Copyright
© 1998, 1999, 2000 by Prentice Hall, Inc.
NOT FOR DISTRIBUTION
Dimensions and Units
Objectives
- Understand the difference between a unit and a dimension.
In engineering practice, physical quantities are quantified by giving a magnitude and a unit of measurement. For example, the playing surface of a football field is one hundred yards long. In this case, the magnitude is the value one hundred, and the yard is the unit of measurement. A unit refers to the arbitrary system of measurement.
Units of measurement are standardized quantities. In the United States, the National Institute of Standards and Technology (NIST) standardizes units of measurement. Standardization of a unit means that within the region where the standardizing organization has jurisdiction, the unit has the same meaning for everyone. For example, the yard in the previous example would imply the same physical quantity of length to a person living in California as it would to a person living in New York. Furthermore, a standard yard would have the same length for both individuals, because the length of a yard is standardized.
Some units are combinations of others. For instance, the unit of force is a combination of the units for mass and acceleration.
A dimension is a measurable quantity of a physical parameter. The yard is a unit of length in a particular system of measurement whose dimension is length. Fundamental dimensions are independent of one another. For example, a set of fundamental dimensions may be mass, length, time, and temperature.
Let us define the dimension of an engineering quantity by placing a bracket around the quantity. Thus, if a is acceleration, then [a] would imply the dimensions of acceleration. Furthermore, let M, L, T and [thgr] represent the fundamental dimensions of mass, length, time, and temperature, respectively. Then, the dimensions of acceleration would be
Therefore, the dimensions of acceleration are length per time squared. The dimension of mass is mass. What are the dimensions of force, F?
The law of Dimensional Homogeneity requires that all terms in an equation have the same dimensions. Newton's second law says that the force on a system is equal to the mass of the system times the acceleration. This may be written as F = ma, where F is the force, m is the mass, and a is the acceleration. Therefore, the dimensions of force must be the same as those of mass times the acceleration in order for Newton's second law to be valid. Then, the dimensions of force are
A nondimensional quantity is a quantity that has dimensions of one. For example, what are the dimensions of F(ma)? The answer is
Nondimensional quantities play an important role in fluid mechanics.
Practice!
You may have noticed that when you pull on a spring, there is a force tending to restore the spring to it's unstretched position. If F is this force and x is the amount that the spring was displaced, engineers write the relationship between the force and spring as
where k is a constant called the spring stiffness. What must the dimensions of k be for this equation to obey the law of dimensional homogeneity?
1 The British Gravitational System
The units of a quantity depend on the arbitrary system of measurement. One of these systems of measurement is the British Gravitational System (BGS). This particular system uses force, as a fundamental dimension, instead of mass. Because of Newton's second law (i.e., F = ma), the dimensions of mass become
In BGS, the unit of force is the pound (lb). Therefore, the unit of mass is related to the unit pound. This unit is called the slug and is defined by
where ft (foot) is the unit of length.
The weight of an object, W, is the mass times the gravitational acceleration constant, g. This may be written as W = mg. In this case, the unit is the pound, as it is a force. Because, on earth, g = 32.174 ft/sec2, a slug weighs
Therefore, in BGS, one slug has a weight of 32.174 pounds.
Additional units for BGS are listed in Table 2-1.
Listed in the Table are Some Common Units for Engineering Parameters. The Parameter in Parentheses is the Symbol for That Unit
Quantity BGS EE SI Mass slug lbm kilogram Length ft ft meter (m) Time sec (s) sec (s) sec (s) Area ft2 ft2 m2 Volume ft3 ft3 m3 Velocity Ft/s Ft/s M/s Acceleration Ft/s2 Ft/s2 M/s2 Density Slug/ft3 Lbm/ft3 Kg/m3 Force Lbf Lbf newton (N) = kg·m/s2 Pressure (or Stress) Lbf/ft2 Lbf/ft2 pascal (Pa) = N/m2 Energy (or Work) ft·Lbf ft·Lbf joule (J) = N·m Power ft·Lbf/s ft·Lbf/s watt (W) = J/s 2 The English Engineering System
The English Engineering (EE) system is defined so that 1 unit of mass has a weight of one pound (Lbf) in a standard gravity. The unit of mass in the EE system is the pound-mass (Lbm), and the unit of force is called the pound-force (Lbf). From Newton's second law, we have that 1 Lbf = 1 Lbm·32.174 ft/sec2. This relation does not have the same units on the right-hand side as the left-hand side of the equation. In order not to violate the law of dimensional homogeneity, the pound-force must be related to the unit of pound mass through the relation:
Thus, in the EE system, when converting between units of mass or force, one must divide or multiply by a conversion factor, gc, equal to 32.174 ft·Lbm/Lbf·sec2. Then, in order to have dimensional homogeneity, Newton's second law is properly written as
EXAMPLE
What is the force (in Lbf) on a 0.1-Lbm body accelerating at 2 ft/ sec2?
SOLUTION
The force is given by Equation 2-3. Therefore, we have
Additional units for the EE system are listed in Table 2-1.
3 The SI System
The Système International (SI), or metric system, was first established in France in about 1800 and has become the worldwide standard. In the United States, voluntary conversion to the metric system began in 1975, even though the system has been recognized by Congress as early as 1866. In time, it is expected that all units will be in SI form.
The unit of mass in SI is the kilogram (kg). The unit of length is the meter (m) and for volume, the liter (L). The system is set up so that one liter of water has a mass of one kilogram. The unit of weight in the SI system is the newton (N). From newton's second law, one newton is defined as
The weight is given by the expression W = mg, where g is 9.81 m/sec2.
A clear advantage of SI over other systems of measurement is that SI is based on the powers of 10. For example, 1/100th of a meter is a centimeter (cm). To convert from meter to centimeter, all one need do is multiply by 100. This is easily memorized, because “centi” means 1/100th. Therefore, centimeter means 1/100th of a meter. Table 2-2 lists the prefixes defined in the SI system.
Prefixes for the SI System
Multiplication Factor Prefix SI Symbol 1012 tera T 109 giga G 106 mega M 103 kilo k 102 hecto h 101 deka da 10 1
deci d 10 centi c 10 milli m 10 micro [mgr] 10 nano n 10 pico p 10 femto f 10 atto a Handling small quantities is very easy in SI. Consider a measurement of 0.03 m. Rather than writing this number in decimal form, multiply it by 100 to get the same measurement in centimeters (i.e., 3 cm). Conversion between SI and the BGS and EE systems is discussed in Section 2.2.
