هکانیک تحلیلی 1 درس اول صحرایی گر ه فیسیک دانشگاه رازی.

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1 هکانیک تحلیلی 1 درس اول صحرایی گر ه فیسیک دانشگاه رازی

2 References: هنابع: naltical Mechanics Grant R. Fowles هکانیک تحلیلی فا لس ترجوو دکتر جعفر قیصری هرکس نشر دانشگاىی چاپ ىشتن Mechanics Keith R. Smon هکانیک سایو ى هکانیک نظری ترجوو تک ک - بير زی - اشپیگل Internet

3 ssessment Mid semester eam ( Sat. 30 ar ): 40% Final eam: 50% Homework and Classroom ctivities: 10% 3

4 فيرست Content فصل ا ل: فصل د م: هفاىین بنیادی بردارىا Fundamental concepts, Vectors هکانیک نی تنی حرکت راست خط رره Newtonian Mechanics, Rectilinear Motion of a particle فصل س م: ن سانگر ىواىنگ The Harmonic Oscillator فصل چيارم: حرکت کلی رره در سو بعذ General Motion of a Particle in Three Dimensions فصل پنجن: دستگاىيای هرجع نالخت Noninertial Reference Sstems فصل ششن: نیر ىای هرکسی هکانیک سوا ی Central Forces and Celestial Mechanics 4

5 What is Mechanics? Mechanics is the science which describes and predicts the conditions of rest or motion of bodies under the action of forces. Categories of Mechanics: - Rigid bodies - Statics - Dnamics - Deformable bodies - Fluids Mechanics is an applied science - it is not an abstract or pure science but does not have the empiricism found in other engineering sciences. Mechanics is the foundation of most engineering sciences and is an indispensable prerequisite to their stud. 5

6 Fundamental Concepts Space - associated with the notion of the position of a point P given in terms of three coordinates measured from a reference point or origin. Time - definition of an event requires specification of the time and position at which it occurred. Mass - used to characterie and compare bodies, e.g., response to earth s gravitational attraction and resistance to changes in translational motion. 6

7 Problem Statement: Includes given data, specification of what is to be determined, and a figure showing all quantities involved. Free-od Diagrams: Create separate diagrams for each of the bodies involved with a clear indication of all forces acting on each bod. Method of Problem Solution Solution Check: - Test for errors in reasoning b verifing that the units of the computed results are correct, - test for errors in computation b substituting given data and computed results into previousl unused equations based on the si principles, - alwas appl eperience and phsical intuition to assess whether results seem reasonable Fundamental Principles: The si fundamental principles are applied to epress the conditions of rest or motion of each bod. The rules of algebra are applied to solve the equations for the unknown quantities. 7

8 VECTOR CLCULUS Some quantities that ou have studied in our earlier Phsics Courses volume mass densit energ pressure displacement velocit acceleration torque electric field 50 kg 40º C 8

9 What are vectors? vector has a both a magnitude and a direction. phsical quantit which has both a magnitude and a direction is represented b a vector. + Displacement Electric Field 9

10 Definition geometric analtic Vector Defined b its lgebra.(operation) *naltical representation. asic method to describe vector is Cartesian coordinate sstem. -D vector. magnitude length unit vector: i j basis vector: Vector: i, j 1 j ˆ [1,0] i P ˆ [0,1] j 0 i 10

11 D Delhi Mumbai C Chennai Kolkata D C 11

12 ddition of Vectors C E D C E ( ) C 1

13 ssociative Law C E C D E E ( C) 13

14 Subtraction of Vectors - C C = = +( ) 14

15 Vectors are geometrical objects independent of an coordinate sstem Look a two dimensional vector in a D Cartesian Coordinate Sstem Y 1 O 1 X 15

16 Y [, ] O X 1 1 iˆ cos cos ˆj The direction cosines of a vector are not independent. The satisf the following relation: cos cos 1 16

17 17 Propert sin, cos j i. 0 j i P k j i 3-D vector. asis vector k j i,, ˆ ˆ ˆ ˆ k j i n nˆ

18 i ( i j j = 1 = cos = 1 = cos Cartesian Coordinates (,, ) k k ) = 1 = cos ( i cos j cos k cos ) nˆ 1 î Z plane plane cos cos cos kˆ 1 ĵ ( 1, 1, 1) 1 [cos cos cos ] 1 Page

19 19 ddition and subtraction of vectors ],, [ ],, [ c c c ) ( Commutative law C C ) ( ) ( ssociative law d c d c ) ( Distributive law ],, [

20 Scalar product( Dot product). ( 4 i + 3 j + k ). ( 3 i - j + k ). = 1 + (- 6) + 4 = 10 0

21 i. i j. j k. k 1 i. j i. k j. k 0. cos 1/ 1/ ( ) ( ) Commutative law Distributive law Tpical eample in Phsics. Work (cos cos sin sin ) cos...( C). C. : w F. d, or, dw F. dr j 0 cos( ) i. Geometric Definition 1

22 Vector Product CROSS PRODUCT Magnitude : rea of the parallelogram generated b &

23 Magnitude : h sin h sin Direction : Perpendicular to both and. 3

24 Right-Hand Rule 4

25 Properties î î ĵ ngle between them 0º ĵ kˆ kˆ 0 j i k cclic î ĵ kˆ ; ĵ kˆ î; kˆ î ĵ ngle between them =90º ĵ î kˆ ;kˆ ĵ î ;î kˆ ĵ 5

26 Vector Product: Determinant î ĵ kˆ 6

27 7 k j i ) ( ) ( ) ( k j i Suppose 0 k ) ( k sin( ) 0 j i k j i ˆ ˆ ˆ k ) sin cos sin (cos

28 8 ). ( cos ) cos (1 sin sin Eample: prove

29 nti-commutative law Distributive law ( C) C Eamples in Phsics The torque produced b a force is r F τ r F 9

30 Eamples in Phsics O r L r p p mv The angular momentum of a particle with respect to O L r p 30

31 pplications Eample: simple cross product ( 5ˆ i ˆj kˆ) (ˆ i 3ˆj kˆ) -5 iˆ ˆj kˆ [( 1 1) 3 9 ] î 5 1 [ 5 1 ( )] ĵ 3 1 [ 53 ( 1)]kˆ 17 31

32 n Eample ( 5ˆ i ˆj kˆ) (ˆ i 3ˆj kˆ) 5iˆ 9 ˆj 17kˆ C H E C K ( ) a & b is perpendicular to (5ˆ i ˆj kˆ).( 5ˆ i 9 ˆj 17kˆ)

33 Eample Find a unit vector perpendicular to the plane containing two vectors & vector perpendicular to and is 33

34 Cross product ( 5ˆ i ˆj kˆ) ( ˆ i 3ˆj kˆ) 5iˆ 9 ˆj 17kˆ Magnitude Unit vector is n Eample Determine a unit vector perpendicular to the plane of and ( 5ˆ i 9 ˆj 17kˆ) 34