Engineering Statics ENGR 2301 Chapter 1 Introduction And Measurement
What is Mechanics? Mechanics is the science which describes and predicts the conditions of rest or motion of bodies under the action of forces. 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 Newton s First Law: If the resultant force on a particle is zero, the particle will remain at rest or continue to move in a straight line. Parallelogram Law Newton s Second Law: A particle will have an acceleration proportional to a nonzero resultant applied force. 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. Principle of Transmissibility Newton s Law of Gravitation: Two particles are attracted with equal and opposite forces, Mm F G 2 r W mg, g GM R 2 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 power of 10, making the number of significant figures much clearer: 2500 = 2.5 10 3 If we write 2.50x10 3, it has three significant figures 0.000036 = 3.6 x 10-5
Significant Figures Round-off error: The last digit in a calculated number may vary depending on how it is calculated, due to rounding 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 hand 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 length, force and time. These units are the foot (ft), the pound (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 weight of a platinum standard, called the standard pound, which is kept at the National Institute of Standards and Technology, outside Washington, the mass 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 kilogram (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 slug. 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 times larger than the mass of standard pound (lb).
Chapter 1: Other U.S. Customary Units Other U.S. customary units frequently used are: mile (mi) = 5280 ft. inch (in) = 1/12 ft kilopound (kip) = force of 1000 lb ton = mass of 2000 lb. Note: In engineering computation, this must be converted into slugs. Conversion into basic units of feet, pounds, seconds 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 velocity 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 unit of Length: The Meter: Originally defined as one ten-millionth of the distance from the equator to either pole, is now defined as 1 650 763.73 wavelengths of the orange-red light corresponding to a certain transition in an atom of krypton-86. 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 water is now defined as mass of a platinum-iridium standard kept at the International Bureau of Weights and Measures at Serves, near Paris, France. earth C A equator B
Chapter 1: SI Units -- Time Base unit of Time: The Second: Originally defined as 1/86 400 of the mean solar day, 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, gram(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 47.2 mm = 47.2 x ³ 10 m
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 Volume. 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: 1 dm = 0.1 m = ¹ 10 m 1 cm = 0.01 m = ² 10 m 1 mm = 0.001 m = ³ 10 m 1 km = 1 000 m = 10³ m Multiple and sub-multiple of the units of Area: 1 dm² = (1 dm)² = ¹ 10 ) m)² = ² 10 m² 1 cm² = (1 cm)² = ² 10 ) m)² = 10-4 m 2 1 mm² = (1 mm)² = ³ 10 ) m)² = 10-6 m 2 Multiple and sub-multiple of the units of Volume: 1 dm³ = (1 dm)³ = ¹ 10 ) m)³ = ³ m³ 10 1 cm³ = (1 cm)³ = ² 10 ) m)³ = 10-6 m 3 1 mm³ = (1 mm)³ = ³ 10 ) m)³ = 10-9 m 3
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 point -- 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: 1 dm = 0.1 m = ¹ 10 m 1 cm = 0.01 m = ² 10 m 1 mm = 0.001 m = ³ 10 m 1 km = 1 000 m = 10³ m Multiple and sub-multiple of the units of Area: 1 dm² = (1 dm)² = ¹ 10 ) m)² = ² 10 m² 1 cm² = (1 cm)² = ² 10 ) m)² = 10-4 m 2 1 mm² = (1 mm)² = ³ 10 ) m)² = 10-6 m 2 Multiple and sub-multiple of the units of Volume: 1 dm³ = (1 dm)³ = ¹ 10 ) m)³ = ³ m³ 10 1 cm³ = (1 cm)³ = ² 10 ) m)³ = 10-6 m 3 1 mm³ = (1 mm)³ = ³ 10 ) m)³ = 10-9 m 3
Chapter 1: SI Units Two ideas However, there have been two ideas as to which metric units should be preferred in science. Scientists working in laboratories, dealing with small quantities and distances, preferred to measure distance in centimeters and mass in grams. Scientists and engineers working in larger contexts preferred larger units: meters for distance and kilograms for mass. Everyone agreed that units of other quantities such as force, pressure, 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 metric system lies in its simplicity and consistency. Despite the advantages of the metric system, English units 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 quantity (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 inches, yards, and miles 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 measurements, 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 chain-link conversion. For example, since there are 60 seconds in one minute, 1min 60s 2min 1 60s, and 1min (2 min) x(1) (2 min) 60s x( 1min ) 120s 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.