Physical Dimensions, Units, and Observable Quantities<\/title><\/p>\n<style>table {\ndisplay: block;\noverflow: scroll;\n}\n<\/style>\n<\/head><body><\/p>\n<div style=\"background-color:#F3F3F3; padding:20px;\"><center><\/p>\n<h1>Physical Dimensions, Units, and Observable Quantities<\/h1>\nSummary:<\/strong> \nIn 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.\n<\/p>\nTABLE OF CONTENTS<\/strong><\/u> \n<a href=\"#1\">What are units and physical dimensions?<\/a> \n<a href=\"#2\">Fundamental units, derived units, and their physical dimensions<\/a> \n<a href=\"#3\">Observables, quantities, and physical units<\/a> \n<a href=\"#4\">Algebra of comparable observables<\/a> \n<a href=\"#5\">Recommended Readings<\/a>\n<\/p>\n<\/center><\/p>\n<center> \n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/WmhuV7rxL_0\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe> \n<\/center>\n<\/div>\n<a name=\"1\"><\/a><\/p>\n<h2>What Are Units and Physical Dimensions?<\/h2>\nDefining precisely what a physical dimension<\/strong> is can be complex. However, it is understood that physics deals with quantities that can be measured. These physical quantities<\/strong> are classified according to their dimension<\/strong> and are quantified by comparison with standard units<\/strong>. There are two main categories of units: the fundamental<\/strong> ones, such as the meter or the kilogram, and the derived<\/strong> 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.<\/p>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: center; background-color: #000000; color: #ffffff;\">Physical Dimensions<\/td>\n<td style=\"text-align: center; background-color: #000000; color: #ffffff;\">Dimensional Symbol<\/td>\n<td style=\"text-align: center; background-color: #000000; color: #ffffff;\">Fundamental Unit<\/td>\n<td style=\"text-align: center; background-color: #000000; color: #ffffff;\">Unit Symbol<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; background-color: #ffffff;\">Mass<\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">M<\/span><\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">kilogram<\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">kg<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; background-color: #ffffff;\">Length<\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">L<\/span><\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">meter<\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">m<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; background-color: #ffffff;\">Time<\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">T<\/span><\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">second<\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">s<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; background-color: #ffffff;\">Electric Current Intensity<\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">I<\/span><\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">ampere<\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">A<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; background-color: #ffffff;\"><a href=\"https:\/\/toposuranos.com\/tag\/temperature\/\" target=\"_blank\" rel=\"noopener\">Thermodynamic Temperature<\/a><\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">\\Theta<\/span><\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">kelvin<\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">K<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; background-color: #ffffff;\"><a href=\"https:\/\/toposuranos.com\/tag\/mol\/\" target=\"_blank\" rel=\"noopener\">Amount of Substance<\/a><\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">N<\/span><\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">mole<\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">mol<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; background-color: #ffffff;\">Luminous Intensity<\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">I_v<\/span><\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">candela<\/td>\n<td style=\"text-align: center; background-color: #ffffff;\">cd<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nIt 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.<\/p>\n<a name=\"2\"><\/a><\/p>\n<h2>Fundamental Units, Derived Units, and Their Physical Dimensions<\/h2>\nEach 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:<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li>The area<\/strong> is the product of two lengths and therefore its dimension is L^2<\/span>, measurable in square meters (m^2<\/span>).<\/li>\n<li>The volume<\/strong>, obtained from three lengths or an area multiplied by a length, has a dimension of L^3<\/span> and is measured in cubic meters (m^3<\/span>).<\/li>\n<li>The speed<\/strong>, defined as distance divided by time, has a dimension of LT^{-1}<\/span> and is expressed in meters per second (m\/s<\/span>).<\/li>\n<li>The acceleration<\/strong> is calculated as speed divided by time, with a dimension of LT^{-2}<\/span> and is measured in meters per second squared (m\/s^2<\/span>).<\/li>\n<li>The force<\/strong> is the result of mass times acceleration, giving rise to a dimension of MLT^{-2}<\/span>. It is commonly measured in Newtons (N<\/span>), represented by the formula:\n\\displaystyle N = \\frac{kg \\cdot m}{s^2}<\/span>\n<\/li>\n<\/ul>\nSimilarly, many other magnitudes and physical dimensions can be derived.<\/p>\n<a name=\"3\"><\/a><\/p>\n<h2>Observables, Quantities, and Physical Units<\/h2>\nWe 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<\/strong>, or simply \u00abobservable\u00bb.<\/p>\nObservables are divided into two categories: comparable and non-comparable. Comparable observables<\/strong> 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.<\/p>\n<a name=\"4\"><\/a><\/p>\n<h2>Algebra of Comparable Observables<\/h2>\nThe logic behind comparable observables is founded on principles of equality and addition:<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li>Equality Criterion:<\/strong> Two comparable observables are equal if the ratio between one and the other equals one (\\frac{A}{B} = 1<\/span>).<\/li>\n<li>Addition Criterion:<\/strong> If we have three comparable observables A<\/span>, B<\/span>, and C<\/span> in relation to a fourth O<\/span>, and the proportions \\frac{A}{O} = n_1<\/span>, \\frac{B}{O} = n_2<\/span>, and \\frac{C}{O} = n_3<\/span> hold true, then we say that A + B = C<\/span> if and only if n_1 + n_2 = n_3<\/span>.<\/li>\n<\/ul>\nWith these principles, it is demonstrated that comparable observables follow the algebraic laws of associativity, distributivity, and commutativity.<\/p>\n<h3>Units of Measurement and Physical Quantities<\/h3>\nA unit of measurement<\/strong> is a comparable observable selected to establish comparisons with other observables of the same dimension. If two observables, A<\/span> and U_A<\/span>, are comparable, there exists a real number \\alpha<\/span> such that A<\/span> equals \\alpha<\/span> times the unit of measurement U_A<\/span>.<\/p>\nA = \\alpha U_A<\/span>\nFor instance, if the length of a beam is 3 meters, we write that the length of the beam is 3 [m]<\/span>. The magnitude of a measurement varies according to the system of units used, which implies that the beam measuring 3<\/span> meters will have an approximate magnitude of 118.11<\/span> if measured in inches.<\/p>\n<h3>Conversion of Units of Measurement<\/h3>\nAs we have seen, an observable can be measured in different units as long as they share the same dimension. If A<\/span> is an observable, and U_1<\/span> and U_2<\/span> are two units of measurement of the same dimension, there will be two corresponding real numbers \\alpha_1<\/span> and \\alpha_2<\/span>.<\/p>\nA = \\alpha_1 U_1<\/span> and A = \\alpha_2 U_2<\/span>\nTherefore, the conversion factor \\gamma^2_1 = \\alpha_2 \/ \\alpha_1<\/span> allows for the transformation of unit U_2<\/span> to U_1<\/span>, and vice versa with \\gamma^1_2 = \\alpha_1 \/ \\alpha_2<\/span>. For example, a rod measuring 5<\/span> inches in length is equivalent to 0.127<\/span> meters, giving us a conversion factor of 0.0254<\/span> meters per inch.<\/p>\n<h3>Vector Quantities<\/h3>\nWe 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<\/strong> and are represented with multiple values. For example, an object positioned at 3<\/span> meters to the right, 5<\/span> meters forward, and 2<\/span> meters up is represented as (3, 5, 2)<\/span> meters.<\/p>\n{position} = (3, 5, 2)<\/span>\nThese 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.<\/p>\n<h2>Recommended Readings<\/h2>\nInternational System of Weights and Measures: <a href=\"https:\/\/www.cem.es\/sites\/default\/files\/siu8edes.pdf\" rel=\"nofollow, noreferrer, noopener\" target=\"_blank\">https:\/\/www.cem.es\/sites\/default\/files\/siu8edes.pdf<\/a><\/p>\nGuide for the Use of the International System of Units (SI): <a href=\"https:\/\/physics.nist.gov\/cuu\/pdf\/sp811.pdf\" rel=\"nofollow, noreferrer, noopener\" target=\"_blank\">https:\/\/physics.nist.gov\/cuu\/pdf\/sp811.pdf<\/a><\/p>\nEnglish System of Units: <a href=\"https:\/\/web.archive.org\/web\/20060427072134\/http:\/\/encyclopedie-es.snyke.com\/articles\/sistema_ingles.html\" rel=\"nofollow, noreferrer, noopener\" target=\"_blank\">https:\/\/web.archive.org\/web\/20060427072134\/http:\/\/encyclopedie-es.snyke.com\/articles\/sistema_ingles.html<\/a><\/p>\n<\/body> \n<\/html><\/p>\n","protected":false},"excerpt":{"rendered":"Physical Dimensions, Units, and Observable Quantities 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 […]<\/p>\n","protected":false},"author":1,"featured_media":25199,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":35,"footnotes":""},"categories":[651,635],"tags":[],"class_list":["post-25190","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-fundamentals-of-mechanics","category-physics"],"yoast_head":"\n<title>Physical Dimensions, Units, and Observable Quantities - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"With this class, you will master the concepts of dimensions and physical quantities. \ud83d\udd0dDiscover how to measure fundamental quantities.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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