1. Engineering Mechanics: Statics Chapter 1 General Principles Chapter 1 General Principles

2. Chapter Objectives To provide an introduction to the basic quantities and idealizations of mechanics. To give a statement of Newton’s Laws of Motion and Gravitation. To review the principles for applying the SI system of units. To examine the standard procedures for performing numerical calculations. To present a general guide for solving problems.

3. Chapter Outline Mechanics Fundamental Concepts Units of Measurement The International System of Units Numerical Calculations General Procedure for Analysis

4. 1.1 Mechanics Mechanics can be divided into 3 branches: - Rigid-body Mechanics - Deformable-body Mechanics - Fluid Mechanics Rigid-body Mechanics deals with - Statics - Dynamics

5. 1.1 Mechanics Statics – Equilibrium of bodies  At rest  Move with constant velocity Dynamics – Accelerated motion of bodies

6. 1.2 Fundamentals Concepts Basic Quantities Length – Locate position and describe size of physical system – Define distance and geometric properties of a body Mass – Comparison of action of one body against another – Measure of resistance of matter to a change in velocity

7. 1.2 Fundamentals Concepts Basic Quantities Time – Conceive as succession of events Force – “push” or “pull” exerted by one body on another – Occur due to direct contact between bodies Eg: Person pushing against the wall – Occur through a distance without direct contact Eg: Gravitational, electrical and magnetic forces

8. 1.2 Fundamentals Concepts Idealizations Particles – Consider mass but neglect size Eg: Size of Earth insignificant compared to its size of orbit Rigid Body – Combination of large number of particles – Neglect material properties Eg: Deformations in structures, machines and mechanism

9. 1.2 Fundamentals Concepts Idealizations Concentrated Force – Effect of loading, assumed to act at a point on a body – Represented by a concentrated force, provided loading area is small compared to overall size Eg: Contact force between wheel and ground

10. 1.2 Fundamentals Concepts Newton’s Three Laws of Motion First Law “A particle originally at rest, or moving in a straight line with constant velocity, will remain in this state provided that the particle is not subjected to an unbalanced force”

11. 1.2 Fundamentals Concepts Newton’s Three Laws of Motion Second Law “A particle acted upon by an unbalanced force F experiences an acceleration a that has the same direction as the force and a magnitude that is directly proportional to the force”

12. 1.2 Fundamentals Concepts Newton’s Three Laws of Motion Third Law “The mutual forces of action and reaction between two particles are equal and, opposite and collinear”

13. 1.2 Fundamentals Concepts Newton’s Law of Gravitational Attraction F = force of gravitation between two particles G = universal constant of gravitation m 1,m 2 = mass of each of the two particles r = distance between the two particles

14. 1.2 Fundamentals Concepts Weight, Letting yields

15. 1.2 Fundamentals Concepts Comparing F = mg with F = ma g is the acceleration due to gravity Since g is dependent on r, weight of a body is not an absolute quantity Magnitude is determined from where the measurement is taken For most engineering calculations, g is determined at sea level and at a latitude of 45 °

16. 1.3 Units of Measurement SI Units Syst è me International d ’ Unit é s F = ma is maintained only if – Three of the units, called base units, are arbitrarily defined – Fourth unit is derived from the equation SI system specifies length in meters (m), time in seconds (s) and mass in kilograms (kg) Unit of force, called Newton (N) is derived from F = ma

17. 1.3 Units of Measurement NameLengthTimeMassForce Internationa l Systems of Units (SI) Meter (m) Second (s) Kilogram (kg) Newton (N)

18. 1.3 Units of Measurement At the standard location, g = 9.806 65 m/s 2 For calculations, we use g = 9.81 m/s 2 Thus, W = mg (g = 9.81m/s 2 ) Hence, a body of mass 1 kg has a weight of 9.81 N, a 2 kg body weighs 19.62 N

19. 1.4 The International System of Units Prefixes For a very large or very small numerical quantity, the units can be modified by using a prefix Each represent a multiple or sub-multiple of a unit Eg: 4,000,000 N = 4000 kN (kilo-newton) = 4 MN (mega- newton) 0.005m = 5 mm (milli-meter)

20. 1.4 The International System of Units Exponential Form PrefixSI Symbol Multiple 1 000 000 00010 9 GigaG 1 000 00010 6 MegaM 1 00010 3 Kilok Sub-Multiple 0.00110 -3 Millim 0.000 00110 -6 Micro μ 0.000 000 00110 -9 nanon

21. 1.4 The International System of Units Rules for Use Never write a symbol with a plural “s”. Easily confused with second (s) Symbols are always written in lowercase letters, except the 2 largest prefixes, mega (M) and giga (G) Symbols named after an individual are capitalized Eg: newton (N)

22. 1.4 The International System of Units Rules for Use Quantities defined by several units which are multiples, are separated by a dot Eg: N = kg.m/s 2 = kg.m.s -2 The exponential power represented for a unit having a prefix refer to both the unit and its prefix Eg: μN 2 = ( μN) 2 = μN. μN

23. 1.4 The International System of Units Rules for Use Physical constants with several digits on either side should be written with a space between 3 digits rather than a comma Eg: 73 569.213 427 In calculations, represent numbers in terms of their base or derived units by converting all prefixes to powers of 10

24. 1.4 The International System of Units Rules for Use Eg: (50kN)(60nm) = [50(10 3 )N][60(10 -9 )m] = 3000(10 -6 )N.m = 3(10 -3 )N.m = 3 mN.m The final result should be expressed using a single prefix

25. 1.4 The International System of Units Rules for Use Compound prefix should not be used Eg: k μs (kilo-micro-second) should be expressed as ms (milli-second) since 1 k μs = 1 (10 3 )(10 -6 ) s = 1 (10 -3 ) s = 1ms With exception of base unit kilogram, avoid use of prefix in the denominator of composite units Eg: Do not write N/mm but rather kN/m Also, m/mg should be expressed as Mm/kg

26. 1.4 The International System of Units Rules for Use Although not expressed in terms of multiples of 10, the minute, hour etc are retained for practical purposes as multiples of second. Plane angular measurements are made using radians. In this class, degrees would be often used where 180 ° = π rad

27. 1.5 Numerical Calculations Dimensional Homogeneity - E ach term must be expressed in the same units Eg: s = vt + ½ at 2 where s is position in meters (m), t is time in seconds (s), v is velocity in m/s and a is acceleration in m/s 2 - Regardless of how the equation is evaluated, it maintains its dimensional homogeneity

28. 1.5 Numerical Calculations Dimensional Homogeneity - All the terms of an equation can be replaced by a consistent set of units, that can be used as a partial check for algebraic manipulations of an equation

29. 1.5 Numerical Calculations Significant Figures - The accuracy of a number is specified by the number of significant figures it contains - A significant figure is any digit including zero, provided it is not used to specify the location of the decimal point for the number Eg: 5604 and 34.52 have four significant numbers

30. 1.5 Numerical Calculations Significant Figures - When numbers begin or end with zero, we make use of prefixes to clarify the number of significant figures Eg: 400 as one significant figure would be 0.4(10 3 ) 2500 as three significant figures would be 2.50(10 3 )

31. 1.5 Numerical Calculations Computers are often used in engineering for advanced design and analysis

32. 1.5 Numerical Calculations Rounding Off Numbers - For numerical calculations, the accuracy obtained from the solution of a problem would never be better than the accuracy of the problem data - Often handheld calculators or computers involve more figures in the answer than the number of significant figures in the data

33. 1.5 Numerical Calculations Rounding Off Numbers - Calculated results should always be “rounded off” to an appropriate number of significant figures

34. 1.5 Numerical Calculations Rules for Rounding to n significant figures - If the n+1 digit is less than 5, the n+1 digit and others following it are dropped Eg: 2.326 and 0.451 rounded off to n = 2 significance figures would be 2.3 and 0.45 - If the n+1 digit is equal to 5 with zero following it, then round nth digit to an even number Eg: 1.245(10 3 ) and 0.8655 rounded off to n = 3 significant figures become 1.24(10 3 ) and 0.866

35. 1.5 Numerical Calculations Rules for Rounding to n significant figures - If the n+1 digit is greater than 5 or equal to 5 with non-zero digits following it, increase the nth digit by 1 and drop the n+1digit and the others following it Eg: 0.723 87 and 565.5003 rounded off to n = 3 significance figures become 0.724 and 566

36. 1.5 Numerical Calculations Calculations - To ensure the accuracy of the final results, always retain a greater number of digits than the problem data - If possible, try work out computations so that numbers that are approximately equal are not subtracted -In engineering, we generally round off final answers to three significant figures

37. 1.5 Numerical Calculations Example 1.1 Evaluate each of the following and express with SI units having an approximate prefix: (a) (50 mN)(6 GN), (b) (400 mm)(0.6 MN) 2, (c) 45 MN 3 /900 Gg Solution First convert to base units, perform indicated operations and choose an appropriate prefix

38. 1.5 Numerical Calculations (a)

39. 1.5 Numerical Calculations (b)

40. 1.5 Numerical Calculations (c)

41. 1.6 General Procedure for Analysis Most efficient way of learning is to solve problems To be successful at this, it is important to present work in a logical and orderly way as suggested: 1) Read problem carefully and try correlate actual physical situation with theory 2) Draw any necessary diagrams and tabulate the problem data

42. 1.6 General Procedure for Analysis 3) Apply relevant principles, generally in mathematics forms 4) Solve the necessary equations algebraically as far as practical, making sure that they are dimensionally homogenous, using a consistent set of units and complete the solution numerically 5) Report the answer with no more significance figures than accuracy of the given data

43. 1.6 General Procedure for Analysis 6) Study the answer with technical judgment and common sense to determine whether or not it seems reasonable

44. 1.6 General Procedure for Analysis When solving the problems, do the work as neatly as possible. Being neat generally stimulates clear and orderly thinking and vice versa.