mechanical engineers must be conversant with both systems. They need to convert quantities from one system to the other, and they must be able to perform calculations equally well in either system. In this textbook, examples and problems are formulated in both systems so that you can learn to work effectively with the USCS and the SI. As we introduce new physical quantities in the following chapters, the corresponding USCS and SI units for each will be described, along with their conversion factors.
Base and Derived Units
Given some perspective from the Mars Climate Orbiter’s loss and the emergency landing of the Air Canada fl ight on the importance of units and their bookkeeping, we now turn to the specifi cs of the USCS and SI. A unit is defi ned as an arbitrary division of a physical quantity, which has a magnitude that is agreed on by mutual consent. Both the USCS and SI are made up of base units and derived units. A base unit is a fundamental quantity that cannot be broken down further or expressed in terms of any simpler elements. Base units are independent of one another, and they form the core building blocks of any unit system. As an example, the base unit for length is the meter (m) in the SI and the foot (ft) in the USCS.
Derived units, as their name implies, are combinations or groupings of several base units. An example of a derived unit is velocity (length/time), which is a combination of the base units for length and time. The liter (which is equivalent to 0.001 m3) is a derived unit for volume in the SI. Likewise, the mile (which is equivalent to 5280 ft) is a derived unit for length in the USCS.
Unit systems generally have relatively few base units and a much larger set of derived units. We next discuss the specifi cs of base and derived units in both the USCS and SI and conversions between them.
International System of Units
In an attempt to standardize the different systems of measurement around the world, in 1960 the International System of Units was named as the United States Customary
System (USCS) and International System of Units (SI)
Base units
Derived units
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measurement standard structured around the seven base units in Table 3.1.
In addition to the mechanical quantities of meters, kilograms, and seconds, the SI includes base units for measuring electric current, temperature, the amount of substance, and light intensity. The SI is colloquially referred to as the metric system, and it conveniently uses powers of ten for multiples and divisions of units.
The base units in the SI are today defi ned by detailed international agreements. However, the units’ defi nitions have evolved and changed slightly as measurement technologies have become more precise. The origins of the meter, for instance, trace back to the eighteenth century. The meter was originally intended to be equivalent to one ten-millionth of the length of the meridian passing from the northern pole, through Paris, and ending at the equator (namely, one-quarter of the Earth’s circumference). Later, the meter was defi ned as the length of a bar that was made from a platinum- iridium metal alloy. Copies of the bar, which are called prototype meters, were distributed to governments and laboratories around the world, and the bar’s length was always measured at the temperature of melting water ice. The meter’s defi nition has been updated periodically to make the SI’s length standard more robust and repeatable, all the while changing the actual length by as little as possible. As of October 20, 1983, the meter is defi ned as the length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second, which in turn is measured to high accuracy by an atomic clock.
In a similar vein, at the end of the eighteenth century, the kilogram was defi ned as the mass of 1000 cm3 of water. Today, the kilogram is determined by the mass of a physical sample that is called the standard kilogram, and like the previously used prototype meter, it is also made of platinum and iridium. The standard kilogram is kept in a vault in Sèvres, France, by the International Bureau of Weights and Measures, and duplicate copies are maintained in other laboratories throughout the world. Although the meter is today based on a reproducible measurement involving the speed of light and time, the kilogram is not. Scientists are researching alternative means to defi ne the kilogram in terms of an equivalent electromagnetic force or in Prototype meter
Standard kilogram
Table 3.1
Base Units in the SI
Quantity SI Base Unit Abbreviation
Length meter m
Mass kilogram kg
Time second s
Electric current ampere A
Temperature Kelvin K
Amount of substance mole mol
Light intensity candela cd
3.3 Unit Systems and Conversions
terms of the number of atoms in a carefully machined silicon sphere, but for the time being the kilogram is the only base unit in the SI that continues to be defi ned by a human-made artifact.
With respect to the other base units in the SI, the second is defi ned in terms of the time required for a certain quantum transition to occur in a cesium-133 atom. The Kelvin [abbreviated K without the degree (°) symbol]
is based on the triple point of pure water, which is a special combination of pressure and temperature where water can exist as a solid, liquid, or gas.
Similar fundamental defi nitions have been established for the ampere, mole, and candela.
A few of the derived units used in the SI are listed in Table 3.2. The newton (N) is a derived unit for force, and it is named after the British physicist Sir Isaac Newton. While his classical laws of motion are presented in more detail in Chapter 4, his second law of motion, F = ma, states that the force F acting on an object is equivalent to the product of its mass m and acceleration a. The newton is therefore defi ned as the force that imparts an acceleration of one meter per second per second to an object having a mass of one kilogram:
1 N (1 kg)
(
1 m __ s2)
1 ______ kg ? ms2 (3.1)By convention, the term “newton” is not capitalized when referring to the SI unit. With the exception of the Kelvin, the units joule ( J), pascal (Pa), watt (W), and others that are named after individuals are not capitalized, although their abbreviations are.
Base and derived units in the SI are often combined with a prefi x so that a physical quantity’s numerical value does not have a power-of-ten exponent that is either too large or too small. Use a prefi x to shorten the representation of a numerical value and to reduce an otherwise excessive number of trailing Second law of motion
Prefi x
Table 3.2
Certain Derived Units in the SI
Quantity SI Derived Unit Abbreviation Defi nition
Length micrometer or
micron
μm 1 μm = 10–6 m
Volume liter L 1 L = 0.001 m3
Force newton N 1 N = 1 (kg ? m)/s2
Torque, or moment of a force
newton-meter N ? m —
Pressure or stress pascal Pa 1 Pa = 1 N/m2
Energy, work, or heat joule J 1 J = 1 N ? m
Power watt W 1 W = 1 J/s
Temperature degree Celsius °C °C = K – 273.15
Although a change in temperature of 1 Kelvin equals a change of 1 degree Celsius, numerical values are converted using the formula.
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zero digits in your calculations. The standard prefi xes in the SI are listed in Table 3.3. For example, modern wind turbines are now producing over 7,000,000 W of power. Because it is cumbersome to write so many trailing zeroes, engineers prefer to condense the powers of ten by using a prefi x. In this case, we describe a turbine’s output as being over 7 MW (megawatt), where the prefi x “mega” denotes a multiplicative factor of 106.
Good practice is not to use a prefi x for any numerical value that falls between 0.1 and 1000. Thus, the “deci,” “deca,” and “hecto” prefi xes in Table 3.3 are rarely used in mechanical engineering. Other conventions for manipulating dimensions in the SI include the following:
1. If a physical quantity involves dimensions appearing in a fraction, a prefi x should be applied to the units appearing in the numerator rather than the denominator. It is preferable to write kN/m in place of N/ mm. An exception to this convention is that the base unit kg can appear in a dimension’s denominator.
2. Placing a dot or hyphen between units that are adjacent in an expression is a good way to keep them visually separated. For instance, in expanding a newton into its base units, engineers write (kg · m)/s2 instead of kgm/s2. An even worse practice would be to write mkg/s2, which is particularly confusing because the numerator could be misinterpreted as a millikilogram!
3. Dimensions in plural form are not written with an “s” suffi x. Engineers write 7 kg rather than 7 kgs because the trailing “s” could be misinterpreted to mean seconds.
SI conventions
Table 3.3
Order-of-Magnitude Prefi xes in the SI
Name Symbol Multiplicative Factor
tera T 1,000,000,000,000 = 1012
giga G 1,000,000,000 = 109
mega M 1,000,000 = 106
kilo k 1000 = 103
hecto h 100 = 102
deca da 10 = 101
deci d 0.1 = 1021
centi c 0.01 = 1022
milli m 0.001 = 1023
micro μ 0.000,001 = 1026
nano n 0.000,000,001 = 1029
pico p 0.000,000,000,001 = 10212
3.3 Unit Systems and Conversions
4. Except for derived units that are named after individuals, dimensions in the SI are written in lowercase.