Engineering Mechanics A science which studies the response of bodies (solids or fluids) or systems of bodies to external forces.

Definitions: Scalar quantities posses only magnitude, examples are time, volume, energy, mass, density, work. Scalars are added by ordinary algebraic method. Vector quantities posses both magnitude and direction, examples are force, displacement, velocity, impulse. A vector is represented by an arrow at the given inclination. The head of the arrow indicates the sense, and the length represents the magnitude of the vector.

Vector quantities are often represented by scaled vector diagrams. Vector diagrams depict a vector by use of an arrow drawn to scale in a specific direction. Vector diagrams were introduced and used in earlier units to depict the forces acting upon an object.

(1) A scale is clearly listed (2) A vector arrow (with arrowhead) is drawn in a specified direction. The vector arrow has a head and

a tail. (3) The magnitude and direction of the vector is clearly labeled. In this case, the diagram shows the

magnitude is 20 m and the direction is (30 degrees West of North).

Basic Quantities: The following are quantities that are used throughout the subject:

Length: Length is needed to locate the position of a point in space and thereby describe the size of the physical system. Unit: meters, feet, inch Time: Time is conceived as succession of events. Unit: seconds, minutes, hours Mass: Mass is the property of matter by which we can compare the action of one body with another body. Unit: kg or lbm Force: Force is considered as “push” or “pull” exerted by one body to another. Unit: N or lbf

Idealization: Used to simplify application of the theory:

Particle: A physical body or portion of a physical body which has mass, the dimension of which are negligible in terms of its surroundings.

Rigid body: Any quantity of matter wherein the particles are definitely fixed and remained unaltered when forces are applied. Concentrated force: Represents the effect of a loading which assumed to act on a point in a body. Newton’s Law of Motion:

First Law: A particle originally at rest, or originally moving in a straight line with constant velocity will remain in this state provided the particle is not subjected to an unbalanced force. Second Law: A particle acted upon by an unbalance force F, experience an acceleration that has the same direction as the force and a magnitude that is directly proportional to the force. F = ma. Third Law: The mutual forces of action and reaction between two particles are equal, opposite and collinear.

VECTOR ADDITION:

A variety of mathematical operations can be performed with and upon vectors. One such operation is the addition of vectors. Two vectors can be added together to determine the result (or resultant). The resultant or the net force of a system of vectors is the least number of vectors that will replace the given system.

A vector directed up and to the right will be added to a vector directed up and to the left. The vector sum will be determined for the more complicated cases shown in the diagrams below.

There are a variety of methods for determining the magnitude and direction of the result of adding two or more vectors. The two methods that will be discussed in this lesson and used throughout the entire unit are:

• The Pythagorean Theorem and trigonometric methods • The head-to-tail method using a scaled vector diagram (Graphical Method)

The Pythagorean Theorem and Trigonometric Methods:

The Pythagorean Theorem is a useful method for determining the result of adding two (and only two) vectors that makes a right angle to each other. The method is not applicable for adding more than two vectors or for adding vectors that are not at 90-degrees to each other.

Example: Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement.

The head-to-tail method using a scaled vector diagram (Graphical Method)

The magnitude and direction of the sum of two or more vectors can also be determined by use of an accurately drawn scaled vector diagram. Using a scaled diagram, the head-to-tail method is employed to determine the vector sum or resultant.

Example:

The head-to-tail method is employed as described above and the resultant is determined (drawn in red). Its magnitude and direction is labeled on the diagram

When added together in this different order, these same three vectors still produce a resultant with the same magnitude and direction as before (20. m, 312 degrees). The order in which vectors are added using the head-to-tail method is insignificant.

References: Schaum’s Outline of Theory and Problems of Engineering Mechanics Statics and Dynamics 5th Edition by E.W. Nelson, C.L. Best and W.G. McLean © 1998 by McGraw-Hill Companies, Inc. http://www.physicsclassroom.com