INTRODUCTION TO ENGINEERING MECHANICS - SPP.pptx

Engineering Mechanics, Mechanics, Scalars, Vectors, Force system, Measurment units, Concept of force, system of forces, Idealized Mechanics, Fundamental Concepts, Scalar and vector Operations, Accuracy of Engineering Calculation, Problem Solving Approach

1. Introduction to Engineering Mechanics Prof. Samirsinh P. Parmar [email protected] , [email protected] Asst. Prof. Dept. of Civil Engineering Dharmasinh Desai University, Nadiad, Gujarat, India Lecture-0 CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 1

2. Content of the presentation General Introduction to Engineering Mechanics SI System of Units Concept of Force System of forces Idealization in Mechanics Fundamental Concepts Scaler and Vector Quantities Accuracy in Calculation Problem Solving Approach Reference BOOKS: CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 2

3. General Introduction to Engineering Mechanics CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 3

4. Engineering Mechanics CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 4

5. 1. Statics:  It is the branch which deals with the forces and their effects on an object or a body at rest. For example, if we have an object or a body at rest and we deal with the forces and their effects that are acting on the body than we are dealing with static branch of engineering mechanics. 2. Dynamics: It is the branch which deals with the forces and their effects on the bodies which are in motion. For example, if we have a body that is moving and we are dealing with the forces and their effects on the moving body than we are dealing with dynamics branch. General Introduction to Engineering Mechanics CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 5

6. Types of Dynamics Dynamics is also divided into two branches and these are: ( i ) . Kinetics: Kinetics is defined as the branch of dynamics which deals with the bodies that are in motion due to the application of forces. (ii) . Kinematics: It is defined as the branch of dynamics which deals with the bodies that are in motion, without knowing the reference of forces responsible for the motion in the body. General Introduction to Engineering Mechanics CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 6

7. SI System of units CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 7

8. Mechanics: Units W  mg F  ma Four Fundamental Quantities → N = kg.m/s 2 → N = kg.m/s 2 1 Newton is the force required to give a mass of 1 kg an acceleration of 1 m/s 2 Quantity Dimensional SI UNIT Symbol U n i t S ymb o l M ass M L e n g t h L K il o g r am Kg Meter M T i me T F o r c e F Second s N e w t o n N Basic Unit CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 8

9. Fundamental units of S.I system CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 9

10. Principal S.I. units CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 10

11. S.I. Prefixes CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 11

12. UNIT CONVERSION CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 12

13. Concept of Force CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 13

14. Concept of Force CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 14

15. The necessity of force: To move a stationary object i.e. to move a body which is at rest. To change the direction of the motion of an object To change the magnitude of the velocity (speed) of the motion of an object To change the shape of an object. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 15

16. Effects of force It may set a body into motion It may bring a body to rest. It may change the magnitude of motion It may change the direction of motion It may change the magnitude and direction of motion It may change the shape of an object CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 16

17. Characteristics of Force It has four characteristics Direction Magnitude Point on which it acts Line of action CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 17

18. Line of Action of force Th e li n e o f action of a f o r c e f i s a g eom e tr i c representation of how the force is applied. It is the line through the point at which the force is applied in the same direction as the vector f→. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 18

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20. System of Forces When t w o a r e or mo r e f o r ces acts act o n a body, they are called system of forces. Coplanar Force system – 2D and Non – Coplanar system – 3D Concurrent and Non – Concurrent Force system Collinear and Non- Collinear Force system Parallel – Like and Unlike CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 20

21. System of Forces CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 21

22. Coplanar Force System – 2D CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 22

23. Non- Coplanar Force System – 3D CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 23

24. Concurrent and Non – Concurrent Force system Concurrent Forces Non- Concurrent Forces CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 24

25. Collinear and Non- Collinear Force system Collinear Forces Non – Collinear Forces CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 25

26. Parallel Force system CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 26

27. Idealization in Mechanics Models or idealizations are used in order to simplify application of the theory of mechanics. Here we will consider three important idealizations Rigid Body Particle Concentrated force CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 27

28. Mechanics: Idealizations To simplify application of the theory Particle : A body with mass but with dimensions that can be neglected Size of earth is insignificant compared to the size of its orbit. Earth can be modeled as a particle when studying its orbital motion CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 28

29. Mechanics: Idealizations Rigid Body : A combination of large number of particles in which all particles remain at a fixed distance (practically) from one another before and after applying a load. Material properties of a rigid body are not required to be considered when analyzing the forces acting on the body. In most cases, actual deformations occurring in structures, machines, mechanisms, etc. are relatively small, and rigid body assumption is suitable for analysis CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 29

30. Mechanics: Idealizations Concentrated Force : Effect of a loading which is assumed to act at a point ( CG ) on a body. Provided the area over which the load is applied is very small compared to the overall size of the body. Ex: Contact Force between a wheel and ground . 40 kN 160 kN CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 30

31. Fundamental Concepts CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 31

32. Mechanics: Fundamental Concepts Length (Space) : needed to locate position of a point in space, & describe size of the physical syste m. → Distances, Geometric Properties Time : measure of succession of even . → basic quantity in Dynamics Mass : quantity of matter in a body → measure of inertia of a body (its resistance to change in velocity) Force : represents the action of one body on another → characterized by its magnitude, direction of its action, and its point of application 🡪 Force is a Vector quantity . CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 32

33. Mechanics: Fundamental Concepts Newtonian Mechanics Length, Time, and Mass are absolute concepts independent of each other Force is a derived concept not independent of the other fundamental concepts. Force acting on a body is related to the mass of the body and the variation of its velocity with time. Force can also occur between bodies that are physically separated (Ex: gravitational, electrical, and magnetic forces) CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 33

34. Mechanics: Fundamental Concepts Remember: Mass is a property of matter that does not change from one location to another. Weight refers to the gravitational attraction of the earth on a body or quantity of mass. Its magnitude depends upon the elevation at which the mass is located Weight of a body is the gravitational force acting on it. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 34

35. Scalar and Vector Quantities CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 35 BASIS FOR COMPARISON SCALAR QUANTITY VECTOR QUANTITY Meaning Any physical quantity that does not include direction is known as a scalar quantity. A vector quantity is one, that has both magnitude and direction. Quantities One-dimensional quantities Multi-dimensional quantities Change It changes with the change in their magnitude. It changes with the change in their direction or magnitude or both. Operations Follow ordinary rules of algebra. Follow the rules of vector algebra. Comparison of two quantities Simple Complex Division Scalar can divide another scalar. Two vectors can never divide.

36. What is a scalar? Scalar quantities are measured with numbers and units. length (e.g. 102 °C) time (e.g. 16 cm) temperature (e.g. 7 s) CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 36

37. What is a vector? Vector quantities are measured with numbers and units, but also have a specific direction . acceleration (e.g. 30 m/s 2 upwards) displacement (e.g. 200 miles northwest) force (e.g. 2 N downwards) CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 37

38. Speed or velocity? Distance is a scalar and displacement is a vector. Similarly, speed is a scalar and velocity is a vector. Speed is the rate of change of distance in the direction of travel. Speedometers in cars measure speed. Velocity is a rate of change of displacement and has both magnitude and direction. average speed average velocity Averages of both can be useful: distance time displacement time = = CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 38

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40. Goa l s for C hapter 1 To l e arn three fun d a m ental q u anti t i e s of p h ysics and the u n i t s to m e a sure them To u n der s t a nd vec t ors and s c a l a r s and h o w to a dd vec t ors grap h i c a l ly To dete rm ine vec t or co m p o ne n ts and h o w to use t hem in c a l c ulat i o n s To u n ders t and u n it vec t ors and h o w to use t hem with co m p o nents to de s c r ibe vec t ors To l e arn two ways of m ultiplying vectors 40 CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad

41. Unit consi s tency and conver s ions An equa t ion m ust be dimens i onal l y cons i st e n t . T er m s to be added or equat e d m ust always have the sa m e uni t s. (Be su r e you’ r e adding “apples to appl e s.”) Al w ays carry units through calculat i ons. Convert to st a ndard uni t s as nece s sa r y . 41 CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad

42. Vectors and sca l ars A sca l ar quan t i t y can be descr i bed by a s i ng l e numbe r . A vector quant i ty has b oth a magnitude and a di r ect i on in space. In th i s book, a vector quant i ty is represen t ed in bold f ace i t al i c type with an arrow over i t : A . The magni t ude of A is w r i t t en as A or | A |. 42 CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad  

43. Dra w ing vectors — Figure 1.10 Draw a ve c tor as a l ine wi t h an a r row h e a d at i t s tip. T h e l e n g th of the l ine s h o ws the vecto r ’ s m a g n i t u d e . T h e di r e c t i on of the l i ne s h o ws the vec t o r ’ s di r e c t i o n . 43 CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad

44. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 44 Adding two vectors graphica l ly T wo vectors m ay be ad ded graphically usin g either the p arallelog ram m etho d or the he a d -to-tail m ethod.

45. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 45 A d ding m o re than t w o v e c tors gr a p h i c all y — T o add several vec t ors, use the hea d -to-tail m ethod. The vectors can be added in any orde r .

46. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 46 The negativ e o f a vec t o r i s define d a s the vec t or tha t , whe n adde d to the origina l vector , give s a result a n t o f zero The negativ e o f the vector wi l l hav e the same magnitude, bu t poin t i n the opposit e direction Represen t ed as • A    A    A Negative of a Vector

47. Subtract i ng vectors CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad Fig u re sho w s h o w to subtra c t ve c tors. 47

48. CL- Engg . Mechanics, DoCL - SPP, DDU, Nadiad 48 Mul t ip l ying a vector by a scal a r I f c is a sca l ar, the product c A has ma g ni t ude | c | A . Mul t ip l ication of a vec t or by a posi t ive sca l ar and a negat i ve sca l ar. 

49. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 49 Addit i on of t w o vec t ors at right angles

50. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 50 Components of a vecto r — Ad ding vectors graphically provides li m ited ac c urac y . V ector co m p o ne nts provide a general m etho d for adding vectors. Any vector can be represented by an x -co m p o ne nt A x and a y - co m p o ne nt A y . Use trigo n o m etry to fin d the co m p o ne nts of a vector: A x = A cos θ and A y = A sin θ , w here θ is m ea s ured from the + x - axis toward the + y - axis.

51. Components of a Vector T h e x - c o mp o n e n t o f a v e ctor i s the projection alon g the x -axis A x  A c o s  T h e y - c o mp o n e n t o f a v e ctor i s the projection alon g the y -axis A y  A s i n  T h i s a s s u mes the a n gl e θ is mea s ure d wit h r es p ec t to t h e po s it i ve directio n o f x -axis If n ot, do n ot use these eq uations, use the sides of the triangle directly CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 51

52. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 52 Components of a Vector, 4 T h e c o mp o n e nt s ar e the leg s o f the right t riangle wh ose h y p otenuse i s the len g th o f A May still ha ve to fin d θ with respect to the p ositive x -axis A  A 2  A 2 a n d   t a n  1 x y A y A x

53. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 53 Posi t ive and negat i ve components — Figure The co m ponents of a vec t or can be posit i ve or negat i ve nu m bers, as shown i n the f igu r e.

54. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 54 Finding componen t s — Figure

55. Componen t s of a Vector, f i nal The com po n ent s can b e positiv e o r n eg ative T h e sig n s o f the c o mp o n e nt s wil l d e p e n d o n the a n gle CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 55

56. Adding Two Vectors U s i ng Their Components R x = A x + B x R y = A y + B y The m ag n itud e an d direction o f resu l tant vectors are: CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 56

57. Adding vectors using their components F o r m ore t han two vectors we c a n use t he co m p o ne n ts of a s e t o f vectors to f i nd the c omp o ne n ts of their s um: R x  A x  B x  C x  , R y  A y  B y  C y  CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 57

58. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 58 Add i ng vectors u s ing t heir comp o nen t s — Ex.

59. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 59 Example 2

60. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 60 Unit vectors in t e r m s o f i ts co m p o ne n ts as A = A x î + A y j + A z k . A u n it v e c tor has a m agnitude of 1 with no u n i t s. T h e u n it vector î p o ints in the + x - dire c t i o n , j poi n ts in the + y - dire c t i o n , and k p o ints in the + z - dire c t i o n . Any vec t or c a n be expre s s e d 

61. Adding vecto r s us i ng un i t - vector nota t ion In th r e e d i mensions if th e n an d so R x = A x + B x , R y = A y + B y , an d R z =A z +B z R   A x ˆ i  A y ˆ j  A z k ˆ    B x ˆ i  B y ˆ j  B z k ˆ  R   A x  B x  ˆ i   A y  B y  ˆ j   A z  B z  k ˆ R  R ˆ i  R ˆ j  R k ˆ x y z R  R 2  R 2  R 2 x y z   c o s  1 R x , e t c . R R  A  B CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 61

62. Unit vector nota t ion , a d di n g v e ctors In two d ime n s i o ns, if then a nd so R x = A x + B x , R y = A y + B y , T h e ma g nitud e a nd dire c tion are R  A  B CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 62

63. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 63 Example 3 I f A = 2 4 i -32 j an d B=24 i +1 j , wh a t i s the magnitu d e an d direction o f the vec t o r C = A - B ?

64. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 64 The sca l ar product The sca l ar p r oduct (a l so ca l led the “dot product”) of two vec t ors is A B  A B c o s  . Figures illus t ra t e the scal a r product.

65. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 65 Dot Products of Unit Vectors ˆ i  ˆ i  ˆ j  ˆ j  k ˆ  k ˆ  1 ˆ i  ˆ j  ˆ i  k ˆ  ˆ j  k ˆ  Us i n g component fo r m w i th vectors:

66. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad Calculat i ng a sc a lar product • Find t h e s c a l ar p r oduct of two v e c t o r s sh o wn in t h e f i g u r e. The m agn i tudes of the vec t ors a r e: A = 4.00, and B = 5.00 66

67. Calculat i ng a sc a lar product – Example 4 • Find t h e s c a l ar p r oduct of two v e c t o r s sh o wn in t h e f i g u r e. The m agn i tudes of the vec t ors a r e: A = 4.00, and B = 5.00 CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 67

68. Finding an ang l e using the scalar product – Ex. 5 Find the angle between the vecto r s. Use equa t ion: CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 68

69. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad Finding an ang l e using the scalar product – Ex. 5 Find the angle between the vecto r s. Use equa t ion: 69

70. The Vector Product D ef i ned Given t w o vectors, A an d B The vect o r (cross) produc t o f A an d B i s defined a s a third vecto r , The magnitude o f vector C i s AB sin   is the angle between A and B C  A  B CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 70

71. More About the Vector Produ c t T h e dire c tion o f C is p erpe n di c ula r to the pla n e formed b y A a n d B T h e b est wa y to d etermine this dire c tion is to use the right -ha nd rule CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 71

72. Using Determinan t s The com po n ent s o f cross produc t can b e calcu l ate d as E x pa n d i n g the d eter m i n ant s g i ves A  B   A y B z  A z B y  ˆ i   A x B z  A z B x  ˆ j   A x B y If A z = an d B z =0 then = ˆ i ˆ j k ˆ A A ˆ i  A x ˆ j  A x k ˆ A  y z B A y B B A z B B x y z y z x y x z x y z A  B  A B B B A y B x  k ˆ CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 72

73. Vector Product Examp l e 6 Given F i nd Resu l t A  2 ˆ i  3 ˆ j ; B   ˆ i  2 ˆ j = = CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 73

74. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad The vector produc t — Summary T he vector product ( “ cross product”) of two ve c tors has m ag nitu de | A  B |  A B s in  and the right- ha nd rule gives its direction. 74

75. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad Calculat i ng the vector produc t — ex. 6 V ec t or has m agni t ude 6 uni t s and is in the direction of t he + x axis. V ector has m agnitude 4 uni t s and l i es in the xy – plane m aking a n angle of 30 with the x axi s . Find the cross p r oduct Use AB sin  to f ind the m agnitude and the r i gh t -hand ru l e to find the di r ec t i o n. 75

76. Accuracy of Calculations CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 76

77. NUMERICAL ACCURACY The accuracy of a solution depends on Accuracy of the given data. Accuracy of the computations performed. The solution cannot be more accurate than the less accurate of these two. The use of hand calculators and computers generally makes the accuracy of the computations much greater than the accuracy of the data. Hence, the solution accuracy is usually limited by the data accuracy. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 77

78. Problem Solving Approach Problem Solving Strategy CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 78

79. Problem Solving Strategy Read the problem Identify the nature of the problem Draw a diagram Some types of problems require very specific types of diagrams CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 79

80. Problem Solving cont. Label the physical quantities Can label on the diagram Use letters that remind you of the quantity Many quantities have specific letters Choose a coordinate system and label it Identify principles and list data Identify the principle involved List the data (given information) Indicate the unknown (what you are looking for) CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 80

81. Problem Solving, cont. Choose equation(s) Based on the principle, choose an equation or set of equations to apply to the problem Substitute into the equation(s) Solve for the unknown quantity Substitute the data into the equation Obtain a result Include units CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 81

82. Problem Solving, final Check the answer Do the units match? Are the units correct for the quantity being found? Does the answer seem reasonable? Check order of magnitude Are signs appropriate and meaningful? CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 82

83. Problem Solving Summary Equations are the tools of physics Understand what the equations mean and how to use them Carry through the algebra as far as possible Substitute numbers at the end Be organized CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 83

84. Reference BOOKS CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 84

85. Pre-Requisite Trigonometry equations Calculation on calculation Unit Conversion Balancing equation for Units CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 85

86. References: https://www.studocu.com/row/document/balochistan-university-of-information-technology-engineering-and-management-sciences/engineering-surveying/engineering-mechanics-17/2544731 https://sites.pitt.edu/~qiw4/Academic/ENGR0135/Chapter2.pdf CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 86

87. CL- Engg. Mechanics, DoCL- SPP, DDU, Nadiad 87