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Mechanics: Scalars and Vectors
• Scalar
– Only magnitude is associated with it
• e.g., time, volume, density, speed, energy, mass etc.
• Vector
– Possess direction as well as magnitude
– Parallelogram law of addition (and the triangle
law)
– e.g., displacement, velocity, acceleration etc.
• Tensor
– e.g., stress (33 components)
1ME101 - Division III Kaustubh Dasgupta
Mechanics: Scalars and Vectors
A Vector V can be written as: V = VnV = magnitude of V
n = unit vector whose magnitude is one and whose direction
coincides with that of V
Unit vector can be formed by dividing any vector, such as the
geometric position vector, by its length or magnitude
Vectors represented by Bold and Non-Italic letters (V)
Magnitude of vectors represented by Non-Bold, Italic letters
(V)
2ME101 - Division III Kaustubh Dasgupta
y
x
z
j
i
k
Types of Vectors: Fixed Vector
• Fixed Vector
– Constant magnitude and direction• Unique point of application
– e.g., force on a deformable body
– e.g., force on a given particle
Local
depression
F F
3ME101 - Division III Kaustubh Dasgupta
Types of Vectors: Sliding Vector
• Sliding Vector
– Constant magnitude and direction
• Unique line of action
– “Slide” along the line of action
• No unique point of application
Force on
coach F
Force on
coach F
4ME101 - Division III Kaustubh Dasgupta
Types of Vectors: Sliding Vector
• Sliding Vector
– Principle of Transmissibility
• Application of force at any point along a particular
line of action
– No change in resultant external effects of the force
5ME101 - Division III Kaustubh Dasgupta
Types of Vectors: Free Vector
• Free Vector
– Freely movable in space
• No unique line of action
– No unique point of application
– e.g., moment of a couple
6ME101 - Division III Kaustubh Dasgupta
Vectors: Rules of addition
• Parallelogram Law
– Equivalent vector represented by the diagonal
of a parallelogram
• V = V1 + V2 (Vector Sum)
• V V1 + V2 (Scalar sum)
7ME101 - Division III Kaustubh Dasgupta
F1
F2
Vectors: Parallelogram law of addition
• Addition of two parallel vectors
– F1 + F2 = R
8ME101 - Division III Kaustubh Dasgupta
F
-F
R
R1R2
R2
R1
Vectors: Parallelogram law of addition
• Addition of 3 vectors
– F1 + F2 + F3 = R
9ME101 - Division III Kaustubh Dasgupta
Vectors: Rules of addition
• Trigonometric Rule
– Law of Sines
– Law of Cosine
BA
C
10ME101 - Division III Kaustubh Dasgupta
Force Systems
Cable
Tension
Force Systems
• Force: Represented by vector– Magnitude, direction, point of application
– P: fixed vector (or sliding vector??)
– External Effect• Applied force; Forces exerted by bracket, bolts,
Foundation (reactive force)
Cable Tension P
12ME101 - Division III Kaustubh Dasgupta
Force Systems
• Rigid Bodies
– External effects only
• Line of action of force is important
– Not its point of application
– Force as sliding vector
13ME101 - Division III Kaustubh Dasgupta
Force Systems
• Concurrent forces
– Lines of action intersect at a point
AF1
F2
R
Plane
Concurrent Forces
F1 and F2
AF1
F2
R
F2
F1
Principle of
Transmissibility
A F1
F2
R
R = F1+F2
R = F1 + F2
14ME101 - Division III Kaustubh Dasgupta
Components and Projections of a Force
• Components and Projections
– Equal when axes are orthogonal
15ME101 - Division III Kaustubh Dasgupta
F1 and F2 are components of R
R = F1 + F2
:Fa and Fb are perpendicular
projections on axes a and b
: R ≠ Fa + Fb unless a and b are
perpendicular to each other
Components of a Force
• Examples
16ME101 - Division III Kaustubh Dasgupta
Components of a Force
• Examples
17ME101 - Division III Kaustubh Dasgupta
Components of a Force
Example 1:
Determine the x and y
scalar components of
F1, F2, and F3 acting
at point A of the bracket
18ME101 - Division III Kaustubh Dasgupta
Components of Force
Solution:
19ME101 - Division III Kaustubh Dasgupta
Components of Force
Alternative Solution: Scalar components of F3 can be
obtained by writing F3 as a magnitude times a unit vector nAB
in the direction of the line segment AB.
Unit vector can be formed by dividing any vector, such as the geometric
position vector by its length or magnitude.
20ME101 - Division III Kaustubh Dasgupta
Components of Force
Example 2: The two forces act on a bolt at A. Determine
their resultant.• Graphical solution –
• Construct a parallelogram with sides in
the same direction as P and Q and
lengths in proportion.
• Graphically evaluate the resultant which
is equivalent in direction and proportional
in magnitude to the diagonal.
• Trigonometric solution
• Use the law of cosines and law of sines
to find the resultant.
21ME101 - Division III Kaustubh Dasgupta
Components of Force
Solution:
• Graphical solution - A parallelogram with
sides equal to P and Q is drawn to scale. The
magnitude and direction of the resultant or of
the diagonal to the parallelogram are
measured,
35N 98 R
• Graphical solution - A triangle is drawn with P
and Q head-to-tail and to scale. The
magnitude and direction of the resultant or of
the third side of the triangle are measured,
35N 98 R
22ME101 - Division III Kaustubh Dasgupta
Components of Force
Trigonometric Solution:
155cosN60N402N60N40
cos222
222 BPQQPR
N73.97R
A
A
R
QBA
R
B
Q
A
20
04.15N73.97
N60155sin
sinsin
sinsin
04.35
23ME101 - Division III Kaustubh Dasgupta
Components of Force
Example 3: Tension in cable BC is 725 N; determine the
resultant of the three forces exerted at point B of beam AB.
Solution:
• Resolve each force into
rectangular components.
• Calculate the magnitude and
direction of the resultant.
• Determine the components of the
resultant by adding the
corresponding force components.
24ME101 - Division III Kaustubh Dasgupta
Components of Force
Solution
• Resolve each force into rectangular components.
• Calculate the magnitude and direction.
25ME101 - Division III Kaustubh Dasgupta
Rectangular Components in Space
• The vector is
contained in the
plane OBAC.
F
• Resolve into
horizontal and vertical
components.
yh FF sin
F
yy FF cos
• Resolve Fh into
rectangular
components
sinsin
sin
cossin
cos
y
hz
y
hx
F
FF
F
FF
26ME101 - Division III Kaustubh Dasgupta
Rectangular Components in Space
• With the angles between and the axes,F
kji
F
kjiF
kFjFiFF
FFFFFF
zyx
zyx
zyx
zzyyxx
coscoscos
coscoscos
coscoscos
• is a unit vector along the line of action of
and are the
direction cosines for
F
F
zyx cos and,cos,cos
27ME101 - Division III Kaustubh Dasgupta
Rectangular Components in Space
Direction of the force is defined by the location of two points:
222111 ,, and ,, zyxNzyxM
d
FdF
d
FdF
d
FdF
kdjdidd
FF
zzdyydxxd
kdjdid
NMd
zz
yy
xx
zyx
zyx
zyx
1
and joining vector
121212
28ME101 - Division III Kaustubh Dasgupta