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Engineering Statics ENGR 2301 Chapter 1

Introduction And What is Mechanics?

• Mechanics is the science which describes and predicts the conditions of rest or motion of bodies under the of .

• Categories of Mechanics: - Rigid bodies - Statics - Dynamics - 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 study.

1 - 2 Fundamental Principles • ’s First Law: If the resultant on a particle is zero, the particle will remain at rest or continue to move in a straight .

• Newton’s Law: A particle will have an proportional to a nonzero resultant applied force. • Parallelogram Law   F  ma • Newton’s Third Law: The forces of action and reaction between two particles have the same magnitude and line of action with opposite sense.

• Newton’s Law of Gravitation: Two particles are attracted with equal and opposite forces, Mm GM • Principle of Transmissibility F  G W  mg, g  r 2 R2 1 - 3 Significant Figures

Scientific Notation • Leading or trailing zeroes can make it hard to determine number of significant figures: 2500, 0.000036 • Each of these has two significant figures • Scientific notation writes these as a number from 1-10 multiplied by a of 10, making the number of significant figures much clearer:

2500 = 2.5 × 103 If we write 2.50x103, it has three significant figures 0.000036 = 3.6 x 10-5 Significant Figures

Round-off error: The digit in a calculated number may vary depending on how it is calculated, due to off of insignificant digits Example: $2.21 + 8% tax = $2.3868, rounds to $2.39 $1.35 + 8% tax = $1.458, rounds to $1.46 Sum: $2.39 + $1.46 = $3.85 $2.21 + $1.35 = $3.56 $3.56 + 8% tax = $3.84 Numerical Accuracy

• The accuracy of a solution depends on 1) accuracy of the given data, and 2) accuracy of the computations performed. The solution cannot be more accurate than the less accurate of these two.

• The use of calculators and computers generally makes the accuracy of the computations much greater than the accuracy of the data. Hence, the solution accuracy is usually limited by the data accuracy.

• As a general rule for engineering problems, the data are seldom known with an accuracy greater than 0.2%. Therefore, 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 lb and 15.58 lb.

1 - 6 Chapter 1: U.S. Customary Units . The base U.S. customary units are the units of , force and . . These units are the (ft), the (lb) and the second (s). . The second (s) is same as corresponding SI unit. . The foot is defined as 0.3048 m. . The pound (lb) is defined as the of a platinum standard, called the standard pound, which is kept at the National Institute of Standards and Technology, outside Washington, the of which is 0.453 592 43 kg. Chapter 1: U.S. Customary Units . Since weight of a body depends on upon the earth gravitational attraction, which varies with location, the U.S. customary units do not form an absolute system of units. . The standard pound (lb) needs to be placed at sea level and at a latitude of 45° to properly defined a force of 1 lb. . On the other hand, SI system of units, the meter (m), the (kg), and the second (s) may be used anywhere on the earth. They may even be used on another planet. They will always have same significance. Hence, they are called absolute system of units. Chapter 1: U.S. Customary Units

. The standard pound also serves as the unit of mass in commercial transactions in the United States, it can not be so used in engineering computations since it will not be consistent with Newton’s second law, F = ma. . So, the unit of mass was derived from basic U.S. system of units. This unit of mass is called the . . F = ma, therefore, 1 lb = (1 slug) (1 ft/s²). And 1 slug = (1 lb) ÷ (1 ft/s² ) = 1 lb · s²/ft . Since acceleration of gravity g is 32.2 ft/s², slug is a mass 32.2 larger than the mass of standard pound (lb).

. Chapter 1: Other U.S. Customary Units . Other U.S. customary units frequently used are: (mi) = 5280 ft. (in) = 1/12 ft kilopound () = force of 1000 lb = mass of 2000 lb. Note: In engineering computation, this must be converted into slugs. . Conversion into basic units of feet, pounds, and slug is often necessary in engineering computation. This is a very involved process in U.S. system of units than in SI system of units. . E.G., to convert of 30mi/h into ft/s, following steps are required: v = (30 mi/hr) (5280 ft/1 mi)(1h/3600s) = 44 ft/s

Chapter 1: System of Units . International System Of Units (SI Units): The universal system used around the world except U.S.A. and a couple of other small countries. SI stands for System Universal, a French word translated in English. . Four fundamental units, called Kinetic Units are units of length, time, mass and force. . Three of these units (Length, Time and Mass) are defined arbitrarily and are referred to as basic units. . The fourth one, the force, is defined by equation F = ma and hence called derived unit.

Chapter 1: SI Units – Length and mass . Base : The Meter: Originally defined as one ten-millionth A of the distance from the equator to either pole, is now defined as 1 650 earth 763.73 wavelengths of the orange-red light corresponding to a certain C transition in an atom of krypton-86. equator B This was changed once again in 1983 to: “The meter is the length of path traveled by light in a vacuum during a time interval of 1/299 792 458 of a second. . Base unit of mass: The Kilogram originally defined as equal to mass of the 0.001 m³ of is now defined as mass of a platinum-iridium standard kept at the International Bureau of and Measures at Serves, near Paris, France.

Chapter 1: SI Units -- Time . Base unit of Time: The Second: Originally defined as 1/86 400 of the mean solar , is now defined as the duration of 9 192 631 770 cycles of the radiation corresponding to the transition between two levels of the fundamental state of the cesium-133 atom. Chapter 1: SI Units -- Force . Base unit of Force: The Newton(N): The unit of force is a derived unit. It is defined as the force which gives an acceleration of 1 m/s² to a mass of 1 kg. . As we know from Newton’s second fundamental law, F = ma . So, 1 N = (1 kg ) (1 m/s² ) = 1 kg · m/s²

Chapter 1: SI Units –Weight . Weight of a body: It is the force of gravity exerted on body. . Like any other force, should be expressed in Newtons, not in kg. . W = mg . I.E., W = ( 1 kg)( 9.81 m/s² ) . I.E. W = 9.81 N . While standard kg also serves as the unit of Weight in commercial transactions, it can not be so used in engineering computations. .

Chapter 1: SI Units – commonly used units

. The most frequently used units are kilometer(km), millimeter(mm), megagram(Mg) which is known as metric ton, (g) and kilonewton(kN). . 1 km = 1000 m 1mm = 0.001 m 1 Mg = 1000 kg 1 g = 0.001 kg 1 kN = 1000 N 3.82 km = 3820 m 47.2 mm = 0.0472 m 3.82 km = 3.82 x 10³ m m ³־mm = 47.2 x 10 47.2

Chapter 1: SI Units – Derived units . There are many other units derived from the basic kinetic units (Length, Mass, Time and Force). . The most common derived units are units of Area and . . The unit of Area is square meter (m²) which represents the area of a square of side 1 m. . The unit of Volume is the cubic meter (m³), equal to the volume of a cube of side 1 m. . The Volume of liquid is measured in cubic decimeter (dm³) is commonly referred as a liter (L). Chapter 1: SI Units – Multiplication factors-Length . Multiple and sub-multiple of the units of Length: m ¹־dm = 0.1 m = 10 1 m ²־cm = 0.01 m = 10 1 m ³־mm = 0.001 m = 10 1 1 km = 1 000 m = 10³ m . Multiple and sub-multiple of the units of Area: m² ²־m)² = 10 ¹־dm² = (1 dm)² = (10 1 m)² = 10 -4m2 ²־cm² = (1 cm)² = (10 1 m)² = 10 -6m 2 ³־mm² = (1 mm)² = (10 1 . Multiple and sub-multiple of the units of Volume: m³·³־m)³ = 10 ¹־dm³ = (1 dm)³ = (10 1 m)³ = 10 -6m 3 ²־cm³ = (1 cm)³ = (10 1 m)³ = 10 -9m 3 ³־mm³ = (1 mm)³ = (10 1

Chapter 1: SI Units – Multiplication factors conventions-

. In order to avoid exceedingly small or large numerical values, many sub-units are defined and used. . When more than four digits are used on either side of the decimal -- as in 427 200 m or 0.002 16 m – spaces, never commas, should be used to separate the digits into groups of three. This is to avoid confusion with the comma used in place of a decimal point, which is the convention in many countries. . Example for use of multiple and sub-multiple of the units of Length: You write 427.2 km instead of 427 200 m You write 2.16 mm instead of 0.002 16 m.

Chapter 1: SI Units – Multiplication factors-Length . Multiple and sub-multiple of the units of Length: m ¹־dm = 0.1 m = 10 1 m ²־cm = 0.01 m = 10 1 m ³־mm = 0.001 m = 10 1 1 km = 1 000 m = 10³ m . Multiple and sub-multiple of the units of Area: m² ²־m)² = 10 ¹־dm² = (1 dm)² = (10 1 m)² = 10 -4m2 ²־cm² = (1 cm)² = (10 1 m)² = 10 -6m 2 ³־mm² = (1 mm)² = (10 1 . Multiple and sub-multiple of the units of Volume: m³·³־m)³ = 10 ¹־dm³ = (1 dm)³ = (10 1 m)³ = 10 -6m 3 ²־cm³ = (1 cm)³ = (10 1 m)³ = 10 -9m 3 ³־mm³ = (1 mm)³ = (10 1

Chapter 1: SI Units – Two ideas . However, there have been two ideas as to which should be preferred in science. Scientists working in laboratories, dealing with small and distances, preferred to measure distance in centimeters and mass in . Scientists and engineers working in larger contexts preferred larger units: meters for distance and for mass. Everyone agreed that units of other quantities such as force, , work, power, and so on should be related in a simple way to the basic units, but which basic units should be used? . The result was two clustering of metric units in science and engineering. One cluster, based on the centimeter, the gram, and the second, is called the CGS system. The other, based on the meter, kilogram, and second, is called the MKS system

Chapter 1: SI units vs. US units . The beauty of the lies in its simplicity and consistency. . Despite the advantages of the metric system, are still in wide use, therefore we must be able to work with all kinds of units. . Unlike the English system, which uses a hodge-podge of units to express the same physical (for example length) with no consistent conversions between them. . the metric system uses a single unit of measure modified by a prefix to change the measurement scale. For example, the English system uses , , and to measure distances of varying scales, which have no consistent conversion factors between them. In contrast, the metric system uses a single unit, the meter, which, with appropriate prefix modifications produce roughly comparable scales: centimeters, meters, and kilometers. Moreover, the metric system uses the same set of prefixes for scaling regardless of the physical quantity under consideration. Chapter 1: SI Units vs. US unit . Hence mass can be measured in centigrams, grams, and kilograms. . it is not true that the US remains the last holdout. . While the rest of the world is pretty much standardized on the metric system of , when it comes to mandatory use, the United States has company in its foot dragging. Great Britain, Liberia and Burma are right there along with the United States. . Some international organizations have threatened to restrict U.S. imports that do not conform to metric standards and rather than trying to maintain dual inventories for domestic and foreign markets, a number of U.S. corporations have chosen to go metric. Chapter 1: SI Units vs. US units . You will be seeing more and more of your customers in the US using the Metric system in their purchases and writing more original specifications in Metric. . Scientists have adopted the metric system to simplify their calculations and promote communication across national boundaries . Some Motor vehicles, farm machinery, and computer equipment are now manufactured to metric specifications.

Chapter 1: Conversion of units . There are many instances when conversion from U.S. system to SI system or vice versa is required. Since the unit of time is the same in both the system, only two kinetic base unit need to be converted. . 1 ft = 0.3048 m and 1 lb = 4.448 N . This makes 1 slug = 14.59 kg. . Since all other kinetic units and conversion factors can be derived from these base units. . E.G. 1 mi = 5280 ft = 5280·(0.3048 m) =1609 m = 1.609km . It is important to solve many problems involving conversion of these units to understand these concepts.

Changing units

Based of the base units, we may need to change the units of a given quantity using the - conversion.

For example, since there are 60 seconds in one , 1min 60s 1  ,and 60s 1min

60s 2min  (2min) x(1)  (2min) x( ) 120s 1min Conversion between one system of units and another can therefore be easily figured out as shown.

The first equation above is often called the “Conversion Factor”.