj  zk
F  Fx i  Fy j  Fzk

x Fx i
 y Fy j
z



Fz k







 yFz  zFy i  xFz  zFx j  xFy  yFx k
 M x i  M y j  M zk

Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

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Rectangular Components of the Moment of a Force
The moment of F applied at A about B:
M B  BA  F  rA / B  F

rA / B  rA  rB
  x A  x B  i   y A  y B  j  z A  z B  k
F  Fx i  Fy j  Fzk

x A  x B  Fx i
M B   y A  y B  Fy j
z A  z B  Fz k

Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

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Rectangular Components of the Moment of a Force
For two-dimensional structures:
M O  M zk  xFy  yFz k





M B  M zk  x A  x B Fy   y A  y B Fz k

Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

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Sample Problem 1
A 100-lb vertical force is applied to the end of
a lever which is attached to a shaft at O.

Determine:
a) moment about O,

b) horizontal force at A which creates the
same moment,
c) smallest force at A which produces the
same moment,
d) location for a 240-lb vertical force to
produce the same moment,
e) whether any of the forces from b, c, and d
is equivalent to the original force.

Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

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Sample Problem 1

a) Moment of force applied at A about O:
M O  Fd
d  24. cos 60  12 in.
M O  100.12

M O  1200 lb  in

Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

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Sample Problem 1

b) Horizontal force at A that produces the
same moment:
d  24.sin 60  20.8 in.
M O  Fd
1200  20.8F
F

Dr. Vo Quoc Thang

1200
20.8

F  57.7 lb

MOMENT OF A FORCE AND A COUPLE

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Sample Problem 1
c) The smallest force A to produce the same
moment occurs when the perpendicular distance
is a maximum or when F is perpendicular to OA:
M O  Fd
1200  24F
F

Dr. Vo Quoc Thang

1200
24

F  50 lb

MOMENT OF A FORCE AND A COUPLE

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Sample Problem 1
d) To determine the point of application of a
240 lb force to produce the same moment:

M O  Fd
1200  240d
1200
 5 in.
240
OB cos60  5 in.
d

Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

OB  10 in.

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Sample Problem 1

e) Although each of the forces in parts b), c), and
d) produces the same moment as the 100 lb
force, none are of the same magnitude and
sense, or on the same line of action. None of
the forces is equivalent to the 100 lb force.

Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

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Sample Problem 2

The rectangular plate is supported by the brackets at A and B and by
a wire CD. Knowing that the tension in the wire is 200 N, determine
the moment about A of the force exerted by the wire at C.

Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

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Sample Problem 2
SOLUTION:
M A  rC A  F

rC A  rC  rA  0.3i  0.08k m 
F  Fλ CD  200

rD C
rD C

 200

 0.3i  0.24 j  0.32k
0 .5

 120 i  96 j  128k  N 

0 .3

 120 i

96 j  7.68 i  28.8 j  28.8k  N.m 
0.08  128 k

MA  0

Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

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Moment of a Force About a Given Axis
• The moment MO of a force F applied at the point A about a point O:

MO  r  F
• Scalar moment MOL about an axis OL is the projection of the moment
vector MO onto the axis OL:

M OL  λ  M O  λ  r  F 

Dr. Vo Quoc Thang

MOMENT OF A FORCE AND A COUPLE

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Moment of a Force About a Given Axis
• Moments of F about the coordinate axes x, y and z:
M x  yFz  zFy
M y  zFx  xFz
M z  xFy  yFx

• Moment of a force about an arbitrary axis:
M BL  λ BL  M B
 λ BL  rA B  F





The result is independent of the point B
along the given axis.
Dr. Vo Quoc Thang

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