Page 33 - 9780077418427.pdf

/Users/user-f465/Desktop
tiL12214_ch01_001-024.indd Page 10 9/1/10 9:51 PM user-f465
tiL12214_ch01_001-024.indd Page 10 9/1/10 9:51 PM user-f465 /Users/user-f465/Desktop

SYMBOLS AND EQUATIONS is called an operational defi nition because a procedure is
In the previous section, the relationship of density, mass, and established that defines a concept as well as tells you how to
volume was written with symbols. Density was represented by measure it. Concepts of what is meant by force, mechanical
ρ, the lowercase letter rho in the Greek alphabet, mass was rep- work, and mechanical power and concepts involved in elec-
resented by m, and volume by V. The use of such symbols is trical and magnetic interactions can be defined by measure-
established and accepted by convention, and these symbols are ment procedures.
like the vocabulary of a foreign language. You learn what the Describing how quantities change relative to each other.
symbols mean by use and practice, with the understanding that The term variable refers to a specific quantity of an object or

each symbol stands for a very specific property or concept. The event that can have different values. Your weight, for exam-
symbols actually represent quantities, or measured properties. ple, is a variable because it can have a different value on dif-
The symbol m thus represents a quantity of mass that is speci- ferent days. The rate of your heartbeat, the number of times
fied by a number and a unit, for example, 16 g. The symbol V you breathe each minute, and your blood pressure are also
represents a quantity of volume that is specified by a number variables. Any quantity describing an object or event can be
3
and a unit, such as 17 cm . considered a variable, including the conditions that result in
such things as your current weight, pulse, breathing rate, or
blood pressure.
Symbols As an example of relationships between variables, consider
Symbols usually provide a clue about which quantity they that your weight changes in size in response to changes in other
represent, such as m for mass and V for volume. However, in variables, such as the amount of food you eat. With all other fac-
some cases, two quantities start with the same letter, such as tors being equal, a change in the amount of food you eat results
volume and velocity, so the uppercase letter is used for one (V in a change in your weight, so the variables of amount of food
for volume) and the lowercase letter is used for the other (v for eaten and weight change together in the same ratio. A graph is
velocity). There are more quantities than upper- and lowercase used to help you picture relationships between variables (see
letters, however, so letters from the Greek alphabet are also used, “Simple Line Graph” on p. 629).
for example, ρ for mass density. Sometimes a subscript is used to When two variables increase (or decrease) together in the
identify a quantity in a particular situation, such as v i for initial, same ratio, they are said to be in direct proportion. When two
or beginning, velocity and v f for final velocity. Some symbols are variables are in direct proportion, an increase or decrease in one
also used to carry messages; for example, the Greek letter delta variable results in the same relative increase or decrease in a sec-
(Δ) is a message that means “the change in” a value. Other mes- ond variable. Recall that the symbol ∝ means “is proportional
sage symbols are the symbol ∴, which means “therefore,” and the to,” so the relationship is
symbol ∝, which means “is proportional to.”
amount of food consumed ∝ weight gain
Variables do not always increase or decrease together in
Equations direct proportion. Sometimes one variable increases while a
Symbols are used in an equation, a statement that describes a rela- second variable decreases in the same ratio. This is an inverse
tionship where the quantities on one side of the equal sign are iden- proportion relationship. Other common relationships include
tical to the quantities on the other side. The word identical refers one variable increasing in proportion to the square or to the
to both the numbers and the units. Thus, in the equation describ- inverse square of a second variable. Here are the forms of these
ing the property of density, ρ = m/V, the numbers on both sides four different types of proportional relationships:
of the equal sign are identical (e.g., 5 = 10/2). The units on both Direct a ∝ b
3
3
sides of the equal sign are also identical (e.g., g/cm = g/cm ).
Inverse a ∝ 1∙b
Equations are used to (1) describe a property, (2) defi ne a 2
Square a ∝ b
concept, or (3) describe how quantities change relative to each 2
Inverse square a ∝ 1∙b
other. Understanding how equations are used in these three
classes is basic to successful problem solving and comprehen-
sion of physical science. Each class of uses is considered sepa- Proportionality Statements
rately in the following discussion. Proportionality statements describe in general how two vari-
Describing a property. You have already learned that the ables change relative to each other, but a proportionality state-
compactness of matter is described by the property called den- ment is not an equation. For example, consider the last time you
sity. Density is a ratio of mass to a unit volume, or ρ = m/V. filled your fuel tank at a service station (Figure 1.12). You could
The key to understanding this property is to understand the say that the volume of gasoline in an empty tank you are fill-
meaning of a ratio and what “per” or “for each” means. Other ing is directly proportional to the amount of time that the fuel
examples of properties that can be defined by ratios are how fast pump was running, or
something is moving (speed) and how rapidly a speed is chang-
volume ∝ time
ing (acceleration).

Defining a concept. A physical science concept is some- This is not an equation because the numbers and units are not
times defined by specifying a measurement procedure. This identical on both sides. Considering the units, for example,

10 CHAPTER 1 What Is Science? 1-10