Chapter 1: General Principles

Engineering Mechanics: Statics Chapter 1: General Principles 1 MECHANICS, UNITS, NUMERICAL CALCULATIONS & GENERAL PROCEDURE FOR ANALYSIS Today’s Objectives: Students will be able to: In-Class activities: a) Explain mechanics/statics. • Reading Quiz b) Work with SI system of units. • What Is Mechanics? c) Round the final answer appropriately. • System of Units d) Apply problem-solving strategies. • Numerical Calculations • Concept Quiz • Problem-Solving Strategy • Attention Quiz READING QUIZ 1. The subject of mechanics deals with what happens to a body when ______ is/are applied to it. A) a magnetic field B) heat D) neutrons E) lasers C) forces 2. ________________ still remains the basis of most of today’s engineering sciences. A) Newtonian Mechanics B) Relativistic Mechanics C) Greek Mechanics C) Euclidean Mechanics WHAT IS MECHANICS? Study of what happens to a “thing” (the technical name is “BODY”) when FORCES are applied to it. Either the body or forces can be large or small. BRANCHES OF MECHANICS Mechanics Rigid Bodies (Things that do not change shape) Statics Dynamics Deformable Bodies (Things that do change shape) Fluids Incompressible Compressible 1.1. Mechanics • Mechanics is the science which describes and predicts the conditions of rest or motion of bodies under the action of forces. 6 1.1 Mechanics Statics: A branch of rigid body mechanics that deals with bodies at rest or forces in equilibrium. In static equilibrium particles or bodies are either ;  At rest  Or move with constant velocity on a straight line. Dynamics –that deals with the accelerated motion of bodies 7 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 8 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 9 1.2 Fundamentals Concepts Idealizations • Particles – Consider mass but neglect size Eg: Size of Earth insignificant compared to its size of its orbit • Rigid Body – Combination of large number of particles – Neglect material properties Eg: Deformations in structures, machines and mechanism 10 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 11 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” 12 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” F  ma 13 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” 14 1.2 Fundamentals Concepts Newton’s Law of Gravitational Attraction F G m1 m 2 r 2 F = force of gravitation between two particles G = universal constant of gravitation =66.73(10-12)(m3/kg.s2) m1,m2 = mass of each of the two particles r = distance between the two particles 15 1.2 Fundamentals Concepts Weight, W Letting G mM e r 2 g  GM e / r yields 2 W  mg 16 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° 17 1.3 Units of Measurement SI Units • Système International d’Unités • 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 18 1.3 Units of Measurement Name Length Time Mass Force International Meter (m) Second Kilogram Newton Systems of (s) (kg) (N) Units (SI)  kg .m   2   s  19 1.3 Units of Measurement • At the standard location, g = 9.806 65 m/s2 • 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 20 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) 21 1.4 The International System of Units Exponential Prefix Form SI Symbol 1 000 000 000 109 Giga G 1 000 000 106 Mega M 1 000 103 Kilo k 0.001 10-3 Milli m 0.000 001 10-6 Micro μ 0.000 000 001 10-9 nano n Multiple Sub-Multiple 22 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) 23 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/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 24 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 25 1.4 The International System of Units Rules for Use Eg: (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 26 1.4 The International System of Units Rules for Use • Compound prefix should not be used Eg: kμs (kilo-micro-second) should be expressed as 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 27 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 28 CONCEPT QUIZ 1. Evaluate the situation in which mass (kg), force (N), and length (m) are the base units and recommend one of the following. A) A new system of units will have to be formulated. B) Only the unit of time have to be changed from second to something else. C) No changes are required. D) The above situation is not feasible. 1.5. Numerical Calculations • Numerical work in engineering practise is most ofen performed by using handheld calculators and computers. It is important, however, that the answers to any problem be reported with justifiable accuracy using appropriate significant figures. • Significant Figures. • The number of significant figures contained in any number determines the accuracy of the number. 30 1.5. Numerical Calculations • For example: The number 4981 contains four significant figures. • If zeros occur at the end of a hole number it may be unclear as to how many significant figures the number represents. For example, • 23400 might have three ( 234) , four ( 2340) or five (23400) significant figures. 31 1.5. Numerical Calculations • To avoid ambiguities engineering notation is used for reporting the results. That is: • Numbers are rounded off to the appropriate number of significant digit and then expressed in multiples of (103), such as (103),(106) or (109). For instance, if 23 400 has five significant figures, it is written as 23.400(103), but if it has only three significant figures, it is written as 23.4 (103). 32 1.5.Numerical Calculations • If zeros occur at the biginning of a number that is less than one , then the zeros are not significant. For example, 0.008 21 has three significant figures. Using engineering notations, this number is expressed as 8.21 (10-3). 33 1.5. Numerical Calculations • Rounding Off numbers. • Rounding off a number is necessary so that the accuracy of the result will be the same as that of the problem data. • As a general rule , any numerical figure ending in a number greater than five is rounded up and a number less than five is not rounded up. For example; suppose the number 3.5587 is to be rounded off to three significant figures. As 3.56. • Likewise 0.5896 becomes 0.590, 9.3866 becomes 9.39, 1.341 becomes 1.34, 0.3762 becomes 0.376. and 9.871 becomes 9.87. 34 1.5. Numerical calculations • There is a special case for any number that ends in a 5. As a general rule, if the digit preceding the five is an even number , then this diğit is not rounded up. If the digit preceding the 5 is an odd number , then it is rounded up. For example, 75.25 rounded off to three significant digits becomes 75.2, 0.1275 becomes 0.128, and 0.2555 becomes 0.256. 35 1.5. Numerical Calculations • Calculations. • When a sequence of calculations is performed , it is best to store the intermediate results in the calculator. i.e. Do not round off calculations until expressing the final result. This procedure maintains precission throughout the series of steps to the final solution. • The accuracy of a solution depends on 1) The accuracy of the known physical data, 2) The accuracy of the physical model 3) The accuracy of the computations performed. 36 1.5. Numerical Calculations • As a general rule for engineering problems, the data are seldom known with an accuracy greater than 0.2%. Therefore, in this course it is usually appropriate to record parameters beginning with “1” with four digits and with three digits in all other cases, i.e., 40.2 N and 15.58 m 37 1.5. Numerical Calculations • Dimensional Homegenity. The terms of any equation used to describe a physical process must be dimensionally homegenous ; that is each term must be expressed in the same units. Consider , for example, the equation s= v0t+1/2 at2, where , in SI units, s is position in meters, m, t is time in seconds, s, v is velocity m/s and a is acceleration in m/s2. Regardless of how this equation is evaluated, it maintains its dimensional homegenity. 38 CONCEPT QUIZ Give the most appropriate reason for using three significant figures in reporting results of typical engineering calculations. A) Historically slide rules could not handle more than three significant figures. B) Three significant figures gives better than one-percent accuracy. C) Telephone systems designed by engineers have area codes consisting of three figures. D) Most of the original data used in engineering calculations do not have accuracy better than one percent. 1.6. General Procedure for Analysis • To be succesuful in this course or in engineering mechanics as a hole, it is important to always present the work in a logical and orderly manner as suggested by the following steps: • Read the problem carefully and try to correlate actual physical situation with the theory studied. • Tabulate the problem data and draw to a large scale any necessary diagrams. 1.6. General Procedure for Analysis • Apply the relevant principles , generally in mathematical form. Check if the equations written are dimensionally homogeneous. • Solve the necessary equations, and report the answer with no more than three significant figures. • Study the answer with technical judgement and common sense to determine whether or not it seems reasonable. 41 PROBLEM SOLVING STRATEGY IPE: A 3-Step Approach 1. Interpret: Read carefully and determine what is given and what is to be found/ delivered. Ask, if not clear. If necessary, make assumptions and indicate them. 2. Plan: Think about major steps (or a road map) that you will take to solve a given problem. Think of alternative/creative solutions and choose the best one. 3. Execute: Carry out your steps. Use appropriate diagrams and equations. Estimate your answers. Avoid simple calculation mistakes. Reflect on and then revise your work, if necessary. ATTENTION QUIZ 1. For a static’s problem, your calculations show the final answer as 12345.6 N. What should you write as your final answer? A) 12345.6 N D) 12.34 kN B) 12.3456 kN C) 12 kN E) 123 kN 2. In the three-step approach to problem solving called IPE, what does P stand for? A) Position B) Plan D) Practical E) Possible C) Problem EXAMPLE 1.1 Convert 2 km/h to m/s. SOLUTION Since 1 km= 1000 m and 1 h =3600s, the factors of conversion are arranged in the following order, so that a cancelletion of the units can be applied. 2 km/h = 2km/h (1000 m/km)(1h/3600 s) = 2000 m/3600 s = 0.556 m/s 44 EXAMPLE1.2 • Evaluate each of the following and express with SI units having an appropriate prefix: (a)(50 mN)(6 GN), (b) (400 mm)(0.6 MN)2, (c)45MN3/900Gg. 45 EXAMPLE 1.2 50mN 6GN          300106 N 2 6 2  1kN  1kN   30010 N  3  3   10 N  10 N   50 103 N 6 109 N  300kN 2 46 EXAMPLE 1.2 (a) 50mN 6GN          300106 N 2 6 2  1kN  1kN   30010 N  3  3   10 N  10 N   50 103 N 6 109 N  300kN 2 47 EXAMPLE 1.2 (b) 400mm 0.6MN  2   m0.610 N   40010 m0.3610 N   14410 m.N 3  400 10 3 12 9  144Gm.N 2 6 2 2 2 48 EXAMPLE 1.2 (c) 45MN 3 / 900Gg  4510 N  900106 kg 6 3  0.051012 N 3 / kg 3  1kN  1  0.051012 N 3  3   10 N  kg  0.05103 kN 3 / kg  50kN 3 / kg 49