# IMPORTANCE OF DIMENSIONS AND UNITS

Any physical quantity is characterized by dimensions . The quantities assigned to the dimensions are called units . Some basic dimensions, such as mass m , length L , time t, and temperature T are selected as primary or fundamental dimensions , while others such as velocity v, energy E, and volume V are expressed in terms of the primary dimensions and are called secondary dimensions. or derived dimensions .

Over the years, several unit systems have been created. Despite the great efforts that the scientific community and engineers have made to unify the world with a single system of units. Two of these are still in common use today: the English system , which is known as the United States Customary System (USCS) and the metric SI (from Le Système International d ‘Unités ), also called the international system. The SI is a simple and logical system based on a decimal relationship between the different units, and is used for scientific and engineering work in most industrialized nations, including England (where the English system of units was invented). However, the English system has no obvious systematic number base and several units of this system relate to each other quite arbitrarily (12 inches = 1 foot, 1 mile = 5,280 feet, 4 quarts = 1 gallon, etc.), which which makes learning confusing and difficult. The United States is the only industrialized country that has not yet adopted the metric system.

Systematic efforts to develop an acceptable universal unit system date back to 1790 when the French National Assembly commissioned the French Academy of Sciences to suggest such a unit system. A first version of the metric system was soon developed in France, but did not find universal acceptance until 1875 when 17 countries, including the United States, prepared and signed the Metric Convention Treaty . In this international agreement, meter and gram were established as the metric units for length and mass, respectively, in addition to establishing that a General Conference on Weights and Measures (GFCM) meets every six years. In 1960 the CGPM produced the SI, which is based on six fundamental quantities, whose units were adopted in 1954 at the Tenth General Conference on Weights and Measures: meter (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A ) for electric current, Kelvin degree (° K) for temperature and candle (cd) for light intensity (amount of light). In 1971, the CGPM added a seventh quantity and fundamental unit: mol (mol) for the quantity of matter.

Based on the notation scheme introduced in 1967, the degree symbol was officially removed from the absolute temperature unit, and all unit names would be lowercase even if derived from proper names (Table 1-1) .

However, the abbreviation for a unit would be capitalized if the unit came from a proper name. For example, the SI unit of force, named after Sir Isaac Newton (1647-1723), is the Newton (not Newton), and is abbreviated as N. Also, the full name of a unit can be pluralized, but not its abbreviation. For example, the length of an object can be 5 m or 5 meters, not 5 ms or 5 meters. Finally, period will not be used in unit abbreviations unless they appear at the end of a sentence. For example, the appropriate abbreviation for meter is m (not m.).

In the United States, the recent shift to the metric system began in 1968 when Congress, in response to what was happening in the rest of the world, passed a Metric Study Decree. Congress continued this push for a voluntary change to the metric system by passing the Metric Conversion Decree in 1975. A commercial law passed in 1988 set September 1992 as the deadline for all federal agencies to move to the metric system. However, the deadlines were relaxed without establishing clear plans for the future (that is, the US uses the units it likes).

Industries with intense participation in international trade (such as automotive, carbonated beverages, and liquor) have been quick to move to the metric system for economic reasons (having a single global design, fewer sizes, and smaller inventories, etc. .). Today almost all cars made in the United States obey the metric system. Most car owners are not likely to notice until they use an inch wrench on a metric screw. However, most industries resist change, which delays the conversion process.

Today, the United States is a dual-system society and will remain so until it completes the transition to the metric system. This adds an extra burden to current engineering students as they are expected to retain their understanding of the English system while learning, thinking, and working in terms of SI. Given the position of engineers in the transition period (in practice in the US, the SI system has no application in daily life), both unit systems are used in this book, with special emphasis on SI units.

As noted, SI is based on a decimal relationship between units. The prefixes used to express the multiples of the various units are listed in Table 1-2, are used as the standard for all of these, and are encouraged to memorize due to their extended use (Fig. 1-6).

Some SI and English units

In SI, the units of mass, length, and time are kilogram (kg), meter (m), and second (s), respectively. The corresponding units in the English system are pound-mass (lbm), foot (ft) and second (s). The pound symbol lb is actually the abbreviation for pound , which was in ancient Rome the unit adapted to express weight. The English system maintained this symbol even after the Roman occupation of Brittany ended in 410. The units of mas and length in the two systems are related to each other by

1 lbm = 0.45359 kg

1 pie = 0.3048 m

In the English system, force is commonly considered as one of the primary dimensions and is assigned a non-derived unit. This is a source of confusion and error that requires the use of a dimensional constant ( c ) in many formulas. To avoid this nuisance, force is considered as a secondary dimension whose unit is derived from Newton’s second law, that is,

Force = (mass) (acceleration)

The

F = ma

In SI, the unit of force is the newton (N), and is defined as the force required to accelerate a mass of 1 kg at a rate of 1 m / s2 . In the English system, the unit of force is the pound-force (lbf) and is defined as the force required to accelerate a mass of 32,174 lbm (1 slug) at the rate of 1 ft / s2 (Fig. 1-7).

A force of 1 N is roughly equivalent to the weight of a small apple ( m = 102 g ), while a force of 1 lbf is roughly equivalent to the weight of four medium apples ( total m = 454 g ), as illustrated in Figure 1-8. Another unit of force in common use in many European countries is the kilogram-force (kgf), which is the weight of 1 kg of mass at sea level ( 1 kgf = 9,807 N ).

The term weight is often used incorrectly to express mass, particularly by weight watchers. Unlike mass, weight W is a force : the gravitational force applied to a body, and its magnitude is determined from Newton’s second law,

W = mg       (N)

where m is the body mass and g is the local gravitational acceleration ( g is 9,807 m / s2 or 32,174 ft / s2 at sea level and latitude 45 °). An ordinary bathroom scale measures the gravitational force acting on a body. The weight of the unit volume of a substance is called the specific weight γ and is determined from γ = ρg , where ρ is the density.

The mass of a body is the same regardless of its location in the universe; however, its weight is modified with a change in gravitational acceleration. A body weighs less on top of a mountain since g decreases with altitude. On the moon’s surface, an astronaut weighs about a sixth of what it weighs on Earth (Fig. 1-9).

At sea level a 1kg mass weighs 9,807 N, as illustrated in Figure 1-10; however, a mass of 1 lbm weighs 1 lbf, leading people to believe that pound-mass and pound-force can be used interchangeably as pound (lb), which is a major source of errors in English system.

It should be noted that the force of gravity acting on a mass is due to the attraction between the masses and, therefore, is proportional to the magnitudes of the masses and inversely proportional to the table of the distance between them. Therefore, the gravitational acceleration g at a location depends on the local density of the Earth’s crust, the distance to the center of the Earth, and, to a lesser degree, the positions of the Moon and the Sun. The value of g varies with the location from 9,832 m / s2 to 1000 km above sea level. However, at altitudes of up to 30 km, the variation of gthe sea level value of 9,807 m / s2 is less than 1 percent. So, for most practical purposes, gravitational acceleration is assumed to be constant at 9.81 m / s2. It is interesting to note that in places located below sea level the value of g increases with distance from sea level, reaches a maximum close to 4500 m and then begins to decrease. (What do you think the value of g is at the center of the Earth?).

The main cause of confusion between mass and weight is that mass is generally measured indirectly by calculating the force of gravity it exerts. With this approach it is also assumed that the forces exerted by other effects such as buoyancy in air and fluid movement are negligible. This is like measuring the distance to a star by measuring its transition to red or determining the altitude of an airplane using barometric pressure – both are indirect measurements. The direct correct way to measure mass is to compare it to a known one. However, this is difficult and is mainly used for calibration and measurement of precious metals.

The work , which is a form of energy, can be simply defined as the force multiplied by the distance; therefore, it has the unit “ newton meter ( N · m )”, called joule ( J ). That is to say,

A more common unit for energy in the SI is the kilojoule (1 kJ = 103 J). In the English system, the energy unit is the Btu (British thermal unit), which is defined as the energy required to raise the temperature of 1 lbm of water to 38 ⁰F by 1 ⁰F. In the metric system, the amount of energy required to raise the temperature of 1 gram of water by 1 ⁰C to 14.5 ⁰C is defined as 1 calorie (cal), and 1 cal = 4.1868 J. The magnitudes of kilojoule and Btu are almost identical (1 Btu = 1.0551 kJ). There is a good way to intuitively appreciate these units: if you light a match and let it consume, it produces approximately one Btu (or one kJ) of energy (Fig. 1-11).

The unit for the energy time ratio is the joul per second (J / s) which is known as the watt (W). In the case of work the energy time ratio is called power. A commonly used unit of power is the horsepower (hp), which is the equivalent of 746 W. Electric power is typically expressed in the unit kilowatt-hour (kWh), which is equivalent to 3600 kJ. An electrical appliance with a nominal power of 1 kW consumes 1 kWh of electricity when it works continuously for one hour. When talking about electric power generation, kW and kWh units are often confused. Note that kW or kJ / s is a unit of power, while kWh is a unit of energy. Therefore, phrases like “the new wind turbine will generate 50kW of electricity per year” are meaningless and incorrect. A correct expression would be something like “the new wind turbine, with a nominal power of 50 kW, will generate 120,000 kWh of electricity per year.”

Dimensional homogeneity

In elementary school you learn that apples and oranges don’t add up, but somehow you manage to do it (by mistake, of course). In engineering, the equations must be dimensionally homogeneous . That is, each term in an equation must have the same unit (Fig. 1-12).

If at any stage of an analysis you are in a position to add two quantities that have different units, it is a clear indication that an error has been made at a previous stage. So checking dimensions can serve as a valuable tool for error detection.

It is known from experience that units can cause terrible headaches if not used carefully when solving a problem. However, with some attention and skill, units can be used profitably. They are used to check formulas and can even be used to deduce formulas, as explained in the following example.

It is important to remember that a formula that is not dimensionally homogeneous is definitely wrong (Fig. 1-15), but a formula with dimensional homogeneity is not necessarily correct.

Unit conversion ratios

Just as it is possible to form non-primary dimensions by suitable combinations of primary dimensions, all non-primary units ( secondary units ) are formed by combinations of primary units. Units of force, for example, can be expressed as

It can also be more conveniently expressed as unit conversion ratios as

Unit conversion ratios are equal to 1 and have no units; therefore, such relationships (or their inverses) can be conveniently inserted into any calculation to properly convert units (Fig. 1-16). It is recommended that we always use unit conversion ratios. In some places, the archaic gravitational constant c defined as c = 32.174 lbm · ft / lbf · s2 = kg · m / N · s2 = 1 is included in the equations in order to match the units of force. This practice produces unnecessary confusion and we consider that it is not advisable. Instead, it is recommended to use unit conversion ratios.