Fundamental dimensions length

Although the units for each of the measurements above are different, they share the same fundamental dimension, length L. Each of these measurements has the dimensions of length. 

The dimension of any of these seven quantities can be written as a power product of the four fundamental dimensions length L, time Z, mass M and temperature T, which are sufficient for describing thermodynamics and heat transfer by physical... 

Length is one of the seven fundamental or base dimensions that we use to properly express what we know of our natural world. In todays globally driven economy, where products are made in one place and assembled somewhere else, there exists an even greater need for a uniform and consistent way of communicating informadon about the fundamental dimension length and other related len dt variables so that parts manufactured in one place can easily be combined on an assembly line with parts made in other places. An automobile is a good example of this concept. It has literally thousands of parts that are manufactured by various companies in diferent parts of the world. 

Finally, it is important to emphasize the fact that all physical variables discussed in this chapter are based on the fundamental dimension length. For example, area has a dimension of (length), volume has a dimension of (length) and the second moment of area has a dimension of (length). In Chapter 8, we will look at time- and length-related parameters in engineering. 

You should understand the significant role the fundamental dimension length plays in engineering problems. You should also realize the importance of area and volume in engineering applications. 

Most laboratory analysis methods (see Section 1.8.1) measure concentration. The choice of units for concentration depends on the medium, on the process that is being measured or described, and sometimes on custom and tradition. In water, a common expression of concentration is mass of chemical per unit volume of water, which has dimensions of [M/ . The letters M, L, and T in square brackets refer to the fimdamental dimensions of mass, length, and time (see Appendix). Many naturally occurring chemicals in water are present at levels of a few milligrams per liter (mg/liter). For clarity in this book, specific units, such as (cm/hr) or (g/m ), either are free-standing or are indicated in parentheses, not in square brackets. Note that the word "liter" is always spelled out in this text, to avoid confusion with the abbreviation [L] for the fundamental dimension length. 

There are two systems of fundamental dimensions in use (with their associated units), which are referred to as scientific and engineering systems. These systems differ basically in the manner in which the dimensions of force is defined. In both systems, mass, length, and time are fundamental dimensions. Furthermore, Newton s second law provides a relation between the dimensions of force, mass, length, and time ... 

These six variables (considering Apfll as a single variable), in = 6. may be expressed in terms of n = 3 fundamental dimensions, i.e. mass, length... 

Mass, M, length, L and time, T, are called fundamental dimensions. With heat transfer problems, temperature, , and heat, H, are introduced as fundamental dimensions (see Section 6.7.4). 

The application of thermodynamics to any real problem starts with the identification of a particular body of matter as the focus of attention. This quantity of matter is called the system, and its thermodynamic state is defined by a few measurable macroscopic properties. These depend on the fundamental dimensions of science, of which length, time, mass, temperature, and amount of substance are of interest here. 

From Newton s second law, as well as Eq. (3.1-8), we know that force has the dimensions of mass times acceleration thus, Eq. (3.1-8) and other fluid-mechanical formulas are most easily expressed by choosing mass (M), length (L), and time (t) as fundamental dimensions, and expressing force (F) in units of ML/t. By following this well-established custom, we avoid... 

Three fundamental dimensions of measurement, mass [M], length [L], and time [T], form the basis for most environmental quantities. A good understanding of these dimensions and of the way in which they are combined to form various units of measurement clarifies many problems of chemical fate and transport. 

Dimensional analysis predicts the various dimensionless parameters which are helpful in correlating experimental data. Certain dimensions must be established as fundamental, with all others expressible in terms of these. One of these fundamental dimensions is length, symbolized L. Thus, area and volume may dimensionally be expressed as L2 and L3, respectively. A second fundamental dimension is time, symbolized t. Velocity and acceleration may be expressed as L/t and Lit1, respectively. Another fundamental dimension is mass, symbolized M. The mole is included in M. An example of a quantity whose dimensional expression involves mass is the density (mass or molar), which would be expressed as Mil . 

In this section, we vinli contider derived physical quantities that are based on die fundamental dimensions of length and time. We will first discuss the concepts of linear speed and acceleration and then define volumetric flow rate. 

As we explained the Btu (British thermal unit) in Chapter 11, one Btu is formally defined as the amount of thermal energy needed to raise the temperature of 1 lb of water by 1°R The calorie is defined as the amount of heat required to raise the temperature of 1 g of water by 1°C. And as you may also recall from our discussion in Chapter 11, in SI units no distinction is made between the units of thermal energy and mechanical eneigy, and therefore the units of thermal energy are defined in terms of fundamental dimensions of mass, length, and time. In the SI system of units, the joule is the unit of energy and is defined as... 

Formulas and equations are often a very important component of a technical report. It is essential that all variables, exponents, parameter, and so on, be clearly defined immediately following the equation. The units of each component of an equation should be stated. Use of SI units is preferred. If the formula is written so that it is valid for any consistent set of units, then the fundamental dimension, for example, length or force, of each symbol should be listed. For lengthy reports including many equations, a list of nomenclature placed normally right after the list of tables and figures or right before the references, is recommended. 

In the English Gravitational system of units, the fundamental dimensions are length in feet (ft), force (lb or Ibp), and time in seconds or hours (s or hr). In this system of units, a one pound force imparts an acceleration of 32 ft/s to a mass of one slug. Newton s second law is then... 

We can generalize this suggestion by considering a physical concept a that we want to quantify. Our first step is to choose a set of fundamental dimensions that will quantify a. For example, let us choose Length, Mass, and Time (LMT) as our fundamental dimension set. We next select the system of units we will use to determine the physical magnitude of a. Since there are many such systems of units, let us choose LjMiTi as our system of units. Thus... 

Step 2. Write down the dimensions cf these variables. The basic dimensions that are commonly chosen in dimensional analysis are those of mass (M), length (L), time (0), and temperature (T). All other quantities are expressed in terms of these fundamental dimensions. Thus, force has the units of MLQ by virtue of Newton s law. Joule (J) is not a fundamental dimension but is instead expressed as ML 0". ... 

Table 1.2 lists the basic quantities as expressed in SI together with the unit abbreviations, Table 1,3 lists the unit prefixes needed for this book, and Table 1.4 lists some of the constants needed in several systems. Finally, Table 1.5 lists the conversion factors into SI for all quantities needed for this book. The boldface letters for each quantity represent the fundamental dimensions F = force, L = length, M = mass, mole = mole, T = temperature, 0 = time. The list of notation at the end of each chapter gives the symbols used, their meaning, and dimensions. 

Before considering the method of proceeding from Eq.(2.24) to Eq. (3.2) in this chapter, a few quantities will be defined. Fundamental or primary dimensions are properties of a system under study that may be considered independent of the other properties of interest. For example, there is one fundamental dimension in any geometry problem and this is length (L). The fundamental dimensions involved in different classes of mechanical problems are listed in Table 3.1 where Z stands for length, F for force, and T for time. Dimensions other than F, L, and T, which are considered fundamental in areas other than mechanics, include temperature... 

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