Download Cross Product, Moment of a Force

CE 201 - Statics
Lecture 10
FORCE SYSTEM RESULTANTS
So far, we know that for a particle to be in equilibrium, the
resultant of the force system acting on the particle is to be
equal to zero. That is:
F=0
In a later stage, we are going to see that  F = 0 is required
for a particle to be in equilibrium, but it is not sufficient to
make that particle in equilibrium since we know that forces
could create momentum which will tend to rotate the particle.
In the coming lectures, moments about a point or axis will be
discussed.
CROSS PRODUCT
If we have two vectors A and B, then the CROSS PRODUCT
of the two vectors is:
C=AB
C
(C is equal to A cross B)
AB sin 
The magnitude of C is:
u
C = AB sin 
The direction of C is perpendicular to

A
the plane containing A and B.
B
Then
C = A  B = ( AB sin  ) uC
C
Laws of Operation
(1) A  B  B  A BUT A  B = - B  A
C =A B

B
A
-C =A B

B
A
Laws of Operation
(2) Multiplying by a Scalar
a (A  B) = (a A)  B = A  (aB) = (A  B) a
(3) A  (B + D) = (A  B) + (A  D)
Cartesian Vector Formulation
Use C = AB sin () to find the cross product of two
vectors.
ij = k; ik = -j ; ii = 0
jk = i; ji = -k; jj = 0
ki = j; kj = -i; kk = 0
ij = ij sin 90 = (1)(1)(1) = 1
ii = ii sin (0) = 0
If A and B are two vectors, then:
A  B = (Ax i + Ay j + Az k)  (Bx i + By j + Bz k)
= AxBx (i  i) +AxBy (i  j) + AxBz (i  k)
+ AyBx (j  i) +AyBy (j  j) + AyBz (j  k)
+ AzBx (k  i) +AzBy (k  j) + AzBz (k  k)
= (AyBz – AzBy) i – (AxBz – AzBx) j + (AxBy – AyBx) k
This equation can be written as:
i
j
k
A  B = Ax Ay Az
Bx By Bz
This is nothing but a determinant whose first raw consists of unit vectors i,
j, and k and its second and third raws represent the x, y, and z components
of the two vectors A and B.
MOMENT OF A FORCE – SCALAR
FORMATION
A moment of a force about a point or
axis is an indication or measure of the
tendency of the force to rotate the
particle about that point or axis.
Fx is perpendicular to the handle of
the wench and located a distance (dy)
from point (O). Fx tends to turn the
pipe about the z-axis. The turning
effect will be larger if Fx or dy gets
larger.
This turning effect is
sometimes referred to as TORQUE,
but most frequently it is called
MOMENT (M0)z
z
O
y
dy
x
Fx
If Fz is applied to the wrench at
dy from O. Fz will tend to
rotate the pipe about the x-axis
and (MO)x is produced.
z
O
y
The force Fz and distance dy lie
in the (x-z) plane, which is
perpendicular to the x-axis ( the
moment axis)
dy
x
Fz
If Fy is applied to the wrench, no moment will be
produced at point (O) since the line of action of Fy
passes through point (O).
z
O
Fy
y
dy
x
Generally, the moment of force F about point
(O) which lies in a plane perpendicular to the
moment axis, has a magnitude and direction.
Therefore, the moment is a vector quantity.
Magnitude
MO = F  d
Where d is called the moment arm,
which is perpendicular distance from
(O) to the line of action of force F.
F
d
O
Direction
MO
d
F
d
O
F
MO
O
Resultant Moment of a System of
Coplannar Forces
F1, F2, and F3 lie in the (x-y) plane. Their distances
from point (O) are d1, d2, and d3, respectively.
+ MO =  F  d
z
F3
d1
O
F1
y
d3
d2
F2
x
Assignment 3
3-52, 54, 57, 60, 64, 65