Physical Dimensions, Units, and Observable Quantities

Summary:
In this class, you will learn to differentiate fundamental quantities such as mass, length, and time, and how they relate to derived units like area and force. You will discover the importance of comparable observable quantities in the laws of algebra and how to convert units between different measurement systems. This class also covers vector quantities, essential for the formulation of physical equations, preparing you for a deeper understanding of measurement in sciences.

TABLE OF CONTENTS
What are units and physical dimensions?
Fundamental units, derived units, and their physical dimensions
Observables, quantities, and physical units
Algebra of comparable observables
Recommended Readings

What Are Units and Physical Dimensions?

Defining precisely what a physical dimension is can be complex. However, it is understood that physics deals with quantities that can be measured. These physical quantities are classified according to their dimension and are quantified by comparison with standard units. There are two main categories of units: the fundamental ones, such as the meter or the kilogram, and the derived ones, which are formed from the fundamental units through algebraic operations. The following table presents some of the fundamental units and their corresponding physical dimensions.

Physical Dimensions	Dimensional Symbol	Fundamental Unit	Unit Symbol
Mass	$M$	kilogram	$kg$
Length	$L$	meter	$m$
Time	$T$	second	$s$
Electric Current Intensity	$I$	ampere	$A$
Thermodynamic Temperature	$\Theta$	kelvin	$K$
Amount of Substance	$N$	mole	$mol$
Luminous Intensity	$I_v$	candela	$cd$

It is a common mistake to directly associate physical quantities with physical dimensions. This association is valid for quantities measured with fundamental units such as mass or time. However, when it comes to quantities that use derived units, like force, the relationship is not direct. Force, for instance, does not have a dimension of its own; instead, it is composed of other basic dimensions.

Fundamental Units, Derived Units, and Their Physical Dimensions

Each fundamental unit corresponds to a unique physical dimension, such as mass, length, or time. The dimensions of derived units result from the algebraic product of the dimensions of fundamental units. Let’s see some examples:

The area is the product of two lengths and therefore its dimension is $L^2$ , measurable in square meters ( $m^2$ ).
The volume, obtained from three lengths or an area multiplied by a length, has a dimension of $L^3$ and is measured in cubic meters ( $m^3$ ).
The speed, defined as distance divided by time, has a dimension of $LT^{-1}$ and is expressed in meters per second ( $m/s$ ).
The acceleration is calculated as speed divided by time, with a dimension of $LT^{-2}$ and is measured in meters per second squared ( $m/s^2$ ).
The force is the result of mass times acceleration, giving rise to a dimension of $MLT^{-2}$ . It is commonly measured in Newtons ( $N$ ), represented by the formula:
$\displaystyle N = \frac{kg \cdot m}{s^2}$

Similarly, many other magnitudes and physical dimensions can be derived.

Observables, Quantities, and Physical Units

We will continue to develop the concepts we have introduced. We refer to any property or phenomenon that can be measured, such as color, length, time, volume, or hardness, as a measurable quantity, or simply “observable”.

Observables are divided into two categories: comparable and non-comparable. Comparable observables are those that can establish a quantitative relationship, for example, when the length of a beam is several times that of a pencil. On the other hand, color cannot be quantitatively compared; thus, while length is a comparable observable, color is non-comparable.

Algebra of Comparable Observables

The logic behind comparable observables is founded on principles of equality and addition:

Equality Criterion: Two comparable observables are equal if the ratio between one and the other equals one ( $\frac{A}{B} = 1$ ).
Addition Criterion: If we have three comparable observables $A$ , $B$ , and $C$ in relation to a fourth $O$ , and the proportions $\frac{A}{O} = n_1$ , $\frac{B}{O} = n_2$ , and $\frac{C}{O} = n_3$ hold true, then we say that $A + B = C$ if and only if $n_1 + n_2 = n_3$ .

With these principles, it is demonstrated that comparable observables follow the algebraic laws of associativity, distributivity, and commutativity.

Units of Measurement and Physical Quantities

A unit of measurement is a comparable observable selected to establish comparisons with other observables of the same dimension. If two observables, $A$ and $U_A$ , are comparable, there exists a real number $\alpha$ such that $A$ equals $\alpha$ times the unit of measurement $U_A$ .

$A = \alpha U_A$

For instance, if the length of a beam is 3 meters, we write that the length of the beam is $3 [m]$ . The magnitude of a measurement varies according to the system of units used, which implies that the beam measuring $3$ meters will have an approximate magnitude of $118.11$ if measured in inches.

Conversion of Units of Measurement

As we have seen, an observable can be measured in different units as long as they share the same dimension. If $A$ is an observable, and $U_1$ and $U_2$ are two units of measurement of the same dimension, there will be two corresponding real numbers $\alpha_1$ and $\alpha_2$ .

$A = \alpha_1 U_1$ and $A = \alpha_2 U_2$

Therefore, the conversion factor $\gamma^2_1 = \alpha_2 / \alpha_1$ allows for the transformation of unit $U_2$ to $U_1$ , and vice versa with $\gamma^1_2 = \alpha_1 / \alpha_2$ . For example, a rod measuring $5$ inches in length is equivalent to $0.127$ meters, giving us a conversion factor of $0.0254$ meters per inch.

Vector Quantities

We have examined observables that are described by a single magnitude. However, there are observables like position in space that require several magnitudes for their complete description. These are known as vector quantities and are represented with multiple values. For example, an object positioned at $3$ meters to the right, $5$ meters forward, and $2$ meters up is represented as $(3, 5, 2)$ meters.

${position} = (3, 5, 2)$

These quantities benefit from vector algebra, which simplifies their handling and application in physical formulas. A common example is force, represented as a vector with magnitude and direction, essential in many physical equations.