chapter two laith batarseh - philadelphia
TRANSCRIPT
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Chapter Two
Laith Batarseh
Definition
Engineering mechanics
Deformable body
mechanics
Rigid body mechanics
Dynamics Statics
Fluid mechanics
Constant Velocity
Variable Velocity
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Definition
Rigid body is the body that has the same volume parameters before and after applying the load
Deformable body is the body changes its volume parameters when the load is applied on it.
Definition
Static Cases:
Dynamic Case:
Velocity = 0P
BA
PVelocity = Constant
BA
PVelocity is changeable
Acceleration or Deceleration
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Definition
P
At rest Acceleration Deceleration Constant velocity
Dynamics Statics
EndStart
Cartesian coordinate system
x
y
x
y
z
y-z plane
x-y plane
x-z
pla
ne
x-y plane
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Definition
x
y
y
P(x,y)
Newtown's Laws of Motion
Newtown's Laws of Motion
First Law: a particle at rest or moves in constant velocity will remain on its state unless it is subjected to unbalance force.
F1F4
F3
F2
0F
F
Second Law: a particle subjected to unbalance force will move at acceleration has the same direction of the force.
maF
Third Law: each acting force has a reaction equal in magnitude and opposite in direction
AAction: Force of A acting on B B Reaction: Force of B acting on A
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Newtown's Law of Gravity
Newtown proved his most famous law at all in the 17th centaury and called itLaw of Universal Gravitation.
Statement: any two objects have a masses (M1 and M2 )and far away from each other by a distance (R) will have attraction force (gravitational) related proportionally with the objects masses product (M1 M2 ) and inversely with the square of the distance between them (R2). Mathematically:
Where: G = 66.73 x 10-12 m3/(kg.s2)
Weight: according to Newtown's Law of Gravity, weight is the gravitational force acting between the body has a mass (m) and the earth (of mass Me) and is given as: W=mg. g is the gravitational acceleration and equal to GMe/R2.
constant isG ;2
21
R
MMGF
Units of Measurements
According to Newton's 2nd law, the unit of force is a combination of the
other three quantities: mass, length and time. So, the force units N
(Newtown) and lbf can be written as:22
.s
ftlbmlbfand
s
mkgN
Units of Measurements
Length
(m or ft)
Mass
(kg or lbm)
Time
(sec)
Force
(N or lbf)
SImKgN
Englishft
lbmlbf
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Scalar versus vectors
Physical
Quantity
Scalar Vector
Is any physical quantity can be described fully by a magnitude
only.
Is any physical quantity needs both magnitude and direction
to be fully described.
Examples: mass, length and time.
Examples: force, position and moment.
Scalar versus vectors
Direction (θ)
Sense of Direction
Fixed axis 30o
This vector has a magnitude of5 units and it tilted from thehorizontal axis by +30o
Example
Vector notations:
1. A
2.
3. V = M∟θA
Sense of Direction
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Vector operations
A
2A
0.5A-A
Multiplication by a scalar
30o
2A =10∟30o
0.5A =2.5∟30o
-A =-5∟30o = 5∟180+30o
Example
Vector operations
A
Vector addition (A+B) Vector subtraction (A-B)
B
R=A+B
A
B
R=A+B
A
B
A
-B
R=A-B
A
-B
R=A-B
A-B
OR
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Vector operations
Example:
Assume the following vectors: A = 10∟30o and B = 7∟-20 o
Find: 1. A+B2. A-B
Solution:
A+B
AB
A-B A
-B
Vector operations
Special cases:Collinear vectors:
Multiple addition:
A+B+C+…: in such case, the addition can be in successive order or in multiple steps.
A B
A + B
A + B
B - AA B
B-A
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Vector operations
Example:
Assume the following vectors: A = 10∟30o ,B = 7∟-20 o and C = 6∟135o . Find: A+B+(A+C)
Solution:
A+B
A+CA
BC
A
A+B+(A+C)
A+B
A+C
Vector operations
Example (cont): Find A+B+(A-C)
A+B
AB A+B
A
A+B+A
A+B+A-C
-C
A+B+A
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Examples
Examples
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Examples
Examples
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Coplanar forces
Coplanar forcesCoplanar forces are the forces that share the same plane.
These forces can be represented by their components on x and yaxes which are called the rectangular components.
The components can be represented by scalar and Cartesian notations.
Scalar notation
Cartesian notation
Fx = F cos(θ)
y
x
Fy
Fx
F
θ
Fy= F sin(θ)
Fy
Fx
F
x
i
y
j F = Fx i + Fy j
Scalar notation cases
y
x
Fy
Fx
F
θ
Fx = F cos(θ)
Fy= F sin(θ)
1 2
y
xFy
Fx
c
Fx /F= a/c
Fy /F= b/c
abF
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Scalar notation cases
Cartesian notation cases
Fy
Fx
F
x
i
y
jF2,y
F2
F2,x
F1,y
F1
F1,x
F4,xF4
F4,y
x
y
F3,x
F3,y
F3
F 1= F1,x i + F1,y j
F 2 = -F2,x i + F2,y j
F 3= -F3,x i - F3,y j F 4= F4,x i - F4,y j
Force summation
For both cases of notations, the magnitude of the resultant force is found by:
And the direction is found by:
FR,x =∑Fx
FR,y =∑Fy
2
yR,
2
xR,R FFF FR,y
FR,x
FR
θ
xR
yR
F
F
,
,1tan
+ve
+ve
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Examples
Examples
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Examples
Examples
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Cartesian Vectors
Forces
Three
dimensions
Two
dimensions
One
dimension
Cartesian Vectors
• In three dimensions system, new component appeared (Az).
• The new dimension direction is represented by unit vector (k)
• The position of the vector has to be located by three angles one from each axis
• The projection of vector on x-y plane represent a new
vector called A’
A
Az
z
y
x
Ax
Ay
A’
Note that
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Cartesian Vectors
Right hand coordinate system
• This method is used to describe the rectangular coordinate system.
• The system is said to be right handed if the thumb points to the positive z-axis and the fingers are curled about this axis and points from the positive x-axis to the positive y-axis.
Cartesian Vectors
Notation • To find the components of a
vector oriented in three dimensions, two successive applications of the parallel-ogram must be done.
• One of the parallelogram applications is to resolve A to A’and Az and the other is used to resolve A’ into Ax and Ay.
• A = A’ + Az = Ax + Ay + Az
A
Az
z
y
x
Ax
A y
A’
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Cartesian Vectors
Angles
• α is the angle between the vector and x-axis
• β is the angle between the vector and y-axis
• γ is the angle between the vector and z-axis
αβ
γ
Cartesian Vectors
Cartesian unit vectors
k
j
i
A = Ax i + Ay j + Az k
Ax = Acos(α)
Ay = Acos(β)
Az = Acos(γ)
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Cartesian Vectors
Magnitude and Direction
Magnitude
Direction
222
zyx AAAA
A
A
A
A
A
A
z
y
x
cos
cos
cos
1coscoscos 222
Examples
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Examples
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Cartesian Vectors
Other direction definition
A = Ax i + Ay j + Az k
kjiuA
A
A
A
A
A
A
zyxA
A
uA = cos(α) i + cos(β) j + cos(γ) k
A = |A| uA
= |A| cos(α) i + |A| cos(β) j + |A| cos(γ) k
= Ax i + Ay j + Az k
Cartesian Vectors
Other direction definition cont
Az = A cos(ϕ)
A’ = A sin(ϕ)
Ax = A’ cos(θ) = A sin(ϕ) cos(θ)
Ay = A’ sin(θ) = A sin(ϕ) sin(θ) θ
ϕ
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Cartesian Vectors
Cartesian Vectors Addition
• A = Ax i + Ay j + Az k
• B = Bx i + By j + Bz k
• R = A+ B
= (Ax + Bx ) i + (Ay + By) j +(Az + Bz ) k
General rule:
FR = ∑F = ∑Fx i + ∑Fy j + ∑Fz k
Cartesian Vectors
Example [1]:Question: assume the following forces:
F1 = 5 i + 6j -4k
F2 = -3i +3j +3k
F3=7i-12j+2k
Find the resultant force F = F1 + F2 +F3 and represent it in both Cartesian and scalar notations
Solution:
Given: F1, F2 and F3
Required: find resultant force F and represent it in both Cartesian and scalar notation
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Cartesian Vectors
Example [1]:
Solution:
F = (5-3+7)i + (6+3-2)j +(-4+3+2)k = 9i +7j +10k (Cartesian notation)
Scalar notation:
Magnitude:
Unit vector u:
NF 17.151079 222
kjikjiF
u 66.046.06.015.17
10 7 9
F
7.48 66.0cos
61.62 46.0cos
13.53 6.0cos
o
o
o
Cartesian Vectors
Position vector
Assume: r = ai + bj + ck
• At this point, r is called position vector.
• Position vector is a vectorlocates a point in space withrespect to other point.
• in this case, vector r is aposition vector relates point P
to point O.
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Cartesian Vectors
Position vector
• Assume vectors rA and rB are used to locate points A and B from the origin (ie. Point 0,0,0).
• Define a position vector r to relate point A to point B.
• rB = r + rA → r = rB – rA
• r = (xB – xA)i + (yB – yA)j + (zB – zA)k
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