{"id":64,"date":"2017-12-21T18:23:17","date_gmt":"2017-12-21T23:23:17","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/chapter\/11-2-density\/"},"modified":"2019-11-06T19:03:05","modified_gmt":"2019-11-07T00:03:05","slug":"11-2-density","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/chapter\/11-2-density\/","title":{"raw":"5.7 Density and Pressure Review","rendered":"5.7 Density and Pressure Review"},"content":{"raw":"<div>\r\n<div class=\"bcc-box bcc-highlight\">\r\n<h3>Summary<\/h3>\r\n<div>\r\n\r\nIn order to use equations for the different types of heat transfer: conduction, convection and radiation it is necessary to review some basic physic principles.\r\n<ul>\r\n \t<li>Define density.<\/li>\r\n \t<li>Calculate the mass of a reservoir from its density.<\/li>\r\n \t<li>Compare and contrast the densities of various substances.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"import-auto-id1596511\">Which weighs more, a ton of feathers or a ton of bricks? This old riddle plays with the distinction between mass and density. A ton is a ton, of course; but bricks have much greater density than feathers, and so we are tempted to think of them as heavier. (See <a href=\"#import-auto-id950378\" class=\"autogenerated-content\">Figure 1<\/a>.)<\/p>\r\n<p id=\"import-auto-id3217118\"><strong><span id=\"import-auto-id2660162\">Density<\/span><\/strong>, as you will see, is an important characteristic of substances. It is crucial, for example, in determining whether an object sinks or floats in a fluid. Density is the mass per unit volume of a substance or object. In equation form, density is defined as<\/p>\r\n\r\n<div style=\"text-align: center\" class=\"equation\" id=\"eip-385\">[latex] \\boldsymbol{ \\rho\\:=\u00a0 \\frac{m}{V} },[\/latex]<\/div>\r\n<p id=\"import-auto-id2677741\">where the Greek letter <strong>\u03c1<\/strong> (rho) is the symbol for density, <strong>m\u00a0<\/strong>is the mass, and <strong>V\u00a0<\/strong>is the volume occupied by the substance.<\/p>\r\n\r\n<div class=\"note\" id=\"fs-id3045369\">\r\n<div class=\"textbox shaded\">\r\n<div class=\"note\">\r\n<h3 class=\"title\">DENSITY<\/h3>\r\nDensity is mass per unit volume.\r\n<div style=\"text-align: center\" class=\"equation\">[latex]\\boldsymbol{\\rho\\:= \\frac{m}{V} },[\/latex]<\/div>\r\n<p id=\"import-auto-id2621355\">where\u00a0<strong>\u03c1\u00a0<\/strong>is the symbol for density, <strong>m\u00a0<\/strong>is the mass, and <strong>V\u00a0<\/strong>is the volume occupied by the substance.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"import-auto-id1011172\">In the riddle regarding the feathers and bricks, the masses are the same, but the volume occupied by the feathers is much greater, since their density is much lower. The SI unit of density is[latex]\\boldsymbol{\\textbf{kg\/m}^3},[\/latex] representative values are given in <a href=\"#fs-id1769034\" class=\"autogenerated-content\">Table 1<\/a>. The metric system was originally devised so that water would have a density of <strong>1 g\/cm<sup>3<\/sup> <\/strong>equivalent to 1000\u00a0 kg\/m<sup>3<\/sup>. Thus the basic mass unit, the kilogram, was first devised to be the mass of 1000 mL of water, which has a volume of 1000 cm<sup>3<\/sup>.<\/p>\r\n\r\n<table id=\"fs-id1769034\" summary=\"The table shows the value of density in units of kilogram per meter cubed for certain solids, liquids, and gases.\">\r\n<thead>\r\n<tr>\r\n<th>Substance<\/th>\r\n<th>\u03c1(10<sup>3 <\/sup>kg\/m<sup>3 <\/sup>or g\/mL)<\/th>\r\n<th>Substance<\/th>\r\n<th>\u03c1(10<sup>3 <\/sup>kg\/m<sup>3 <\/sup>or g\/mL)<\/th>\r\n<th>Substance<\/th>\r\n<th>\u03c1(10<sup>3 <\/sup>kg\/m<sup>3 <\/sup>or g\/mL)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td><strong>Solids<\/strong><\/td>\r\n<td><\/td>\r\n<td><strong>Liquids<\/strong><\/td>\r\n<td><\/td>\r\n<td><strong>Gases<\/strong><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Aluminum<\/td>\r\n<td>2.7<\/td>\r\n<td>Water (4\u00baC)<\/td>\r\n<td>1.000<\/td>\r\n<td>Air<\/td>\r\n<td>[latex]\\boldsymbol{1.29\\times10^{-3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Brass<\/td>\r\n<td>8.44<\/td>\r\n<td>Blood<\/td>\r\n<td>1.05<\/td>\r\n<td>Carbon dioxide<\/td>\r\n<td>[latex]\\boldsymbol{1.98\\times10^{-3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Copper (average)<\/td>\r\n<td>8.8<\/td>\r\n<td>Sea water<\/td>\r\n<td>1.025<\/td>\r\n<td>Carbon monoxide<\/td>\r\n<td>[latex]\\boldsymbol{1.25\\times10^{-3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Gold<\/td>\r\n<td>19.32<\/td>\r\n<td>Mercury<\/td>\r\n<td>13.6<\/td>\r\n<td>Hydrogen<\/td>\r\n<td>[latex]\\boldsymbol{0.090\\times10^{-3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Iron or steel<\/td>\r\n<td>7.8<\/td>\r\n<td>Ethyl alcohol<\/td>\r\n<td>0.79<\/td>\r\n<td>Helium<\/td>\r\n<td>[latex]\\boldsymbol{0.18\\times10^{-3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Lead<\/td>\r\n<td>11.3<\/td>\r\n<td>Petrol<\/td>\r\n<td>0.68<\/td>\r\n<td>Methane<\/td>\r\n<td>[latex]\\boldsymbol{0.72\\times10^{-3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Polystyrene<\/td>\r\n<td>0.10<\/td>\r\n<td>Glycerin<\/td>\r\n<td>1.26<\/td>\r\n<td>Nitrogen<\/td>\r\n<td>[latex]\\boldsymbol{1.25\\times10^{-3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Tungsten<\/td>\r\n<td>19.30<\/td>\r\n<td>Olive oil<\/td>\r\n<td>0.92<\/td>\r\n<td>Nitrous oxide<\/td>\r\n<td>[latex]\\boldsymbol{1.98\\times10^{-3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Uranium<\/td>\r\n<td>18.70<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>Oxygen<\/td>\r\n<td>[latex]\\boldsymbol{1.43\\times10^{-3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Concrete<\/td>\r\n<td>2.30\u20133.0<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>Steam 100\u00ba C<\/td>\r\n<td>[latex]\\boldsymbol{0.60\\times10^{-3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Cork<\/td>\r\n<td>0.24<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Glass, common (average)<\/td>\r\n<td>2.6<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Granite<\/td>\r\n<td>2.7<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Earth\u2019s crust<\/td>\r\n<td>3.3<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Wood<\/td>\r\n<td>0.3\u20130.9<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Ice (0\u00b0C)<\/td>\r\n<td>0.917<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Bone<\/td>\r\n<td>1.7\u20132.0<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<tbody>\r\n<tr>\r\n<td colspan=\"6\"><strong>Table 1.<\/strong> Densities of Various Substances<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<figure id=\"import-auto-id950378\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"375\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/uploads\/sites\/307\/2017\/12\/Figure_12_02_01a.jpg\" alt=\"A pile of feathers measuring a ton and a ton of bricks are placed on either side of a plank that is balanced on a small support.\" height=\"320\" width=\"375\" \/> <strong>Figure 1.<\/strong> A ton of feathers and a ton of bricks have the same mass, but the feathers make a much bigger pile because they have a much lower density.[\/caption]<\/figure>\r\n<p id=\"import-auto-id1418776\">As you can see by examining <a href=\"#fs-id1769034\" class=\"autogenerated-content\">Table 1<\/a>, the density of an object may help identify its composition. The density of gold, for example, is about 2.5 times the density of iron, which is about 2.5 times the density of aluminum. Density also reveals something about the phase of the matter and its substructure. Notice that the densities of liquids and solids are roughly comparable, consistent with the fact that their atoms are in close contact. The densities of gases are much less than those of liquids and solids, because the atoms in gases are separated by large amounts of empty space.<\/p>\r\n\r\n<div class=\"note\" id=\"fs-id2971617\">\r\n<div class=\"textbox shaded\">\r\n<div class=\"note\">\r\n<h3 class=\"title\">TAKE-HOME EXPERIMENT: SUGAR AND SALT<\/h3>\r\n<p id=\"import-auto-id1381758\">A pile of sugar and a pile of salt look pretty similar, but which weighs more? If the volumes of both piles are the same, any difference in mass is due to their different densities (including the air space between crystals). Which do you think has the greater density? What values did you find? What method did you use to determine these values?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<div class=\"example\" id=\"fs-id2449267\">\r\n<h3 id=\"import-auto-id2688762\">Example 1: Calculating the Mass of a Reservoir From Its Volume<\/h3>\r\nA reservoir has a surface area of\u00a0 <strong>50.0 km<sup>2<\/sup>\u00a0<\/strong>and an average depth of <strong>40.0 m<\/strong>. What mass of water is held behind the dam? (See <a href=\"#import-auto-id1371721\" class=\"autogenerated-content\">Figure 2<\/a> for a view of a large reservoir\u2014the Three Gorges Dam site on the Yangtze River in central China.)\r\n<p id=\"import-auto-id1861411\"><strong>\u00a0Strategy <\/strong><\/p>\r\n<p id=\"import-auto-id2392409\">We can calculate the volume <strong>V<\/strong> of the reservoir from its dimensions, and the density of water which is 1000 kg\/m<sup>3<\/sup>.\u00a0 Then the mass <strong>m<\/strong>\u00a0 can be found from the definition of density<\/p>\r\n\r\n<div class=\"equation\" id=\"eip-48\" style=\"text-align: center\">[latex]\\boldsymbol{\\rho\\:= \\frac{m}{V} }.[\/latex]<\/div>\r\n<p id=\"import-auto-id1933185\"><strong>Solution<\/strong><\/p>\r\nSolving equation for mass = (density) (volume)\r\n<p id=\"import-auto-id1425212\">The volume\u00a0<strong>V\u00a0<\/strong> of the reservoir is its surface area\u00a0<strong>A<\/strong> times its average depth\u00a0<strong>h<\/strong>.<\/p>\r\n\r\n<div class=\"equation\" style=\"text-align: center\">[latex]\\begin{array}{lcl} \\boldsymbol{V} &amp; \\boldsymbol{=} &amp; \\boldsymbol{Ah=(50.0\\textbf{ km}^2)(40.0\\textbf{ m})} \\\\ {} &amp; \\boldsymbol{=} &amp; \\boldsymbol{[(50.0\\textbf{ km}^2)(\\frac{10^3\\textbf{ m}}{1\\textbf{ km}})^2](40.0\\textbf{ m})=2.00\\times10^9\\textbf{ m}^3} \\end{array}[\/latex]<\/div>\r\n<p id=\"import-auto-id2671000\">The density of water\u00a0<strong>\u03c1\u00a0<\/strong>from <a href=\"#fs-id1769034\" class=\"autogenerated-content\">Table 1<\/a> is 1.000 x 10<sup>3<\/sup> kg\/m<sup>3<\/sup>.\u00a0 Substituting <strong>V\u00a0<\/strong>and\u00a0<strong>\u03c1\u00a0<\/strong>\u00a0into the expression for mass gives<\/p>\r\n\r\n<div class=\"equation\" style=\"text-align: center\">[latex]\\begin{array}{lcl} \\boldsymbol{m} &amp; \\boldsymbol{=} &amp; \\boldsymbol{(1.00\\times10^3\\textbf{ kg\/m}^3)(2.00\\times10^9\\textbf{ m}^3)} \\\\ {} &amp; \\boldsymbol{=} &amp; \\boldsymbol{2.00\\times10^{12}\\textbf{ kg.}} \\end{array}[\/latex]<\/div>\r\n<p id=\"import-auto-id2423221\"><strong>Discussion<\/strong><\/p>\r\n<p id=\"import-auto-id2009193\">A large reservoir contains a very large mass of water. In this example, the weight of the water in the reservoir is\u00a0 <strong>mg = 1.96 x 10<sup>13<\/sup> N\u00a0<\/strong> where <strong>g<\/strong> is the acceleration due to the Earth\u2019s gravity, about <strong>9.80 m\/s<sup>2<\/sup>\u00a0<\/strong>. It is reasonable to ask whether the dam must supply a force equal to this tremendous weight. The answer is no. As we shall see in the following sections, the force the dam must supply can be much smaller than the weight of the water it holds back.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<figure id=\"import-auto-id1371721\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"350\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/uploads\/sites\/307\/2017\/12\/Figure_12_02_02a.jpg\" alt=\"Photograph of the Three Gorges Dam in central China.\" height=\"593\" width=\"350\" \/> <strong>Figure 2.<\/strong> Three Gorges Dam in central China. When completed in 2008, this became the world\u2019s largest hydroelectric plant, generating power equivalent to that generated by 22 average-sized nuclear power plants. The concrete dam is 181 m high and 2.3 km across. The reservoir made by this dam is 660 km long. Over 1 million people were displaced by the creation of the reservoir. (credit: Le Grand Portage)[\/caption]<\/figure>\r\n<section id=\"fs-id1198403\" class=\"section-summary\">\r\n<h1>Section Summary<\/h1>\r\n<ul id=\"fs-id3080578\">\r\n \t<li id=\"import-auto-id2438220\">Density is the mass per unit volume of a substance or object. In equation form, density is defined as\r\n<div style=\"text-align: center\" class=\"equation\">[latex]\\boldsymbol{ \\rho\\:= \\frac{m}{V} }.[\/latex]<\/div><\/li>\r\n \t<li id=\"import-auto-id1588322\">The SI unit of density is\u00a0 <strong>kg\/m<sup>3<\/sup>.\u00a0<\/strong><\/li>\r\n<\/ul>\r\n<\/section><section id=\"fs-id2677348\" class=\"conceptual-questions\">\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Conceptual Questions<\/h3>\r\n<div class=\"exercise\" id=\"fs-id1417224\">\r\n<div class=\"problem\" id=\"fs-id2435831\">\r\n<p id=\"import-auto-id2929368\"><strong>1: <\/strong>Approximately how does the density of air vary with altitude?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\" id=\"fs-id1613737\">\r\n<div class=\"problem\" id=\"fs-id1859324\">\r\n<p id=\"import-auto-id1474822\"><strong>2: <\/strong>Give an example in which density is used to identify the substance composing an object. Would information in addition to average density be needed to identify the substances in an object composed of more than one material?<\/p>\r\n<strong>3: <\/strong><a href=\"#import-auto-id1988132\" class=\"autogenerated-content\">Figure 3<\/a> shows a glass of ice water filled to the brim. Will the water overflow when the ice melts? Explain your answer.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\" id=\"fs-id1397150\">\r\n<div class=\"problem\" id=\"fs-id2397130\">\r\n<figure id=\"import-auto-id1988132\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"150\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/uploads\/sites\/307\/2017\/12\/Figure_12_02_03a.jpg\" alt=\"A glass filled to the brim with water and ice cubes.\" height=\"759\" width=\"150\" \/> <strong>Figure 3.<\/strong>[\/caption]<\/figure>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id2681028\" class=\"problems-exercises\">\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Problems &amp; Exercises<\/h3>\r\n<div class=\"exercise\" id=\"fs-id1081259\">\r\n<div class=\"problem\" id=\"fs-id2622699\">\r\n\r\n<strong>You will have to look up the values for densities using the table earlier in this chapter or some other reference.\u00a0 <\/strong>\r\n<p style=\"text-align: left\"><strong>1:\u00a0<\/strong> Gold is sold by the troy ounce (31.103 g). What is the volume of 1 troy ounce of pure gold?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\" id=\"fs-id2056422\">\r\n<div class=\"problem\" id=\"fs-id1216339\">\r\n<p id=\"import-auto-id3175349\"><strong>2: <\/strong>Mercury is commonly supplied in flasks containing 34.5 kg (about 76 lb). What is the volume in litres of this much mercury?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\" id=\"fs-id2639547\">\r\n<div class=\"problem\" id=\"fs-id1901135\">\r\n<p id=\"import-auto-id3115592\"><strong>3: <\/strong>(a) What is the mass of a deep breath of air having a volume of 2.00 L? (b) Discuss the effect taking such a breath has on your body\u2019s volume and density.\u00a0 \u00a0Remember that 1 litre = a cube that is 10 cm x 10 cm x 10 cm.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\" id=\"fs-id1561911\">\r\n<div class=\"problem\" id=\"fs-id3176934\">\r\n<p id=\"import-auto-id1282324\"><strong>4: <\/strong>A straightforward method of finding the density of an object is to measure its mass and then measure its volume by submerging it in a graduated cylinder. What is the density of a 240-g rock that displaces\u00a0 89.0 cm<sup>3<\/sup> or millilitres of water? (Note that the accuracy and practical applications of this technique are more limited than a variety of others that are based on Archimedes\u2019 principle.)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\" id=\"fs-id1381125\">\r\n<div class=\"problem\" id=\"fs-id2066861\">\r\n<p id=\"import-auto-id1245988\"><strong>5: <\/strong>Suppose you have a coffee mug with a circular cross section and vertical sides (uniform radius). What is its inside radius if it holds 375 g of coffee when filled to a depth of 7.50 cm? Assume coffee has the same density as water.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\" id=\"fs-id1157183\">\r\n<div class=\"problem\" id=\"fs-id3135333\">\r\n<p id=\"import-auto-id3418304\"><strong>6: <\/strong>(a) A rectangular gasoline tank can hold 50.0 kg of gasoline when full. What is the depth of the tank if it is 0.500-m wide by 0.900-m long? Hint:\u00a0 find the volume of the tank first from the density and remember that volume = width x length x depth.\u00a0 (b) Discuss whether this gas tank has a reasonable volume for a passenger car.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\" id=\"fs-id1602523\">\r\n<div class=\"problem\" id=\"fs-id2588384\">\r\n<p id=\"import-auto-id2444694\"><strong>7: <\/strong>A trash compactor can reduce the volume of its contents to 0.350 their original value. Neglecting the mass of air expelled, by what factor is the density of the rubbish increased?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\" id=\"fs-id1599362\">\r\n<div class=\"problem\" id=\"fs-id1824721\">\r\n<p id=\"import-auto-id1614056\"><strong>8: <\/strong>A 2.50-kg steel gasoline can holds 20.0 L of gasoline when full. What is the average density of the full gas can, taking into account the volume occupied by steel as well as by gasoline?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\" id=\"fs-id1848674\">\r\n<div class=\"problem\" id=\"fs-id2591615\">\r\n<p id=\"import-auto-id3055502\"><strong>9: <\/strong>What is the density of 18.0-karat gold that is a mixture of 18 parts gold, 5 parts silver, and 1 part copper? (These values are parts by mass, not volume.) Assume that this is a simple mixture having an average density equal to the weighted densities of its constituents.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\" id=\"fs-id3101625\">\r\n<div class=\"problem\" id=\"fs-id1849112\">\r\n<p id=\"import-auto-id2616385\"><strong>10: <\/strong>There is relatively little empty space between atoms in solids and liquids, so that the average density of an atom is about the same as matter on a macroscopic scale\u2014approximately 1000 kg\/m3. The nucleus of an atom has a radius about\u00a0 10<sup>-5<\/sup> that of the atom and contains nearly all the mass of the entire atom. (a) What is the approximate density of a nucleus? (b) One remnant of a supernova, called a neutron star, can have the density of a nucleus. What would be the radius of a neutron star with a mass 10 times that of our Sun (the radius of the Sun is 7 x10<sup>8<\/sup> m?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"import-auto-id3125373\" class=\"definition\">\r\n \t<dt>density<\/dt>\r\n \t<dd id=\"fs-id957039\">the mass per unit volume of a substance or object<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Solutions<\/h3>\r\n<strong>Problems &amp; Exercises<\/strong>\r\n\r\n<strong>1:\u00a0\u00a0<\/strong>1.610 cm<sup>3<\/sup>\r\n\r\n<strong>3:\u00a0<\/strong><span style=\"font-size: 1rem;text-indent: 0px\">(a) 2.58 g\u00a0<\/span><span style=\"font-size: 1rem;text-indent: 0px\">(b) The volume of your body increases by the volume of air you inhale. The average density of your body decreases when you take a deep breath, because the density of air is substantially smaller than the average density of the body before you took the deep breath.<\/span>\r\n\r\n<strong>4:\u00a0\u00a0<\/strong>2.70 g\/cm<sup>3\u00a0\u00a0<\/sup>or 2700 kg\/m<sup>3\u00a0<\/sup>\r\n\r\n<strong>6:\u00a0<\/strong><span style=\"font-size: 1rem;text-indent: 0px\">(a) 0.163 m\u00a0<\/span><span style=\"font-size: 1rem;text-indent: 0px\">(b) Equivalent to 19.4 gallons, which is reasonable<\/span>\r\n\r\n<strong>8:\u00a0\u00a0<\/strong>7.9 x10<sup>2\u00a0<\/sup> kg\/m<sup>3<\/sup>\r\n\r\n<strong>9:\u00a0<\/strong>15.6 g\/cm<sup>3<\/sup>\r\n\r\n<strong>10:\u00a0<\/strong><span style=\"font-size: 1rem;text-indent: 0px\">(a)\u00a0 10<sup>18\u00a0<\/sup>\u00a0kg\/m<sup>3\u00a0<\/sup><\/span><span style=\"font-size: 1rem;text-indent: 0px\">(b)\u00a0 2 x 10<sup>4<\/sup> m<\/span>\r\n\r\n<\/div>","rendered":"<div>\n<div class=\"bcc-box bcc-highlight\">\n<h3>Summary<\/h3>\n<div>\n<p>In order to use equations for the different types of heat transfer: conduction, convection and radiation it is necessary to review some basic physic principles.<\/p>\n<ul>\n<li>Define density.<\/li>\n<li>Calculate the mass of a reservoir from its density.<\/li>\n<li>Compare and contrast the densities of various substances.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"import-auto-id1596511\">Which weighs more, a ton of feathers or a ton of bricks? This old riddle plays with the distinction between mass and density. A ton is a ton, of course; but bricks have much greater density than feathers, and so we are tempted to think of them as heavier. (See <a href=\"#import-auto-id950378\" class=\"autogenerated-content\">Figure 1<\/a>.)<\/p>\n<p id=\"import-auto-id3217118\"><strong><span id=\"import-auto-id2660162\">Density<\/span><\/strong>, as you will see, is an important characteristic of substances. It is crucial, for example, in determining whether an object sinks or floats in a fluid. Density is the mass per unit volume of a substance or object. In equation form, density is defined as<\/p>\n<div style=\"text-align: center\" class=\"equation\" id=\"eip-385\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/ql-cache\/quicklatex.com-e2694ebba215f60c1dea10f79d23344f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#32;&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#32;&#92;&#114;&#104;&#111;&#92;&#58;&#61;&#32;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#125;&#123;&#86;&#125;&#32;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"65\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id2677741\">where the Greek letter <strong>\u03c1<\/strong> (rho) is the symbol for density, <strong>m\u00a0<\/strong>is the mass, and <strong>V\u00a0<\/strong>is the volume occupied by the substance.<\/p>\n<div class=\"note\" id=\"fs-id3045369\">\n<div class=\"textbox shaded\">\n<div class=\"note\">\n<h3 class=\"title\">DENSITY<\/h3>\n<p>Density is mass per unit volume.<\/p>\n<div style=\"text-align: center\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/ql-cache\/quicklatex.com-e40feec7552694551dd0f3a5be149d0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#92;&#114;&#104;&#111;&#92;&#58;&#61;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#125;&#123;&#86;&#125;&#32;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"65\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id2621355\">where\u00a0<strong>\u03c1\u00a0<\/strong>is the symbol for density, <strong>m\u00a0<\/strong>is the mass, and <strong>V\u00a0<\/strong>is the volume occupied by the substance.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"import-auto-id1011172\">In the riddle regarding the feathers and bricks, the masses are the same, but the volume occupied by the feathers is much greater, since their density is much lower. The SI unit of density is<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/ql-cache\/quicklatex.com-32ff77aa45dad69e76ed7fdbeb893610_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#92;&#116;&#101;&#120;&#116;&#98;&#102;&#123;&#107;&#103;&#47;&#109;&#125;&#94;&#51;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"61\" style=\"vertical-align: -4px;\" \/> representative values are given in <a href=\"#fs-id1769034\" class=\"autogenerated-content\">Table 1<\/a>. The metric system was originally devised so that water would have a density of <strong>1 g\/cm<sup>3<\/sup> <\/strong>equivalent to 1000\u00a0 kg\/m<sup>3<\/sup>. Thus the basic mass unit, the kilogram, was first devised to be the mass of 1000 mL of water, which has a volume of 1000 cm<sup>3<\/sup>.<\/p>\n<table id=\"fs-id1769034\" summary=\"The table shows the value of density in units of kilogram per meter cubed for certain solids, liquids, and gases.\">\n<thead>\n<tr>\n<th>Substance<\/th>\n<th>\u03c1(10<sup>3 <\/sup>kg\/m<sup>3 <\/sup>or g\/mL)<\/th>\n<th>Substance<\/th>\n<th>\u03c1(10<sup>3 <\/sup>kg\/m<sup>3 <\/sup>or g\/mL)<\/th>\n<th>Substance<\/th>\n<th>\u03c1(10<sup>3 <\/sup>kg\/m<sup>3 <\/sup>or g\/mL)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><strong>Solids<\/strong><\/td>\n<td><\/td>\n<td><strong>Liquids<\/strong><\/td>\n<td><\/td>\n<td><strong>Gases<\/strong><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Aluminum<\/td>\n<td>2.7<\/td>\n<td>Water (4\u00baC)<\/td>\n<td>1.000<\/td>\n<td>Air<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/ql-cache\/quicklatex.com-f3d07bbde47a616a9672d5758db56233_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#49;&#46;&#50;&#57;&#92;&#116;&#105;&#109;&#101;&#115;&#49;&#48;&#94;&#123;&#45;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"100\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Brass<\/td>\n<td>8.44<\/td>\n<td>Blood<\/td>\n<td>1.05<\/td>\n<td>Carbon dioxide<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/ql-cache\/quicklatex.com-bbf2264679f4cef10c1825c7a5f90ab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#49;&#46;&#57;&#56;&#92;&#116;&#105;&#109;&#101;&#115;&#49;&#48;&#94;&#123;&#45;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"100\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Copper (average)<\/td>\n<td>8.8<\/td>\n<td>Sea water<\/td>\n<td>1.025<\/td>\n<td>Carbon monoxide<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/ql-cache\/quicklatex.com-9b8187bf7b1d7b07919b4b01a9e142c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#49;&#46;&#50;&#53;&#92;&#116;&#105;&#109;&#101;&#115;&#49;&#48;&#94;&#123;&#45;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"100\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Gold<\/td>\n<td>19.32<\/td>\n<td>Mercury<\/td>\n<td>13.6<\/td>\n<td>Hydrogen<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/ql-cache\/quicklatex.com-c26fdb46f3f30d5a38b813921433bc5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#48;&#46;&#48;&#57;&#48;&#92;&#116;&#105;&#109;&#101;&#115;&#49;&#48;&#94;&#123;&#45;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"111\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Iron or steel<\/td>\n<td>7.8<\/td>\n<td>Ethyl alcohol<\/td>\n<td>0.79<\/td>\n<td>Helium<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/ql-cache\/quicklatex.com-736875c0e8d939f4a1c6f12ee04e3df1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#48;&#46;&#49;&#56;&#92;&#116;&#105;&#109;&#101;&#115;&#49;&#48;&#94;&#123;&#45;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"101\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Lead<\/td>\n<td>11.3<\/td>\n<td>Petrol<\/td>\n<td>0.68<\/td>\n<td>Methane<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/ql-cache\/quicklatex.com-92d3cecd14ce8aa6cc99dbe4737a7ef5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#48;&#46;&#55;&#50;&#92;&#116;&#105;&#109;&#101;&#115;&#49;&#48;&#94;&#123;&#45;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"101\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Polystyrene<\/td>\n<td>0.10<\/td>\n<td>Glycerin<\/td>\n<td>1.26<\/td>\n<td>Nitrogen<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/ql-cache\/quicklatex.com-9b8187bf7b1d7b07919b4b01a9e142c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#49;&#46;&#50;&#53;&#92;&#116;&#105;&#109;&#101;&#115;&#49;&#48;&#94;&#123;&#45;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"100\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Tungsten<\/td>\n<td>19.30<\/td>\n<td>Olive oil<\/td>\n<td>0.92<\/td>\n<td>Nitrous oxide<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/ql-cache\/quicklatex.com-bbf2264679f4cef10c1825c7a5f90ab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#49;&#46;&#57;&#56;&#92;&#116;&#105;&#109;&#101;&#115;&#49;&#48;&#94;&#123;&#45;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"100\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Uranium<\/td>\n<td>18.70<\/td>\n<td><\/td>\n<td><\/td>\n<td>Oxygen<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/ql-cache\/quicklatex.com-e449ca5f94134577f6b2d56df006833e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#49;&#46;&#52;&#51;&#92;&#116;&#105;&#109;&#101;&#115;&#49;&#48;&#94;&#123;&#45;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"100\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Concrete<\/td>\n<td>2.30\u20133.0<\/td>\n<td><\/td>\n<td><\/td>\n<td>Steam 100\u00ba C<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/ql-cache\/quicklatex.com-04f37b4aa83d5ae9619b9ba386ed219f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#48;&#46;&#54;&#48;&#92;&#116;&#105;&#109;&#101;&#115;&#49;&#48;&#94;&#123;&#45;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"101\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Cork<\/td>\n<td>0.24<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Glass, common (average)<\/td>\n<td>2.6<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Granite<\/td>\n<td>2.7<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Earth\u2019s crust<\/td>\n<td>3.3<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Wood<\/td>\n<td>0.3\u20130.9<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Ice (0\u00b0C)<\/td>\n<td>0.917<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Bone<\/td>\n<td>1.7\u20132.0<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<tbody>\n<tr>\n<td colspan=\"6\"><strong>Table 1.<\/strong> Densities of Various Substances<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<figure id=\"import-auto-id950378\">\n<figure style=\"width: 375px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/uploads\/sites\/307\/2017\/12\/Figure_12_02_01a.jpg\" alt=\"A pile of feathers measuring a ton and a ton of bricks are placed on either side of a plank that is balanced on a small support.\" height=\"320\" width=\"375\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong> A ton of feathers and a ton of bricks have the same mass, but the feathers make a much bigger pile because they have a much lower density.<\/figcaption><\/figure>\n<\/figure>\n<p id=\"import-auto-id1418776\">As you can see by examining <a href=\"#fs-id1769034\" class=\"autogenerated-content\">Table 1<\/a>, the density of an object may help identify its composition. The density of gold, for example, is about 2.5 times the density of iron, which is about 2.5 times the density of aluminum. Density also reveals something about the phase of the matter and its substructure. Notice that the densities of liquids and solids are roughly comparable, consistent with the fact that their atoms are in close contact. The densities of gases are much less than those of liquids and solids, because the atoms in gases are separated by large amounts of empty space.<\/p>\n<div class=\"note\" id=\"fs-id2971617\">\n<div class=\"textbox shaded\">\n<div class=\"note\">\n<h3 class=\"title\">TAKE-HOME EXPERIMENT: SUGAR AND SALT<\/h3>\n<p id=\"import-auto-id1381758\">A pile of sugar and a pile of salt look pretty similar, but which weighs more? If the volumes of both piles are the same, any difference in mass is due to their different densities (including the air space between crystals). Which do you think has the greater density? What values did you find? What method did you use to determine these values?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"example\" id=\"fs-id2449267\">\n<h3 id=\"import-auto-id2688762\">Example 1: Calculating the Mass of a Reservoir From Its Volume<\/h3>\n<p>A reservoir has a surface area of\u00a0 <strong>50.0 km<sup>2<\/sup>\u00a0<\/strong>and an average depth of <strong>40.0 m<\/strong>. What mass of water is held behind the dam? (See <a href=\"#import-auto-id1371721\" class=\"autogenerated-content\">Figure 2<\/a> for a view of a large reservoir\u2014the Three Gorges Dam site on the Yangtze River in central China.)<\/p>\n<p id=\"import-auto-id1861411\"><strong>\u00a0Strategy <\/strong><\/p>\n<p id=\"import-auto-id2392409\">We can calculate the volume <strong>V<\/strong> of the reservoir from its dimensions, and the density of water which is 1000 kg\/m<sup>3<\/sup>.\u00a0 Then the mass <strong>m<\/strong>\u00a0 can be found from the definition of density<\/p>\n<div class=\"equation\" id=\"eip-48\" style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/ql-cache\/quicklatex.com-7c7bb19088fbf8d54e3778f2fb12fd5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#92;&#114;&#104;&#111;&#92;&#58;&#61;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#125;&#123;&#86;&#125;&#32;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"65\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id1933185\"><strong>Solution<\/strong><\/p>\n<p>Solving equation for mass = (density) (volume)<\/p>\n<p id=\"import-auto-id1425212\">The volume\u00a0<strong>V\u00a0<\/strong> of the reservoir is its surface area\u00a0<strong>A<\/strong> times its average depth\u00a0<strong>h<\/strong>.<\/p>\n<div class=\"equation\" style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/ql-cache\/quicklatex.com-dff38146f613c859622138641c42ec9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#99;&#108;&#125;&#32;&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#86;&#125;&#32;&#38;&#32;&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#61;&#125;&#32;&#38;&#32;&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#65;&#104;&#61;&#40;&#53;&#48;&#46;&#48;&#92;&#116;&#101;&#120;&#116;&#98;&#102;&#123;&#32;&#107;&#109;&#125;&#94;&#50;&#41;&#40;&#52;&#48;&#46;&#48;&#92;&#116;&#101;&#120;&#116;&#98;&#102;&#123;&#32;&#109;&#125;&#41;&#125;&#32;&#92;&#92;&#32;&#123;&#125;&#32;&#38;&#32;&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#61;&#125;&#32;&#38;&#32;&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#091;&#40;&#53;&#48;&#46;&#48;&#92;&#116;&#101;&#120;&#116;&#98;&#102;&#123;&#32;&#107;&#109;&#125;&#94;&#50;&#41;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#94;&#51;&#92;&#116;&#101;&#120;&#116;&#98;&#102;&#123;&#32;&#109;&#125;&#125;&#123;&#49;&#92;&#116;&#101;&#120;&#116;&#98;&#102;&#123;&#32;&#107;&#109;&#125;&#125;&#41;&#94;&#50;&#093;&#40;&#52;&#48;&#46;&#48;&#92;&#116;&#101;&#120;&#116;&#98;&#102;&#123;&#32;&#109;&#125;&#41;&#61;&#50;&#46;&#48;&#48;&#92;&#116;&#105;&#109;&#101;&#115;&#49;&#48;&#94;&#57;&#92;&#116;&#101;&#120;&#116;&#98;&#102;&#123;&#32;&#109;&#125;&#94;&#51;&#125;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"466\" style=\"vertical-align: -20px;\" \/><\/div>\n<p id=\"import-auto-id2671000\">The density of water\u00a0<strong>\u03c1\u00a0<\/strong>from <a href=\"#fs-id1769034\" class=\"autogenerated-content\">Table 1<\/a> is 1.000 x 10<sup>3<\/sup> kg\/m<sup>3<\/sup>.\u00a0 Substituting <strong>V\u00a0<\/strong>and\u00a0<strong>\u03c1\u00a0<\/strong>\u00a0into the expression for mass gives<\/p>\n<div class=\"equation\" style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/ql-cache\/quicklatex.com-734cd4e3a4e8162eead63b3e3afac95b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#99;&#108;&#125;&#32;&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#109;&#125;&#32;&#38;&#32;&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#61;&#125;&#32;&#38;&#32;&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#40;&#49;&#46;&#48;&#48;&#92;&#116;&#105;&#109;&#101;&#115;&#49;&#48;&#94;&#51;&#92;&#116;&#101;&#120;&#116;&#98;&#102;&#123;&#32;&#107;&#103;&#47;&#109;&#125;&#94;&#51;&#41;&#40;&#50;&#46;&#48;&#48;&#92;&#116;&#105;&#109;&#101;&#115;&#49;&#48;&#94;&#57;&#92;&#116;&#101;&#120;&#116;&#98;&#102;&#123;&#32;&#109;&#125;&#94;&#51;&#41;&#125;&#32;&#92;&#92;&#32;&#123;&#125;&#32;&#38;&#32;&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#61;&#125;&#32;&#38;&#32;&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#50;&#46;&#48;&#48;&#92;&#116;&#105;&#109;&#101;&#115;&#49;&#48;&#94;&#123;&#49;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#98;&#102;&#123;&#32;&#107;&#103;&#46;&#125;&#125;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"373\" style=\"vertical-align: -15px;\" \/><\/div>\n<p id=\"import-auto-id2423221\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2009193\">A large reservoir contains a very large mass of water. In this example, the weight of the water in the reservoir is\u00a0 <strong>mg = 1.96 x 10<sup>13<\/sup> N\u00a0<\/strong> where <strong>g<\/strong> is the acceleration due to the Earth\u2019s gravity, about <strong>9.80 m\/s<sup>2<\/sup>\u00a0<\/strong>. It is reasonable to ask whether the dam must supply a force equal to this tremendous weight. The answer is no. As we shall see in the following sections, the force the dam must supply can be much smaller than the weight of the water it holds back.<\/p>\n<\/div>\n<\/div>\n<figure id=\"import-auto-id1371721\">\n<figure style=\"width: 350px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/uploads\/sites\/307\/2017\/12\/Figure_12_02_02a.jpg\" alt=\"Photograph of the Three Gorges Dam in central China.\" height=\"593\" width=\"350\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong> Three Gorges Dam in central China. When completed in 2008, this became the world\u2019s largest hydroelectric plant, generating power equivalent to that generated by 22 average-sized nuclear power plants. The concrete dam is 181 m high and 2.3 km across. The reservoir made by this dam is 660 km long. Over 1 million people were displaced by the creation of the reservoir. (credit: Le Grand Portage)<\/figcaption><\/figure>\n<\/figure>\n<section id=\"fs-id1198403\" class=\"section-summary\">\n<h1>Section Summary<\/h1>\n<ul id=\"fs-id3080578\">\n<li id=\"import-auto-id2438220\">Density is the mass per unit volume of a substance or object. In equation form, density is defined as\n<div style=\"text-align: center\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/ql-cache\/quicklatex.com-02cadc28c435ee4784122d262b24d218_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#111;&#108;&#100;&#115;&#121;&#109;&#98;&#111;&#108;&#123;&#32;&#92;&#114;&#104;&#111;&#92;&#58;&#61;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#125;&#123;&#86;&#125;&#32;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"65\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id1588322\">The SI unit of density is\u00a0 <strong>kg\/m<sup>3<\/sup>.\u00a0<\/strong><\/li>\n<\/ul>\n<\/section>\n<section id=\"fs-id2677348\" class=\"conceptual-questions\">\n<div class=\"bcc-box bcc-info\">\n<h3>Conceptual Questions<\/h3>\n<div class=\"exercise\" id=\"fs-id1417224\">\n<div class=\"problem\" id=\"fs-id2435831\">\n<p id=\"import-auto-id2929368\"><strong>1: <\/strong>Approximately how does the density of air vary with altitude?<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\" id=\"fs-id1613737\">\n<div class=\"problem\" id=\"fs-id1859324\">\n<p id=\"import-auto-id1474822\"><strong>2: <\/strong>Give an example in which density is used to identify the substance composing an object. Would information in addition to average density be needed to identify the substances in an object composed of more than one material?<\/p>\n<p><strong>3: <\/strong><a href=\"#import-auto-id1988132\" class=\"autogenerated-content\">Figure 3<\/a> shows a glass of ice water filled to the brim. Will the water overflow when the ice melts? Explain your answer.<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\" id=\"fs-id1397150\">\n<div class=\"problem\" id=\"fs-id2397130\">\n<figure id=\"import-auto-id1988132\">\n<figure style=\"width: 150px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-content\/uploads\/sites\/307\/2017\/12\/Figure_12_02_03a.jpg\" alt=\"A glass filled to the brim with water and ice cubes.\" height=\"759\" width=\"150\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3.<\/strong><\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id2681028\" class=\"problems-exercises\">\n<div class=\"bcc-box bcc-info\">\n<h3>Problems &amp; Exercises<\/h3>\n<div class=\"exercise\" id=\"fs-id1081259\">\n<div class=\"problem\" id=\"fs-id2622699\">\n<p><strong>You will have to look up the values for densities using the table earlier in this chapter or some other reference.\u00a0 <\/strong><\/p>\n<p style=\"text-align: left\"><strong>1:\u00a0<\/strong> Gold is sold by the troy ounce (31.103 g). What is the volume of 1 troy ounce of pure gold?<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\" id=\"fs-id2056422\">\n<div class=\"problem\" id=\"fs-id1216339\">\n<p id=\"import-auto-id3175349\"><strong>2: <\/strong>Mercury is commonly supplied in flasks containing 34.5 kg (about 76 lb). What is the volume in litres of this much mercury?<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\" id=\"fs-id2639547\">\n<div class=\"problem\" id=\"fs-id1901135\">\n<p id=\"import-auto-id3115592\"><strong>3: <\/strong>(a) What is the mass of a deep breath of air having a volume of 2.00 L? (b) Discuss the effect taking such a breath has on your body\u2019s volume and density.\u00a0 \u00a0Remember that 1 litre = a cube that is 10 cm x 10 cm x 10 cm.<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\" id=\"fs-id1561911\">\n<div class=\"problem\" id=\"fs-id3176934\">\n<p id=\"import-auto-id1282324\"><strong>4: <\/strong>A straightforward method of finding the density of an object is to measure its mass and then measure its volume by submerging it in a graduated cylinder. What is the density of a 240-g rock that displaces\u00a0 89.0 cm<sup>3<\/sup> or millilitres of water? (Note that the accuracy and practical applications of this technique are more limited than a variety of others that are based on Archimedes\u2019 principle.)<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\" id=\"fs-id1381125\">\n<div class=\"problem\" id=\"fs-id2066861\">\n<p id=\"import-auto-id1245988\"><strong>5: <\/strong>Suppose you have a coffee mug with a circular cross section and vertical sides (uniform radius). What is its inside radius if it holds 375 g of coffee when filled to a depth of 7.50 cm? Assume coffee has the same density as water.<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\" id=\"fs-id1157183\">\n<div class=\"problem\" id=\"fs-id3135333\">\n<p id=\"import-auto-id3418304\"><strong>6: <\/strong>(a) A rectangular gasoline tank can hold 50.0 kg of gasoline when full. What is the depth of the tank if it is 0.500-m wide by 0.900-m long? Hint:\u00a0 find the volume of the tank first from the density and remember that volume = width x length x depth.\u00a0 (b) Discuss whether this gas tank has a reasonable volume for a passenger car.<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\" id=\"fs-id1602523\">\n<div class=\"problem\" id=\"fs-id2588384\">\n<p id=\"import-auto-id2444694\"><strong>7: <\/strong>A trash compactor can reduce the volume of its contents to 0.350 their original value. Neglecting the mass of air expelled, by what factor is the density of the rubbish increased?<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\" id=\"fs-id1599362\">\n<div class=\"problem\" id=\"fs-id1824721\">\n<p id=\"import-auto-id1614056\"><strong>8: <\/strong>A 2.50-kg steel gasoline can holds 20.0 L of gasoline when full. What is the average density of the full gas can, taking into account the volume occupied by steel as well as by gasoline?<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\" id=\"fs-id1848674\">\n<div class=\"problem\" id=\"fs-id2591615\">\n<p id=\"import-auto-id3055502\"><strong>9: <\/strong>What is the density of 18.0-karat gold that is a mixture of 18 parts gold, 5 parts silver, and 1 part copper? (These values are parts by mass, not volume.) Assume that this is a simple mixture having an average density equal to the weighted densities of its constituents.<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\" id=\"fs-id3101625\">\n<div class=\"problem\" id=\"fs-id1849112\">\n<p id=\"import-auto-id2616385\"><strong>10: <\/strong>There is relatively little empty space between atoms in solids and liquids, so that the average density of an atom is about the same as matter on a macroscopic scale\u2014approximately 1000 kg\/m3. The nucleus of an atom has a radius about\u00a0 10<sup>-5<\/sup> that of the atom and contains nearly all the mass of the entire atom. (a) What is the approximate density of a nucleus? (b) One remnant of a supernova, called a neutron star, can have the density of a nucleus. What would be the radius of a neutron star with a mass 10 times that of our Sun (the radius of the Sun is 7 x10<sup>8<\/sup> m?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<div>\n<h2>Glossary<\/h2>\n<dl id=\"import-auto-id3125373\" class=\"definition\">\n<dt>density<\/dt>\n<dd id=\"fs-id957039\">the mass per unit volume of a substance or object<\/dd>\n<\/dl>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Solutions<\/h3>\n<p><strong>Problems &amp; Exercises<\/strong><\/p>\n<p><strong>1:\u00a0\u00a0<\/strong>1.610 cm<sup>3<\/sup><\/p>\n<p><strong>3:\u00a0<\/strong><span style=\"font-size: 1rem;text-indent: 0px\">(a) 2.58 g\u00a0<\/span><span style=\"font-size: 1rem;text-indent: 0px\">(b) The volume of your body increases by the volume of air you inhale. The average density of your body decreases when you take a deep breath, because the density of air is substantially smaller than the average density of the body before you took the deep breath.<\/span><\/p>\n<p><strong>4:\u00a0\u00a0<\/strong>2.70 g\/cm<sup>3\u00a0\u00a0<\/sup>or 2700 kg\/m<sup>3\u00a0<\/sup><\/p>\n<p><strong>6:\u00a0<\/strong><span style=\"font-size: 1rem;text-indent: 0px\">(a) 0.163 m\u00a0<\/span><span style=\"font-size: 1rem;text-indent: 0px\">(b) Equivalent to 19.4 gallons, which is reasonable<\/span><\/p>\n<p><strong>8:\u00a0\u00a0<\/strong>7.9 x10<sup>2\u00a0<\/sup> kg\/m<sup>3<\/sup><\/p>\n<p><strong>9:\u00a0<\/strong>15.6 g\/cm<sup>3<\/sup><\/p>\n<p><strong>10:\u00a0<\/strong><span style=\"font-size: 1rem;text-indent: 0px\">(a)\u00a0 10<sup>18\u00a0<\/sup>\u00a0kg\/m<sup>3\u00a0<\/sup><\/span><span style=\"font-size: 1rem;text-indent: 0px\">(b)\u00a0 2 x 10<sup>4<\/sup> m<\/span><\/p>\n<\/div>\n","protected":false},"author":9,"menu_order":8,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-64","chapter","type-chapter","status-publish","hentry"],"part":148,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-json\/pressbooks\/v2\/chapters\/64","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-json\/wp\/v2\/users\/9"}],"version-history":[{"count":8,"href":"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-json\/pressbooks\/v2\/chapters\/64\/revisions"}],"predecessor-version":[{"id":2411,"href":"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-json\/pressbooks\/v2\/chapters\/64\/revisions\/2411"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-json\/pressbooks\/v2\/parts\/148"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-json\/pressbooks\/v2\/chapters\/64\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-json\/wp\/v2\/media?parent=64"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-json\/pressbooks\/v2\/chapter-type?post=64"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-json\/wp\/v2\/contributor?post=64"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/introductorygeneralphysics2phys1207\/wp-json\/wp\/v2\/license?post=64"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}