BSBSCPH- 101 B. Sc. I YEAR MECHANICS DEPEPARTMENT OF PHYSICS SCHOOL OF SCIENCES UTTARARAKHAND OPEN UNIVERSIRSITY Board of Studies and Programme Coordinator Board of Studies Prof. S. C. Garg Prof. R.P. Gairola ,ice Chancellor, Usha Martin University , Ranchi, Department of Physics, HNBG University, -har hand Shrinagar, Garhwal Prof S.R. (ha Prof. H. M. Agrawal Professor of Physics , School of Sciences Professor and Head, Department of Physics, ..G.N.O.U, Maidan Garhi, New Delhi CBSH, G.B.P.U.A.01. Pantnagar, .ndia Prof. P D Pant Dr. $amal Devlal Director ./C, School of Sciences Department of Physics Uttara hand Open University, Haldwani School of Sciences, Uttara hand Open University Programme Coordinator Dr. $amal Devlal Department of Physics School of Sciences, Uttara hand Open University Haldwani, Nainital Unit writing and Editing Editing Writing Prof. R. P. Gairola 1. Prof. Madan Singh Department of Physics Department of Physics and Electronics HNB Garhwal University (Central National University of Lesotho, Southern Africa 2. Dr. L. P. erma University), Associate Prof., Department Physics Shrinagar, Uttara hand M.B.G.P.G.C. Haldwani, Nainital, Uttara hand 3. Dr. Mahipal Singh Asstt. Professor ,Department Physics R.H.G.P.G.C. Kashipur, US Nagar, Uttara hand 4. Dr. $amal Devlal Department of Physics School of Sciences, Uttara hand Open University 5. Dr. Tara Bhatt Asstt Professor, Department Physics M.B. G.P.G.C. Haldwani, Nainital, Uttara hand Course Title and Code , Machanics .BSCPH 1011 2SBN , Copyright , Uttara5hand Open University Edition , 2017 Published By , Uttara5hand Open University, Haldwani, Nainital9 263139 Printed By , BSCPH9101 Mechanics DEPARTMENT OF PHYS2CS SCHOOL OF SC2ENCES UTTARA$HAND OPEN UN2 ERS2TY Phone No. 059469261122, 261123 Toll free No. 18001804025 Fax No. 059469264232, E. mail [email protected] htpp,AAuou.ac.in Contents / a / ./tI / . Ü t . ! !"# Ç !$ $ / 5 5 $!% D ' { D( D { )!"# D . $ a * t "%!+ ) ,-( - /. / / b "%!#" " P / * . * / #0!" 0 1 0!)0 ! + a * / / a / / )+!+ .2h/4 5 * 1 . +)!$# # a * 9! a +)!#" t aL & w % C * * C #0!%% ( ) t / Y + , $!$# .2h/4 ) D . t/ * $%!$ D . D D $%!$$# D $ 7 / . . / 9 D $$%!$" D /. * . 8 - / - $")!$+0 , C , t , / . L ! Y + ) 7 I+ . $++!$%% , " 9 * * . .!$ ./tI a9/I!bL/ UNIT 1: VECTOR STRUCTURE: 1.0 Objective 1.1 Introduction 1.2 Vector representation 1.2.1 Unit Vector 1.2.2 Zero Vector 1.2.3 Graphic representation of vector 1.2.4 Addition and subtraction of vectors 1.2. Resolution of a vector 1.2.6 Direction cosines 1.2.7 Position vector 1.3 Multiplication of vector 1.3.1 Multiplication and division of a vector by scalar 1.3.2 Product of two vectors 1.3.2.1 Scalar Product or dot product 1.3.2.2 Vector product or cross product 1.3.3 Product of three vectors 1.3.3.1 scalar triple product 1.3.3.2 vector triple product 1.4 Solved examples 1. Summary 1.6 Glossary 1.7 Self Assessment -uestions 1.. References, su00ested readin0 1.1 2erminal -uestions 1.1.1 Short answer type -uestions 1.1.2 Essay type -uestion 1.1.3 Numerical -uestion 0°• ./tI a9/I!bL/ 1.0 Objective: After readin0 this unit you will be able to understand5 6 Definin0 vector. 6Vector representation, addition, subtraction 6Ortho0onal representation 6Multiplication of vectors 6Scalar product, vector product 6Scalar triple product and vector triple product 1.1 Introduction: On the basis of direction, the physical -uantities may be divided into two main classes. 1.1.1 Scalar -uantities5 2he physical -uantities which do not re-uire direction for their representation. 2hese -uantities re-uire only ma0nitude and unit and are added accordin0 to the usual rules of al0ebra. Examples of these -uantities are5 mass, len0th, area, volume, distance, time speed, density, electric current, temperature, wor7 etc. 1.1.2 Vector -uantities5 2he physical -uantities which re-uire both ma0nitude and direction and which can be added accordin0 to the vector laws of addition are called vector -uantities or vector. 2hese -uantities re-uire ma0nitude, unit and direction. Examples are wei0ht, displacement, velocity, acceleration, ma0netic field, current density, electric field, momentum an0ular velocity, force etc. 1.2 Vector representation5 Any vector -uantity say A, is represented by puttin0 a small arrow above the physical -uantity li7e In case of print text a vector -uantity is represented by bold type letter li7e A. 2he vector can be represented by both capital and small letters. 2he ma0nitude of a vector -uantity A is denoted by or mod A or some time li0ht forced italic letter A. 8e should understand followin0 types of vectors and their representations. 1.2.1 Unit vector A unit vector of any vector -uantity is that vector which has unit ma0nitude. Suppose then unit vector is defined as 0°• ./tI a9/I!bL/ 2he unit vector is denoted by and read as 9A unit vector or A hat‘. It is clear that the ma0nitude of unit vector is always 1. A unit vector merely indicates direction only. In Cartesian coordinate system, the unit vector alon0 x, y and x axis are represented by , and respectively as shown in fi0ure 1.1. y x z igure 1.1 Any vector in Cartesian coordinate system can be represented as 8here , and are unit vector alon0 x, y z axis! and, ! , !, are the ma0nitudes projections or components of alon0 x, y, z axis respectively.! ! ! 2he unit vector in Cartesian coordinate system can be 0iven as5 ! " # $ & & & % ' ( 1.2.2. "ero vector or Null vector: A vector with zero ma0nitude is called zero vector or null vector. 2he condition for null vector is = 0 1.2.3 Equal vectors: If two vectors have same ma0nitude and same direction, the vectors are called e-ual vector. 0°• ./tI a9/I!bL/ 1.2.4 Li)e vectors: If two or more vectors have same direction, but may have different ma0nitude, then the vectors are called li7e vectors. 1.2.5 Negative vector: A vector is called ne0ative vector with reference to another one, if both have same ma0nitude but opposite directions. 1.2.6 Collinear vectors: All the vectors parallel to each other are called collinear vectors. Basically collinear means the line of action is alon0 the same line. 1.2.6 Coplanar vector: All the vectors whose line of action lies on a same plane are called coplanar vectors. Basically coplanar means lies on the same plane. 1.2.,. -rap.ical representation of vectors: Graphically a vector -uantity is represented by an arrow shaped strai0ht line, with suitable len0th which represents ma0nitude, and the direction of arrow represents direction of vector -uantity. For example, if a force is directed towards east and another force is directed toward north-west then these forces can be represented as shown in fi0ure )*1.2. N = 1 N )*** 8 E =10 N *** S igure 1.2 1.2.0 Addition and subtraction of vectors: 0°• ./tI a9/I!bL/ 2he addition of two vectors can be performed by followin0 two laws. 1A2 T.e parallelogram law: Accordin0 to this law, if two vectors and are represented by two adjacent sides of a parallelo0ram as show in fi0ure 1.3, then the)* sum of these two vectors or resultant is represented by the dia0onal of Parallelo0ram. +* )*** )*** **+* *** *** igure 1.3 If vector and are represented by the sides of a parallelo0ram as shown in fi0ure 1.4 and the an0le between )* and is , and resultant ma7es an0le with vector then ma0nitude of is )* , +* - +* / / + . ) 0) 123 , 2he an0le is 0iven as - 56 ) 378 , - 4 ) , You should notice that all three vectors , and are concurrent i.e. vectors actin0 on the same point O. )* +* )*** **+* O A , *** igure 1.4 0°• ./tI a9/I!bL/ 152 Triangle law: Accordin0 to this law if a vector is placed at the head of another vector, and these two vectors represent two sides of a trian0le then the third side or a vector drawn for the tail end of first to the head end of second represents the resultant of these two vectors. If vectors and are two vector as shown in fi0ure 1. , then resultant can be obtained by applyin0 trian0le)* law. +* )*** **+* )*** *** *** igure 1.5 1c2 Polygon law of vector addition: 2his law is used for the addition of more than two vectors. Accordin0 to this law if we have a lar0e number of vectors, place the tail end of each successive vector at the head end of previous one. 2he resultant of all vectors can be obtained by drawin0 a vector from the tail end of first to the head end of the last vector. Fi0ure 1.6 shows the polynomial law of vector addition different vectors , , , , etc. and is resultant vector. )* 9 :** ;* +* ;*** ;*** :*** +* :*** 9*** 9*** )*** )*** *** *** igure 1.6 0°• ./tI a9/I!bL/ 1.2.8 Resolution of vector: A vector can be resolved into two or more vectors and these vectors can be added in accordance with the poly0on law of vector addition, and finally ori0inal vector can be obtained. If a vector is resolved into three components which are mutually perpendicular to each other then these are called rectan0ular components or mutual perpendicular components of a vector. 2hese components are alon0 the three coordinate axes x, y and z respectively as show in fi0ure 1.7. y < x = > z igure 1., If the unit vectors alon0 x, y and x axis are represented by , and respectively then any vector can be 0ive as ! ! ! constitutes the dia0onal of a parallelepiped, and , and are the ed0es alon0 x, y and z axes respectively. is polynomial addition ! !of vectors! , and . 2he rectan0ular components , and can be considered as ortho0onal ! ! projections! of ! ! ! 0°• ./tI a9/I!bL/ vector on x, y and z axis respectively.