MECE 701 Fundamentals of Mechanical Engineering

MECE 701 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(m 1 m 2 )/r 2 F =force of gravitation btw two particles G =Universal constant of gravitation 66.73( )m 3 /(kg.s 2 ) m 1,m 2 =mass of each of the two particles r = distance between two particles

Newton’s Laws of Motion Weight W=weight m 2 =mass of earth r = distance btw earth’s center and the particle g=gravitational acceleration g=Gm 2 /r 2 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) 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 A R

Trigonometry Sine Law A B C a b c Cosine Law

Cartesian Vectors Right Handed Coordinate System A=A x +A y +A z

Cartesian Vectors Unit Vector A unit vector is a vector having a magnitude of 1. Unit vector is dimensionless.

Cartesian Vectors Cartesian Unit Vectors A= A x i+A y j+A z k

Cartesian Vectors Magnitude of a Cartesian Vector Direction of a Cartesian Vector DIRECTION COSINES

Cartesian Vectors Unit vector of A

Cartesian Vectors Addition and Subtraction of Cartesian Vectors R=A+B=(A x +B x )i+(A y +B y )j+(A z +B z )k R=A-B=(A x -B x )i+(A y -B y )j+(A z -B z )k

Dot Product Result is a scalar. Result is the magnitude of the projection vector of A on B.

Dot Product Laws of Operation Commutative law: Multiplication by a scalar: Distributive law:

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 Commutative law is not valid: Multiplication by a scalar: Distributive law: a(AxB) = (aA)xB = Ax(aB) = (AxB)a Ax(B+D) = (AxB) + (AxD)

Cross Product