Engineering Mechanics: StaticsChapter 1General Principles

Text BookENGINEERING MECHANICS; STATICS,R.C. HIBBELER 7 RUKUN/KEWAJIBAN AKADEMIKTO TEACHTO MENTORTO DISCOVERTO PUBLISHTO REACH BEYOND THE WALLTO CHANGETO TELL THE TRUTH

Chapter ObjectivesTo provide an introduction to the basic quantities and idealizations of mechanics.To give a statement of Newtons 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.

Chapter OutlineMechanicsFundamental ConceptsUnits of MeasurementThe International System of UnitsNumerical CalculationsGeneral Procedure for Analysis

1.1 MechanicsMechanics can be divided into 3 branches:

- Rigid-body Mechanics- Deformable-body Mechanics- Fluid MechanicsRigid-body Mechanics deals with

- Statics - Dynamics

1.1 MechanicsStatics Equilibrium of bodies

At restMove with constant velocityDynamics Accelerated motion of bodies

1.2 Fundamentals Concepts Basic QuantitiesLength

Locate position and describe size of physical system Define distance and geometric properties of a bodyMass

Comparison of action of one body against another Measure of resistance of matter to a change in velocity

1.2 Fundamentals Concepts Basic QuantitiesTime

Conceive as succession of eventsForce

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

1.2 Fundamentals ConceptsIdealizationsParticles

Consider mass but neglect sizeEg: Size of Earth insignificant compared to its size of orbitRigid Body

Combination of large number of particles Neglect material propertiesEg: Deformations in structures, machines and mechanism

1.2 Fundamentals ConceptsIdealizationsConcentrated 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 sizeEg: Contact force between wheel and ground

1.2 Fundamentals ConceptsNewtons Three Laws of MotionFirst 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

1.2 Fundamentals ConceptsNewtons Three Laws of MotionSecond 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

1.2 Fundamentals ConceptsNewtons Three Laws of MotionThird Law

The mutual forces of action and reaction between two particles are equal and, opposite and collinear

1.2 Fundamentals ConceptsNewtons Law of Gravitational Attraction

F = force of gravitation between two particlesG = universal constant of gravitationm1,m2 = mass of each of the two particlesr = distance between the two particles

1.2 Fundamentals Concepts

Weight,

Letting yields

1.2 Fundamentals ConceptsComparing F = mg with F = mag is the acceleration due to gravitySince g is dependent on r, weight of a body is not an absolute quantityMagnitude is determined from where the measurement is takenFor most engineering calculations, g is determined at sea level and at a latitude of 45

1.3 Units of MeasurementSI UnitsSystme International dUnitsF = ma is maintained only if

Three of the units, called base units, are arbitrarily defined Fourth unit is derived from the equationSI 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

1.3 Units of Measurement

NameLengthTimeMassForceInternational Systems of Units (SI)Meter (m)Second (s)Kilogram (kg)Newton (N)

1.3 Units of MeasurementAt the standard location,

g = 9.806 65 m/s2For calculations, we use

g = 9.81 m/s2Thus,

W = mg(g = 9.81m/s2)Hence, a body of mass 1 kg has a weight of 9.81 N, a 2 kg body weighs 19.62 N

1.4 The International System of UnitsPrefixesFor a very large or very small numerical quantity, the units can be modified by using a prefixEach 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)

1.4 The International System of Units

Exponential Form PrefixSI SymbolMultiple1 000 000 000109GigaG1 000 000106MegaM1 000103KilokSub-Multiple0.00110-3Millim0.000 00110-6Micro0.000 000 00110-9nanon

1.4 The International System of UnitsRules for UseNever 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)

1.4 The International System of UnitsRules for UseQuantities defined by several units which are multiples, are separated by a dot

Eg: N = kg.m/s2 = kg.m.s-2The exponential power represented for a unit having a prefix refer to both the unit and its prefix

Eg: N2 = (N)2 = N. N

1.4 The International System of UnitsRules for UsePhysical constants with several digits on either side should be written with a space between 3 digits rather than a comma

Eg: 73 569.213 427In calculations, represent numbers in terms of their base or derived units by converting all prefixes to powers of 10

1.4 The International System of UnitsRules for UseEg: (50kN)(60nm) = [50(103)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

1.4 The International System of UnitsRules for UseCompound prefix should not be used

Eg: ks (kilo-micro-second) should be expressed as ms (milli-second) since 1 ks = 1 (103)(10-6) s = 1 (10-3) s = 1msWith exception of base unit kilogram, avoid use of prefix in the denominator of composite units

Eg: Do not write N/mm but rather kN/mAlso, m/mg should be expressed as Mm/kg

1.4 The International System of UnitsRules for UseAlthough 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 = rad

1.5 Numerical CalculationsDimensional Homogeneity

- Each term must be expressed in the same unitsEg: s = vt + at2 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/s2

- Regardless of how the equation is evaluated, it maintains its dimensional homogeneity

1.5 Numerical CalculationsDimensional 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

1.5 Numerical CalculationsSignificant 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 numberEg: 5604 and 34.52 have four significant numbers

1.5 Numerical CalculationsSignificant 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(103) 2500 as three significant figures would be 2.50(103)

1.5 Numerical CalculationsComputers are often used in engineering foradvanced design and analysis

1.5 Numerical CalculationsRounding 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

1.5 Numerical CalculationsRounding Off Numbers

- Calculated results should always be rounded off to an appropriate number of significant figures

1.5 Numerical CalculationsRules for Rounding to n significant figures

- If the n+1 digit is less than 5, the n+1 digit and others following it are droppedEg: 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 numberEg: 1.245(103) and 0.8655 rounded off to n = 3 significant figures become 1.24(103) and 0.866

1.5 Numerical CalculationsRules 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 itEg: 0.723 87 and 565.5003 rounded off to n = 3 significance figures become 0.724 and 566

1.5 Numerical CalculationsCalculations

- 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

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 MN3/900 Gg

Solution First convert to base units, perform indicated operations and choose an appropriate prefix

1.5 Numerical Calculations(a)

1.5 Numerical Calculations(b)

1.5 Numerical Calculations(c)

1.6 General Procedure for AnalysisMost 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 theory2) Draw any necessary diagrams and tabulate the problem data

1.6 General Procedure for Analysis3) Apply relevant principles, generally in mathematics forms4) 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 numerically5) Report the answer with no more significance figures than accuracy of the given data

1.6 General Procedure for Analysis6) Study the answer with technical judgment and common sense to determine whether or not it seems reasonable

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