MECE 701 Fundamentals of Mechanical Engineering

MECE 701

MECE701

Engineering Mechanics

Machine Elements&

Machine Design

Mechanics of Materials

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 particlesG =Universal constant of gravitation

66.73(10-12)m3/(kg.s2)

m1,m2 =mass of each of the two particlesr = 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|

A2A -1.5A

Basic Vector Operations

Vector Addition

Resultant (R)= A+B = B+A

(commutative)

A

B

A

B

R=A+B

Parallelogram Law

A

B

R=A+B

Triangle Construction

A

B

R=A+B

Basic Vector Operations

Vector Subtraction

R= A-B = A+(-B)

Resolution of a Vector

b

a

B

AR

Trigonometry

Sine Law

c

C

b

B

a

A

sinsinsinA B

C

ab

c

Cosine Law

cABBAC cos222

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.

AuA

A

Cartesian Vectors

Cartesian Unit Vectors

A= Axi+Ayj+Azk

Cartesian Vectors

Magnitude of a Cartesian Vector

222zyx AAAA

Direction of a Cartesian Vector

A

AxcosA

AycosA

Azcos

DIRECTION COSINES

Cartesian Vectors

Unit vector of A

kA

Aj

A

Ai

A

A

A

Au zyxA ||||||||

kjiuA coscoscos

1coscoscos 222

kAjAiA coscoscos A

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.

cosABBA

Result is the magnitude of the projection vector of A on B.

Dot Product

Laws of Operation

ABBA Commutative law:

Multiplication by a scalar:

Distributive law:

aBAaBABaABAa )()()()(

)()()( DABADBA

Cross Product

The cross product of two vectors A and B yields the vector C

C = A x B

Magnitude:

C = ABsinθ

Cross Product

Laws of Operation

ABBA Commutative law is not valid:

Multiplication by a scalar:

Distributive law:

ABBA

a(AxB) = (aA)xB = Ax(aB) = (AxB)a

Ax(B+D) = (AxB) + (AxD)

Cross Product

)()( kBjBiBkAjAiABA zyxzyx

kBABAjBABAiBABABA xyyxxzzxyzzy )()()(

Cross Product

zyx

zyx

BBB

AAA

kji

BA

kBABAjBABAiBABABA xyyxxzzxyzzy )()()(