ME221 Statics
LECTURE # 10
Sections 3.4 & 3.6
ME221
Lecture 10
1
Homework #4
• Chapter 3 problems:
–
–
–
–
1, 4, 8, 11, 17, 25, 26, 28, 35 & 40
To be solved using hand calculations
May check work using MathCAD, etc.
Due Friday, September 26
Quiz #3
• Wednesday, September 24
ME221
Lecture 10
2
Last Lecture
• Cross Product
• Rigid Bodies
• Moment of a Force about a Point
Today
• Moment about an Axis
• Moment of Couple
• Equivalent Force Couple System
ME221
Lecture 10
3
Vector Moment Definition
The moment about point O of a force acting at point A is:
F
O
A
MO = rA/O x F
rA/O
Compute the cross product with whichever
method you prefer.
ME221
Lecture 10
4
200 N
Example
Method # 1
O.4
0.2
60 o
tan 60°=0.2m/x
x=0.115m
sin 60°=d/0.285m
60 o
0.285
A
x
d
d = 0.247 m
MA =200N *0.247m= 49.4 Nm
ME221
Lecture 10
5
200 sin 60
200 N
Method # 2
O.4
60 o
200 cos60
0.2
A
+
M =200N (sin 60)(0.4m)- 200N (cos 60)(0.2m)
= 49.4 Nm
ME221
Lecture 10
6
200 N
Method # 3
O.4
0.2
60 o
r
A
F=200N cos 60 i + 200N sin 60 j
r =0.4 i + 0.2 j
^i
j^
0.2
MA= 0.4
200cos60 200sin60
ME221
^k
0 =200 (sin 60)(0.4) - 200 (cos 60)(0.2)
0 = 49.4 Nm
Lecture 10
7
200 N
Method # 4
O.4
60 o
0.2
r =0.285 i
A
F=200N cos 60 i + 200N sin 60 j
r =0.285 i
i
j
k
0
0 = 49.4 Nm
MA= 0.285
200cos60 200sin60 0
ME221
Lecture 10
8
Moment of a Force about an Axis
y ^
n
|Mn| =MA·n^
A
^
=n·(rB/A x F )
Same as the projection of MA
along n
rAB=rB/A
F
B
O
x
z
nx
r
|Mn|= B/Ax
Fx
ME221
ny
r B/A y
Fy
nz
r B/Az
Fz
Lecture 10
9
Resolve the vector MA into MA
two vectors one parallel and
one perpendicular to n.
y Mp
Mn
A
Mn=|Mn|n^
n^
F
rAB=rB/A B
O
Mp = MA - Mn
x
z
^
^
=n x [(r B/A x F) x n]
ME221
Lecture 10
10
Moment of a Couple
Let F1 = -F2
B
y
|C|=|F1| d
ME221
d
rAB=rB/A
F2
Mo=rA x F2+ rB x F1
=(rB - rA ) x F1
=rAB x F1= C
F1
A
rB
rA
O
x
The Moment of two equal
and opposite forces is called
a couple
z
Lecture 10
11
Moment of a Couple (continued)
• The two equal and opposite forces form a couple
(no net force, pure moment)
•The moment depends only on the relative positions
of the two forces and not on their position with
respect to the origin of coordinates
ME221
Lecture 10
12
Moment of a Couple (continued)
•Since the moment is independent of the origin, it
can be treated as a free vector, meaning that it is the
same at any point in space
•The two parallel forces define a plane, and the
moment of the couple is perpendicular to that plane
ME221
Lecture 10
13
Example
ME221
Lecture 10
14