6161103 1 general principles

Engineering Mechanics: StaticsEngineering Mechanics: Statics

Chapter 1

General Principles

Chapter 1

General Principles

Chapter ObjectivesChapter 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.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 OutlineChapter Outline

� Mechanics

� Fundamental Concepts

� Units of Measurement� Units of Measurement

� The International System of Units

� Numerical Calculations

� General Procedure for Analysis

1.1 Mechanics1.1 Mechanics

� Mechanics can be divided into 3 branches:

- Rigid-body Mechanics

- Deformable-body Mechanics

- Fluid Mechanics

� Rigid-body Mechanics deals with

- Statics

- Dynamics

1.1 Mechanics1.1 Mechanics

� Statics – Equilibrium of bodies

� At rest

� Move with constant velocity� Move with constant velocity

Dynamics – Accelerated motion of bodies

1.2 Fundamentals Concepts 1.2 Fundamentals Concepts

Basic Quantities� Length

– Locate position and describe size of physical systemsystem– 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

1.2 Fundamentals Concepts 1.2 Fundamentals Concepts

Basic Quantities� Time

– Conceive as succession of events

� Force� 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

1.2 Fundamentals Concepts1.2 Fundamentals Concepts

Idealizations� Particles

– Consider mass but neglect sizeEg: Size of Earth insignificant compared to its Eg: Size of Earth insignificant compared to its size of orbit

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


Idealizations� Concentrated Force

– Effect of loading, assumed to act at a point on a bodypoint on a body

– Represented by a concentrated force, provided loading area is small compared to overall size

Eg: Contact force between wheel and ground


Newton’s Three Laws of Motion

� First Law

“A particle originally at rest, or moving in a straight line with constant velocity, will straight line with constant velocity, will remain in this state provided that the particle is not subjected to an unbalanced

force”



� Second Law

“A particle acted upon by an unbalanced force F experiences an acceleration a that force F experiences an acceleration a that has the same direction as the force and a magnitude that is directly proportional to the

force”

maF =



� Third Law

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


Newton’s Law of Gravitational Attraction

221

mmGF =

F = force of gravitation between two particles

G = universal constant of gravitation

m1,m2 = mass of each of the two particles

r = distance between the two particles

2rGF =


Weight, 2r

mMGW e=

Letting yields2/ rGMg e=

mgW =


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 quantitynot 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°

1.3 Units of Measurement1.3 Units of Measurement

SI Units� Système International d’Unités

� F = ma is maintained only if

– Three of the units, called base units, are – 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


Name Length Time Mass Force

International Systems of

Meter (m)

Second (s)

Kilogram (kg)

Newton (N)

2.

s

mkg

l Systems of Units (SI)

(m) (s) (kg) (N)


� At the standard location,

g = 9.806 65 m/s2

� For calculations, we use � For calculations, we use

g = 9.81 m/s2

� Thus,

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 Units


Prefixes� For a very large or very small numerical

quantity, the units can be modified by using a prefixusing 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)



Exponential Form

Prefix SI Symbol

Multiple

1 000 000 000 109 Giga G1 000 000 000 10 Giga G

1 000 000 106 Mega M

1 000 103 Kilo k

Sub-Multiple

0.001 10-3 Milli m

0.000 001 10-6 Micro µ

0.000 000 001 10-9 nano n



Rules for Use� Never write a symbol with a plural “s”.

Easily confused with second (s)

Symbols are always written in � 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)



Rules for Use� Quantities defined by several units

which are multiples, are separated by a dotdot

Eg: N = kg.m/s2 = kg.m.s-2

� The exponential power represented for a unit having a prefix refer to both the unit and its prefix

Eg: µN2 = (µN)2 = µN. µN



Rules for Use� Physical constants with several digits on

either side should be written with a space between 3 digits rather than a commabetween 3 digits rather than a commaEg: 73 569.213 427

� In calculations, represent numbers in terms of their base or derived units by converting all prefixes to powers of 10



Rules for UseEg: (50kN)(60nm) = [50(103)N][60(10-9)m]

= 3000(10-6)N.m = 3(10-3)N.m= 3(10-3)N.m= 3 mN.m

� The final result should be expressed using a single prefix



Rules for Use� Compound prefix should not be used

Eg: kµs (kilo-micro-second) should be expressed as ms (milli-second) since ms (milli-second) since

1 kµs = 1 (103)(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



Rules for Use� Although not expressed in terms of

multiples of 10, the minute, hour etc are 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

1.5 Numerical Calculations1.5 Numerical Calculations

� Dimensional Homogeneity- Each term must be expressed in the same unitsEg: s = vt + ½ at2 where s is position Eg: s = vt + ½ 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


� Dimensional Homogeneity

- All the terms of an equation can be replaced by a consistent set of units, replaced by a consistent set of units, that can be used as a partial check for algebraic manipulations of an equation


� 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 numberEg: 5604 and 34.52 have four significant numbers


� Significant Figures- When numbers begin or end with zero, we make use of prefixes to clarify the number of significant figuresfigures

Eg: 400 as one significant figure would be 0.4(103)

2500 as three significant figures would be 2.50(103)


Computers are often used in engineering for

advanced design and analysis


� 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 datathe problem data

- Often handheld calculators or computers involve more figures in the answer than the number of significant figures in the data


� Rounding Off Numbers- Calculated results should always be “rounded off” to an appropriate number of significant figures


� 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 droppedEg: 2.326 and 0.451 rounded off to n = 2 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 numberEg: 1.245(103) and 0.8655 rounded off to n = 3 significant figures become 1.24(103) and 0.866


� 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, 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


�Calculations- To ensure the accuracy of the final results, always retain a greater number of digits than the problem datadigits 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


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)(6 GN), (b) (400 mm)(0.6 MN) , (c) 45 MN3/900 Gg

Solution

First convert to base units, perform indicated

operations and choose an appropriate prefix


(a) ( )( )( )[ ] ( )[ ]( )( )

26

93

11

10300

1061050

650

kNkN

N

NN

GNmN

=

= −

( )2

3326

300

10

1

10

110300

kN

N

kN

N

kNN

=

=


(b) ( )( )( )[ ] ( )[ ]( )[ ] ( )[ ]( )

2123

263

2

1036.010400

106.010400

6.0400

Nm

Nm

MNmm

=

=−

−

( )[ ] ( )[ ]( )

2

29

.144

.10144

kNGm

Nm

=

=


( )( )( )

kg

N

GgMN

10900

1045

900/45

6

36

3

=

(c)

( )( )

( )kgkN

kgkN

kgN

kNN

kgN

/50

/1005.0

1

10

11005.0

/1005.0

3

33

3312

312

=

=

=

=

1.6 General Procedure for Analysis


� Most efficient way of learning is to solve problems

� To be successful at this, it is important to � 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



3) Apply relevant principles, generally in mathematics forms

4) Solve the necessary equations algebraically as far as practical, making sure 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



6) Study the answer with technical judgment and common sense to determine whether or not it seems reasonable



When solving the problems, do the work as neatly as possible. Being neat generally stimulates clear and orderly thinking and vice versa.

6161103 1 general principles

Education

body of mass

unit of force

contact force

newton n

size of physical system

force of gravitation

body mass comparison

si system ofunits