mechanics: scalars and vectors
TRANSCRIPT
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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)
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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)
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y
x
z
j
i
k
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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
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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
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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
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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
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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)
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F1
F2
Vectors: Parallelogram law of addition
• Addition of two parallel vectors
– F1 + F2 = R
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F
-F
R
R1R2
R2
R1
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Vectors: Parallelogram law of addition
• Addition of 3 vectors
– F1 + F2 + F3 = R
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Vectors: Rules of addition
• Trigonometric Rule
– Law of Sines
– Law of Cosine
BA
C
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Force Systems
Cable
Tension
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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
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Force Systems
• Rigid Bodies
– External effects only
• Line of action of force is important
– Not its point of application
– Force as sliding vector
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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
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Components and Projections of a Force
• Components and Projections
– Equal when axes are orthogonal
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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
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Components of a Force
• Examples
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Components of a Force
• Examples
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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
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Components of Force
Solution:
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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.
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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.
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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
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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
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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.
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Components of Force
Solution
• Resolve each force into rectangular components.
• Calculate the magnitude and direction.
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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
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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
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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
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