CHAPTER 2 – FORCE SYSTEMS
Scalars and Vectors
Scalar: A physical quantity that is completely characterized by a real number (or
by its numerical value) is called a scalar. In other words, a scalar possesses only a
magnitude. Mass, density, volume, temperature, time, energy, area, speed and
length are examples to scalar quantities.
Vector: Several quantities that occur in mechanics require a description in terms
of their direction as well as the numerical value of their magnitude. Such
quantities behave as vectors. Therefore, vectors possess both magnitude and
direction; and they obey the parallelogram law of addition. Force, moment,
displacement, velocity, acceleration, impuls and momentum are vector quantities.
+
v
V
Line of action
v
V
V, magnitude
direction
, slope
Types of Vectors:
Physical quantities that are vectors fall into one of the three classifications as
free, sliding or fixed.
A free vector is one whose action is not confined to or associated with a unique
line in space. For example if a body is in translational motion, velocity of any point
in the body may be taken as a vector and this vector will describe equally well the
velocity of every point in the body. Hence, we may represent the velocity of such
a body by a free vector. In statics, couple moment is a free vector.
v
v
B
v
v
C
v z
v
A
v
v
v
v
y
x
A sliding vector is one for which a unique line in space must be maintained along
which the quantity acts. When we deal with the external action of a force on a
rigid body, the force may be applied at any point along its line of action without
changing its effect on the body as a whole and hence, considered as a sliding
vector.
v
F
v
F
A
line of
action of
force
B
rigid body
rigid
body
v
F
v
F
v
F
A fixed vector is one for which a unique point of application is specified and
therefore the vector occupies a particular position in space. The action of a force
on a deformable body must be specified by a fixed vector.
elastic bar
F
F
A
B
Principle of Transmissibility: The external effect of a force on a rigid body will
remain unchanged if the forced is moved to act on its line of action. In other
words, a force may be applied at any point on its given line of action without
altering the resultant external effects on the rigid body on which it acts.
push
pull
Equality and Equivalence of vectors
Two vectors are equal if they have the same dimensions, magnitudes and
directions.
Two vectors are equivalent in a certain capacity if each produces the very same
effect in this capacity.
To sum up, the equality of two vectors is determined by the vectors themselves,
but the equivalence between two vectors is determined by the situation at hand.
B
B
v
F1
v
F1
1m
1m
v
F2
A
1m
A
1m
v
F2
O
O
F1' = 5 N, F2 ' = 10 N
v
v
F1 and F2 are not equal but they are
equivalent since their moments about the
base O are equal (they create the same
twisting motion at the base)
F1 = F2 = 10 N
v
v
F1 and F2 are equal but not
equivalent
PROPERTIES OF VECTORS
Addition of Vectors is done according to the parallelogram law of vector addition.
v v
U +V = W
v v v v
U +V = V +U
or
v
v
v
v
v
v
(U + V ) + M = U + (V + M )
v
V
v
W
v
U
v
V
v
U
Subtraction of Vectors is done according to the parallelogram law.
v
+V
v
U
v v v
v
v
U V =U + ( V )= Z
v
Z
v
V
Multiplication of a Scalar and a Vector
v
v
a U = aU
v
v
a(bU ) = abU
v
v
v
(a + b )U = aU + bU
v v
v
v
a(U + V ) = aU + aV
Unit Vector A unit vector is a free vector having a magnitude of 1 (one) as
v
v
v U U
e= v =
. It describes direction. The most convenient way to describe a vector
U U
in a certain direction is to multiply its magnitude with its unit vector.
v
v
U = Ue
U
v
U
v
e
1
v
v
U and U have the same unit, hence the unit vector is dimensionless. Therefore, U
may be expressed in terms of both its magnitude and direction separately. U (a
v
scalar) expresses the magnitude and e (a dimensionless vector) expresses the
v
directional sense of U .
v
v
v
v
Vector Components and Resultant Vector Let the sum of U and V be W . Here, U
v
v
and V are named as the components and W is named as the resultant.
Sine theorem
Cosine theorem
U
sin
W 2 =U 2 +V 2
=
V
sin
=
W
sin
v
W
v
V
2UV cos
v
U
Cartesian Coordinates Cartesian coordinate system is composed of 90°
(orthogonal) axes. It consists of x and y axes in two dimensional (planer) case, x, y
and z axes in three dimensional (spatial) case. x-y axes are generally taken within
the plane of the paper, their positive directions can be selected arbitrarily; the
positive direction of the z axes must be determined in accordance with the right
hand rule.
z
y
z
y
x
z
x
y
x
rotation direction of screw
translation direction of screw
Vector Components in Two Dimensional (Planer) Cartesian Coordinates
y
v v
v
U =Ux +U y
v
j
v
Uy
tan =
v
U
v
Ux
2
U = Ux +Uy
v
i
2
Uy
Ux
x
v
v
unit vector along the x axis, i , unit vector along the y axis j ,
v
v
U x = U xi
v
v
v
V = Vx i + V y j
,
v
v
v
v
v
U y =U y j
U = U xi + U y j
v v
v
v
v
v
v
v
U + V = U x i + U y j + V x i + V y j = (U x + V x )i + U y + V y j
(
)
Vector Components in Three Dimensional (Spatial) Cartesian Coordinates
v
unit vector along the x axis, i ,
v
unit vector along the y axis, j ,
z
v
v
k
v
U
unit vector along the z axis, k
y
v
j
(
v
v
v
v
2
2
2
U = U xi + U y j + U z k
U = Ux +Uy +Uz
v
v
v
v
V = Vx i + V y j + Vz k
v
v v
v
v
U + V = (U x + V x )i + U y + V y j + (U z + V z )k
v
Uy
v
Uz
v
Ux
(
x
v
i
)
Position Vector It is the vector that describes the location of one point with
respect to another point.
In two dimensional case
y
v
v
v
v
v
rB/A = rB/A x + rB/A y = rB/A x i + rB/A y j
"v
rB/A =
j
yB
(yB- yA)
yA
A (xA, yA)
xA
xB
2
+ rB/A y 2
)
,
rB/A y = y B
v
v
x A )i + ( y B y A ) j
xA
yA
x
v
i
(xB- xA)
B/A x
rB/A x = x B
v
rB/A = ( x B
B (xB, yB)
v
rB / A
(r
In three dimensional case
z
v
v
v
v
v
v
v
rB/A = rB/A x + rB/A y + rB/A z = rB/A x i + rB/A y j + rB/A z k
B (xB, yB, zB)
v
k
y
v
rB / A
v
j
(r
B/A x
rB/A x = x B
v
rB/A = ( x B
A (xA, yA, zA)
v
i
rB/A =
x
2
2
+ rB/A y + rB/A z
2
)
,
rB/A y = y B y A ,
v
v
v
x A )i + ( y B y A ) j + ( z B z A )k
xA
rB/A z = z B
zA
)
Dot (Scalar) Product A scalar quantity is obtained from the dot product of two
vectors.
v v
v v
U V = a order of multiplication is irrelevant V U = a
v v
v v v v
U V
cos = v v
U V = U V cos
UV
v
V
In terms of unit vectors in Cartesian Coordinates,
v
U
v
i
v
i
v
U
v
U
v vv
v
i = i i cos 0 = 1
,
j
v v v
v
j = i j cos 90 = 0
,
j
v
v
v
= U xi + U y j + U z k
v
V = U xV x + U yV y + U zV z
v
j = 1,
v
k = 0,
v v
k k =1
v v
k i =0
v
v
v
v
V = Vx i + V y j + Vz k
Normal and Parallel Components of a Vector with respect to a Line
Magnitude of parallel component
L
v
e
L
v
U //
v v v v
U e = U e cos = U cos
{
v
U
,
U // = U cos
v v
U // = U e
1
v
U
Parallel component
Normal (Orthogonal) component
v
v v v
U // = (U e ) e
v
v v
U = U U //
Cross (Vector) Product The multiplication of two vectors in cross product results
in a vector. This multiplication vector is normal to the plane containing the other
two vectors. Its direction is determined by the right hand rule. The order of
multiplication is important.
v v v
U ×V = W
v v
v
V ×U = W
,
v v
U ×V
sin = v v
UV
v v
v v
U × V = U V sin
v
V
v v
v v v
v
a(U × V ) = (aU ) × V = U × (aV )
v v v
v v v v
U × (V + Y ) = U × V + U × Y
v
U
In terms of unit vectors in Cartesian
v v vv
v v
i × i = i i sin 0 = 0
,
j × j = 0,
v v v v
v v v
i × j = i j sin 90 = 1
, i× j =k
,
v
v v
j ×i = k
,
v
W
v
V
Coordinates,
v v
k ×k = 0
v v v
j ×k = i
,
v v
v
k× j = i
,
v
U
v
i
z
v
k
+
y
v
j
v
i
(
x
v
j
v
k
) (
)
v
v
v v
v
v
v
v
U × V = U xi + U y j + U zk × Vxi + V y j + Vzk
v
v
v
v
v
i
j
k
i
j
v v
U ×V = U x U y U z U x U y
Vx
Vy
Vz Vx Vy
v
v
v
v
v
v
= i (U y V z ) + j (U z V x ) + k (U x V y ) - j (U x V z ) - i (U z V y ) - k (U y V x )
v
v v
v
v
U × V = U y V z U z V y i + [U z V x U xV z ] j + U xV y U y V x k
[
]
[
]
v v
k ×i =
v v
i ×k =
v
j
v
j
or
v
v
v
v
v
v
v
v
U × V = U x V y k - (U x V z ) j
U y V x k + U y V z i + (U z V x ) j
U zV y i
v
v
v
= U y V z U z V y i + [U z V x U x V z ] j + U x V y U y V x k
(
[
)
]
(
)
(
[
)
]
(
)
Mixed Triple Product It is used when taking the moment of a force about a line.
v
v
v
v
U = U xi + U y j + U z k
v
v
v
v
V = Vx i + V y j + Vz k
v
v
v
v
W = Wx i + W y j + Wz k
v
v
i
j
v
v v v
v
v
U (V × W ) = U x i + U y j + U z k V x V y
Wx W y
(
)
or
Ux Uy
v v v
U (V × W ) = V x V y
Wx W y
Uz
Vz
Wz
v
k
Vz
Wz