1-Eng2P04-DS-Notes - ENGINEERING 2P04 STATICS Edited and...

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ENGINEERING 2P04 STATICS Edited and modified by Dieter Stolle July 2010
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ENG2P04 2
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ENG2P04 1 A BRIEF INTRODUCTION TO "STATICS" This set of notes was originally prepared by Dr. Bob Sowerby of Mechanical Engineering during the early 1970‟s to supplement the textbook by Popov. They are not comprehensive but provide useful background. These notes, along with the lectures, tutorial classes and assignments, should be sufficient to provide the basic principles of the subject. Additional material on mathematics has been added, as well as on mechanics concepts from the lecture notes of Dr. Gunhard Oravas. It is assumed that the student has a basic knowledge of vector Algebra and calculus, obtained from their Level I courses. The first part of this course will deal with the resolution of vectors, vector addition and finding the resultant of a system of vectors. These vectors represent mechanical entities called forces and moments. MATHEMATICAL PRELIMINARIES (Stolle) Coordinate System and Vectors Students at this stage should be familiar with notions such as temperature and density. These measures can be described in terms of a single number, referred to as a scalar quantity. When dealing with notions such as velocity, force or displacement, it is necessary to work with quantities referred to as vectors . A vector, which has both a magnitude and direction associated with it, is independent of a coordinate system. However, if one wishes to perform meaningful calculations it then becomes necessary to refer to a coordinate system. Vector notation provides a convenient compact means for writing equations used to solve boundary-valued problems. Given that we restrict ourselves to a Cartesian coordinate system, we can define a position vector r in a number of ways. In long form r may be defined as r xi yj zk or by the ordered set (coordinate) representation   ,, x y z r where the unit vector directions i j k are implied. Note that bolded characters are sometimes used to denote vectors (or matrices). Magnitude of vector: r r r  or r r r
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ENG2P04 2 Alternatively, one can use index (or indicial) notation, in which 1 1 2 2 3 3 ii r x e x e x e x e with summation being implied by the repeated index i . Another popular representation is given by matrix notation in which we define a vector 1 2 3 T x x x x with the superscript T implying the transpose of a vector or matrix. To demonstrate the use of these notations and the connection between them, consider the following examples: (a) Dot Product (Work done) – W = cos F u F u  , where is the angle between the two vectors. Given Cartesian vectors written as   1 2 3 ,, F F F F and   1 2 3 u u u u , we have Indicial Notation 1 2 2 3 3 i i 1 F u Fu F u Matrix Notation 1 2 1 2 2 3 3 3 W = = = T 1 2 3 1 u F F F u F u F u F u u      ab Although this example is for three-dimensional (3-D) space, the rules of operation also apply to n -dimensional space.
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This note was uploaded on 09/21/2011 for the course ENGINEERIN 2P04 taught by Professor Ss during the Spring '11 term at McMaster University.

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1-Eng2P04-DS-Notes - ENGINEERING 2P04 STATICS Edited and...

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