MECE 701
Fundamentals of
Mechanical
Engineering
MECE 701
Engineering Mechanics
Mechanics of Materials
MECE701
Machine Elements
&
Machine Design
Materials Science
Fundamental Concepts
Idealizations:
Particle:
A particle has a mass but its size can be
neglected.
Rigid Body:
A rigid body is a combination of a large number
of particles in which all the particles remain at
a fixed distance from one another both before
and after applying a load
Fundamental Concepts
Concentrated Force:
A concentrated force represents the
effect of a loading which is assumed to
act at a point on a body
Newton’s Laws of Motion

First Law:
A particle originally at rest, or moving in a
straight line with constant velocity, will
remain in this state provided that the particle
is not subjected to an unbalanced force.
Newton’s Laws of Motion

Second Law
A particle acted upon by an unbalanced
force F experiences an acceleration a that
has the same direction as the force and a
magnitude that is directly proportional to the
force.
F=ma
Newton’s Laws of Motion

Third Law
The mutual forces of action and reaction
between two particles are equal, opposite,
and collinear.
Newton’s Laws of Motion

Law of Gravitational Attraction
F=G(m1m2)/r2
F =force of gravitation btw two particles
G =Universal constant of gravitation
66.73(10-12)m3/(kg.s2)
m1,m2 =mass of each of the two particles
r = distance between two particles
Newton’s Laws of Motion

Weight
W=weight
m2=mass of earth
r = distance btw earth’s center and the particle
g=gravitational acceleration
g=Gm2/r2
W=mg
Scalars and Vectors
Scalar:
A quantity characterized by a positive or
negative number is called a scalar. (mass,
volume, length)
 Vector:
A vector is a quantity that has both a
magnitude and direction. (position, force,
momentum)

Basic Vector Operations

Multiplication and Division of a Vector
by a Scalar:
The product of vector A and a scalar a
yields a vector having a magnitude of
|aA|
A
2A
-1.5A
Basic Vector Operations
Vector Addition
Resultant (R)= A+B = B+A
(commutative)

Parallelogram Law
A
Triangle Construction
B
R=A+B
A
R=A+B
A
A
B
B
B
R=A+B
Basic Vector Operations

Vector Subtraction
R= A-B = A+(-B)

Resolution of a Vector
a
A
R
B
b
Trigonometry

A
c
Sine Law
A
B
C


sin a sin b sin c
B
a
b

Cosine Law
C
C  A  B  2 AB cos c
2
2
Cartesian Vectors
Right Handed Coordinate System
A=Ax+Ay+Az
Cartesian Vectors

Unit Vector
A unit vector is a vector having a magnitude of 1.
Unit vector is dimensionless.
A
uA 
A
Cartesian Vectors

Cartesian Unit Vectors
A= Axi+Ayj+Azk
Cartesian Vectors
Magnitude of a Cartesian Vector

A

Ax  Ay  Az
2
2
2
Direction of a Cartesian Vector
Ax
cos  
A
cos  
Ay
A
DIRECTION COSINES
cos  
Az
A
Cartesian Vectors

Unit vector of A
Ay
Ax
A
Az
uA 

i
j
k
| A| | A| | A|
| A|
u A  cos i  cos j  cos k
cos 2   cos 2   cos 2   1
A  A cos i  A cos j  A cos k
Cartesian Vectors

Addition and Subtraction of Cartesian Vectors
R=A+B=(Ax+Bx)i+(Ay+By)j+(Az+Bz)k
R=A-B=(Ax-Bx)i+(Ay-By)j+(Az-Bz)k
Dot Product
Result is a scalar.
A  B  AB cos
Result is the magnitude of the projection
vector of A on B.
Dot Product

Laws of Operation
Commutative law:
A B  B  A
Multiplication by a scalar:
a( A  B)  (aA)  B  A  (aB)  ( A  B)a
Distributive law:
A  ( B  D)  ( A  B)  ( A  D)
Cross Product
The cross product of two vectors A and B yields the
vector C
C=AxB
Magnitude:
C = ABsinθ
Cross Product

Laws of Operation
Commutative law is not valid:
A B  B  A A B  B  A
Multiplication by a scalar:
a(AxB) = (aA)xB = Ax(aB) = (AxB)a
Distributive law:
Ax(B+D) = (AxB) + (AxD)
Cross Product
A  B  ( Ax i  Ay j  Az k )  ( Bx i  B y j  Bz k )
A  B  ( Ay Bz  Az B y )i  ( Ax Bz  Az Bx ) j  ( Ax B y  Ay Bx )k
Cross Product
i
A  B  Ax
Bx
j
Ay
By
k
Az
Bz
A  B  ( Ay Bz  Az B y )i  ( Ax Bz  Az Bx ) j  ( Ax B y  Ay Bx )k