KAHIBARO
Table of Contents
Introduction
Physics describes how the natural world behaves using quantities that can be measured. These measurable aspects of nature are called physical quantities. Every calculation and prediction in physics is built from them. In this chapter we introduce what a physical quantity is, how it is described, and how it differs from everyday vague descriptions.
What is a Physical Quantity?
A physical quantity is any property of a system or phenomenon that can be measured and expressed as a number and a unit. Temperature, length, mass, time, electric charge, and speed are all examples of physical quantities.
To say that something is a physical quantity, two conditions must be met. First, you must be able to compare it meaningfully with a standard, such as a meter stick or a clock. Second, the result of that comparison must be expressible as a specific numerical value together with a unit. For example, saying "the table is 1.2 m long" specifies both a number, 1.2, and a unit, meter, for the physical quantity length.
Vague statements like "the table is long" do not describe a physical quantity in the scientific sense, because they lack a numerical value and a defined unit.
Numerical Value and Unit
Every measured physical quantity is written in the form
$$
\text{physical quantity} = \text{numerical value} \times \text{unit}.
$$
If the length $L$ of a rod is measured to be 2.35 meters, you can write
$$
L = 2.35\,\text{m}.
$$
The numerical value, here 2.35, tells you how many times the chosen unit fits into the quantity you are describing. The unit, here meter, defines the scale of measurement. Changing the unit changes the numerical value, even though the physical quantity itself has not changed. For the same length you could write
$$
L = 235\,\text{cm},
$$
and the length of the rod has not physically changed at all.
The separation of a physical quantity into a number and a unit is fundamental. Later chapters on units, conversions, and dimensional analysis will build directly on this idea, so here you only need to recognize that both parts are always present in a proper physical description.
Fundamental and Derived Physical Quantities
In physics, some physical quantities are treated as basic starting points, while others are built from them. The basic ones are called fundamental or base quantities. Examples include length, mass, time, electric current, temperature, amount of substance, and luminous intensity. They are chosen so that other quantities can be expressed in terms of them.
Derived quantities are constructed from fundamental quantities by mathematical relationships. For example, area is length multiplied by length and volume is length multiplied by length multiplied by length. Speed is distance divided by time. Although the units of these quantities will be handled in later chapters, it is useful now to see that the idea of "derived" already exists at the level of the quantities themselves.
The table below gives some examples of fundamental and derived physical quantities without going into their units.
| Type of quantity | Example | Relationship (conceptual) |
|---|---|---|
| Fundamental | Length | Basic measure of distance |
| Fundamental | Mass | Basic measure of amount of matter |
| Fundamental | Time | Basic measure of duration |
| Derived | Area | Length × length |
| Derived | Volume | Length × length × length |
| Derived | Speed | Distance ÷ time |
| Derived | Acceleration | Change in speed ÷ time |
| Derived | Force | Mass × acceleration |
Later chapters on SI base units and derived units will formalize this distinction. For now, you only need to recognize that some quantities are taken as basic building blocks and others are formed from them.
Physical Quantities vs Pure Numbers
A pure number has no unit. The number 3, by itself, does not tell you what is being counted or measured. By contrast, a physical quantity always carries meaning through its unit and its physical interpretation.
For example, 3 by itself is ambiguous. It might be 3 meters of rope, 3 seconds of time, or 3 apples. When you write a physical quantity, you must attach the relevant unit and understand what physical property you are describing:
$$
t = 3\,\text{s}, \quad L = 3\,\text{m}.
$$
In equations of physics, you will often see both pure numbers and physical quantities. Mathematical constants, such as $\pi$ or the number 2 in a factor like $2E$, are unitless. Physical symbols such as $t$ for time or $v$ for speed represent quantities that carry units. Keeping these roles distinct helps avoid confusion. A ratio of two identical quantities, such as distance divided by distance, can give a pure number, which is then dimensionless.
Classification of Physical Quantities
Physical quantities are classified in several ways. One especially important classification, treated fully in a later chapter, is the difference between scalars and vectors. Here we only describe the idea in words without formal definitions or notation.
Some physical quantities are completely described by a single number and a unit, with no reference to direction. Temperature, mass, and time are of this type. Other physical quantities also require a direction in space for a complete description. Displacement, velocity, and force have this directional character. The first group will be called scalars and the second group will be called vectors in a later chapter.
Another useful distinction is between extensive and intensive quantities, which arises especially in thermodynamics. Extensive quantities depend on the size or amount of the system, while intensive quantities do not.
| Type | Example quantities | Depends on system size? |
|---|---|---|
| Extensive | Mass, volume, total energy | Yes |
| Intensive | Temperature, pressure, density | No, for a uniform sample |
While this book will not rely heavily on that terminology in the early parts, you will encounter it in thermodynamics and material science, so it is helpful to recognize that physical quantities can be grouped according to how they behave when systems are combined or divided.
Measuring Physical Quantities
To use physical quantities scientifically, you must be able to measure them. Measurement is the process of comparing an unknown quantity with a known standard. For example, to measure length you might place a ruler beside an object and count how many unit segments match the object. To measure time you might count the number of ticks of a clock between two events.
Each specific physical quantity requires an appropriate measurement method and instrument. Length can be measured with rulers, calipers, or laser range finders. Time can be measured with stopwatches or atomic clocks. Mass can be measured with balances. Temperature can be measured with thermometers.
The act of measurement is never perfect. Later chapters on measurement uncertainty and significant figures will explain how to represent and handle this imperfection. At this stage, it is enough to understand that a measured physical quantity expresses an approximation to the true value, not an absolutely exact number.
Relationships Between Physical Quantities
One of the main aims of physics is to discover how different physical quantities are related. These relationships are written as mathematical equations in which each symbol represents a physical quantity.
For example, you might encounter a relationship such as
$$
v = \frac{d}{t},
$$
where $v$ is speed, $d$ is distance traveled, and $t$ is time taken. This equation relates three physical quantities. If you know two of them, you can solve for the third.
Later chapters will introduce many such relationships and will explain how to use algebra and calculus to manipulate them. Here you should notice that in physics, symbols like $v$, $d$, and $t$ are not just abstract mathematical variables. Each represents a specific quantity that can, at least in principle, be measured in an experiment.
In any correct physical equation, both sides must represent the same physical quantity. They must have the same type and the same combination of units, even if those units are written in different but equivalent forms.
This idea, called dimensional consistency, will be explored systematically in the chapter on dimensional analysis. For now, simply recognize that you cannot meaningfully equate unrelated physical quantities, such as a time and a mass.
Symbols for Physical Quantities
To work compactly, physics uses symbols to represent physical quantities. For example:
| Symbol | Physical quantity |
|---|---|
| $t$ | Time |
| $x$ | Position or length |
| $m$ | Mass |
| $T$ | Temperature or period |
| $v$ | Speed or velocity magnitude |
| $a$ | Acceleration magnitude |
| $F$ | Force magnitude |
| $E$ | Energy |
The same letter can represent different quantities in different contexts, so the meaning is always determined by the surrounding text or by convention in that part of physics. As you progress, you will learn common conventions, but it is always wise to check how a symbol is defined in a particular chapter or problem.
It is important to distinguish between the symbol for a physical quantity and its numerical value in a particular unit. When you write $F = 5\,\text{N}$, the symbol $F$ stands for the general concept of force in that situation, while $5\,\text{N}$ is a specific measured or calculated value of that force.
Visualizing Physical Quantities
Many physical quantities can be pictured. For non directional quantities such as temperature or pressure in a gas, you can imagine regions of space where the quantity is larger or smaller and represent them with shaded diagrams or contour lines. For directional quantities such as displacement or force, arrows are used to represent both magnitude and direction.
Below is a simple technical drawing that suggests a thermometer measuring the temperature of a liquid in a container. The exact numbers are not important yet. The drawing serves only to show that a physical quantity like temperature can be associated with a physical system and measured by an instrument.
In later chapters, such diagrams will be extended to show vectors, fields, and more complex distributions of physical quantities in space and time.
Summary
A physical quantity is a measurable property of a physical system that can be expressed as a numerical value multiplied by a unit. Fundamental quantities serve as building blocks, while derived quantities are formed from them through mathematical relationships. Physical quantities differ from pure numbers and must always carry units and physical meaning. They can be measured, represented with symbols, and related through equations. With this foundation, you are ready to study the specific units associated with these quantities and the rules for working with them in a systematic way.