{"id":150,"date":"2018-02-06T15:13:09","date_gmt":"2018-02-06T15:13:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/?post_type=chapter&#038;p=150"},"modified":"2019-01-17T18:36:40","modified_gmt":"2019-01-17T18:36:40","slug":"1-3-unit-conversion","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/chapter\/1-3-unit-conversion\/","title":{"raw":"1.3 Unit Conversion","rendered":"1.3 Unit Conversion"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Use conversion factors to express the value of a given quantity in different units.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1168327934938\">It is often necessary to convert from one unit to another. For example, if you are reading a European cookbook, some quantities may be expressed in units of liters and you need to convert them to cups. Or perhaps you are reading walking directions from one location to another and you are interested in how many miles you will be walking. In this case, you may need to convert units of feet or meters to miles.<\/p>\r\n<p id=\"fs-id1168328205340\">Let\u2019s consider a simple example of how to convert units. Suppose we want to convert 80 m to kilometers. The first thing to do is to list the units you have and the units to which you want to convert. In this case, we have units in <em>meters<\/em> and we want to convert to <em>kilometers<\/em>. Next, we need to determine a conversion factor relating meters to kilometers. A<strong> conversion factor<\/strong> is a ratio that expresses how many of one unit are equal to another unit. For example, there are 12 in. in 1 ft, 1609 m in 1 mi, 100 cm in 1 m, 60 s in 1 min, and so on. Refer to <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/back-matter\/conversion-factors\/\">Appendix B<\/a> for a more complete list of conversion factors. In this case, we know that there are 1000 m in 1 km. Now we can set up our unit conversion. We write the units we have and then multiply them by the conversion factor so the units cancel out, as shown:<\/p>\r\n\r\n<div id=\"fs-id1168328201234\" class=\"unnumbered\">$$80\\,\\overline{)\\text{m}}\\,\u00d7\\,\\frac{1\\,\\text{km}}{1000\\,\\overline{)\\text{m}}}=0.080\\,\\text{km}.$$<\/div>\r\n<p id=\"fs-id1168327941758\">Note that the unwanted meter unit cancels, leaving only the desired kilometer unit. You can use this method to convert between any type of unit. Now, the conversion of 80 m to kilometers is simply the use of a metric prefix, as we saw in the preceding section, so we can get the same answer just as easily by noting that<\/p>\r\n\r\n<div id=\"fs-id1168328191070\" class=\"unnumbered\">$$80\\,\\text{m}=8.0\\,\u00d7\\,{10}^{1}\\text{m}=8.0\\,\u00d7\\,{10}^{-2}\\text{km}=0.080\\,\\text{km,}$$<\/div>\r\n<p id=\"fs-id1168328144040\">since \u201ckilo-\u201d means 10<sup>3<\/sup> (see <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/chapter\/1-2-units-and-standards#T1.2\">(Figure)<\/a>) and $$ 1=-2+3. $$ However, using conversion factors is handy when converting between units that are not metric or when converting between derived units, as the following examples illustrate.<\/p>\r\n\r\n<div id=\"fs-id1168328168078\" class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\n<h4>Converting Nonmetric Units to Metric<\/h4>\r\nThe distance from the university to home is 10 mi and it usually takes 20 min to drive this distance. Calculate the average speed in meters per second (m\/s). (<em>Note:<\/em> Average speed is distance traveled divided by time of travel.)\r\n<h4>Strategy<\/h4>\r\nFirst we calculate the average speed using the given units, then we can get the average speed into the desired units by picking the correct conversion factors and multiplying by them. The correct conversion factors are those that cancel the unwanted units and leave the desired units in their place. In this case, we want to convert miles to meters, so we need to know the fact that there are 1609 m in 1 mi. We also want to convert minutes to seconds, so we use the conversion of 60 s in 1 min.\r\n<h4>Solution<\/h4>\r\n<ol id=\"fs-id1168327949701\" type=\"1\">\r\n \t<li>Calculate average speed. Average speed is distance traveled divided by time of travel. (Take this definition as a given for now. Average speed and other motion concepts are covered in later chapters.) In equation form,\r\n<div id=\"fs-id1168328070162\" class=\"unnumbered\">$$\\text{Average speed}=\\frac{\\text{Distance}}{\\text{Time}}.$$<\/div><\/li>\r\n \t<li>Substitute the given values for distance and time:\r\n<div id=\"fs-id1168327942214\" class=\"unnumbered\">$$\\text{Average speed}=\\frac{10\\,\\text{mi}}{20\\,\\text{min}}=0.50\\,\\frac{\\text{mi}}{\\text{min}}.$$<\/div><\/li>\r\n \t<li>Convert miles per minute to meters per second by multiplying by the conversion factor that cancels miles and leave meters, and also by the conversion factor that cancels minutes and leave seconds:\r\n<div id=\"fs-id1168327891322\" class=\"unnumbered\">$$0.50\\,\\frac{\\overline{)\\text{mile}}}{\\overline{)\\text{min}}}\\,\u00d7\\,\\frac{1609\\,\\text{m}}{1\\,\\overline{)\\text{mile}}}\\,\u00d7\\,\\frac{1\\,\\overline{)\\text{min}}}{60\\,\\text{s}}=\\frac{(0.50)(1609)}{60}\\,\\text{m\/s}=13\\,\\text{m\/s}.$$<\/div><\/li>\r\n<\/ol>\r\n<h4>Significance<\/h4>\r\nCheck the answer in the following ways:\r\n<ol id=\"fs-id1168328187129\" type=\"1\">\r\n \t<li>Be sure the units in the unit conversion cancel correctly. If the unit conversion factor was written upside down, the units do not cancel correctly in the equation. We see the \u201cmiles\u201d in the numerator in 0.50 mi\/min cancels the \u201cmile\u201d in the denominator in the first conversion factor. Also, the \u201cmin\u201d in the denominator in 0.50 mi\/min cancels the \u201cmin\u201d in the numerator in the second conversion factor.<\/li>\r\n \t<li>Check that the units of the final answer are the desired units. The problem asked us to solve for average speed in units of meters per second and, after the cancellations, the only units left are a meter (m) in the numerator and a second (s) in the denominator, so we have indeed obtained these units.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1168327874278\" class=\"textbox exercises check-understanding\">\r\n<h3>Check Your Understanding<\/h3>\r\n<div id=\"fs-id1168328327747\" class=\"problem textbox\">\r\n<div id=\"fs-id1168327988106\">\r\n<p id=\"fs-id1168328242532\">Light travels about 9 Pm in a year. Given that a year is about $$ 3\\,\u00d7\\,{10}^{7}\\text{s}, $$ what is the speed of light in meters per second?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1168328199684\" class=\"solution\">\r\n\r\n[reveal-answer q=\"fs-id1168328199684\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"fs-id1168328199684\"]\r\n<p id=\"fs-id1168327986853\">$$3\\,\u00d7\\,{10}^{8}\\text{m\/s}$$<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168328289500\" class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\n<h4>Converting between Metric Units<\/h4>\r\nThe density of iron is $$ 7.86\\,{\\text{g\/cm}}^{3} $$ under standard conditions. Convert this to kg\/m<sup>3<\/sup>.\r\n<h4>Strategy<\/h4>\r\nWe need to convert grams to kilograms and cubic centimeters to cubic meters. The conversion factors we need are $$ 1\\,\\text{kg}={10}^{3}\\text{g} $$ and $$ 1\\,\\text{cm}={10}^{-2}\\text{m}\\text{.} $$ However, we are dealing with cubic centimeters $$ {\\text{(cm}}^{3}=\\text{cm}\\,\u00d7\\,\\text{cm}\\,\u00d7\\,\\text{cm),} $$ so we have to use the second conversion factor three times (that is, we need to cube it). The idea is still to multiply by the conversion factors in such a way that they cancel the units we want to get rid of and introduce the units we want to keep.\r\n<h4>Solution<\/h4>\r\n<div id=\"fs-id1168328194417\" class=\"unnumbered\">$$7.86\\,\\frac{\\overline{)\\text{g}}}{{\\overline{)\\text{cm}}}^{3}}\\,\u00d7\\,\\frac{\\text{kg}}{{10}^{3}\\overline{)\\text{g}}}\\,\u00d7\\,{(\\frac{\\overline{)\\text{cm}}}{{10}^{-2}\\text{m}})}^{3}=\\frac{7.86}{({10}^{3})({10}^{-6})}\\,{\\text{kg\/m}}^{3}=7.86\\,\u00d7\\,{10}^{3}{\\,\\text{kg\/m}}^{3}$$<\/div>\r\n<h4>Significance<\/h4>\r\nRemember, it\u2019s always important to check the answer.\r\n<ol id=\"fs-id1168327984172\" type=\"1\">\r\n \t<li>Be sure to cancel the units in the unit conversion correctly. We see that the gram (\u201cg\u201d) in the numerator in 7.86 g\/cm<sup>3<\/sup> cancels the \u201cg\u201d in the denominator in the first conversion factor. Also, the three factors of \u201ccm\u201d in the denominator in 7.86 g\/cm<sup>3<\/sup> cancel with the three factors of \u201ccm\u201d in the numerator that we get by cubing the second conversion factor.<\/li>\r\n \t<li>Check that the units of the final answer are the desired units. The problem asked for us to convert to kilograms per cubic meter. After the cancellations just described, we see the only units we have left are \u201ckg\u201d in the numerator and three factors of \u201cm\u201d in the denominator (that is, one factor of \u201cm\u201d cubed, or \u201cm<sup>3<\/sup>\u201d). Therefore, the units on the final answer are correct.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1168327874671\" class=\"textbox exercises check-understanding\">\r\n<h3>Check Your Understanding<\/h3>\r\n<div id=\"fs-id1168327876191\" class=\"problem textbox\">\r\n<div id=\"fs-id1168328204361\">\r\n<p id=\"fs-id1168327929521\">We know from <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/chapter\/1-1-the-scope-and-scale-of-physics#1.4\">(Figure)<\/a> that the diameter of Earth is on the order of 10<sup>7<\/sup> m, so the order of magnitude of its surface area is 10<sup>14<\/sup> m<sup>2<\/sup>. What is that in square kilometers (that is, km<sup>2<\/sup>)? (Try doing this both by converting 10<sup>7<\/sup> m to km and then squaring it and then by converting 10<sup>14<\/sup> m<sup>2<\/sup> directly to square kilometers. You should get the same answer both ways.)<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1168327864339\" class=\"solution\">\r\n\r\n[reveal-answer q=\"fs-id1168327864339\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"fs-id1168327864339\"]\r\n<p id=\"fs-id1168328228670\">$$1{0}^{8}{\\,\\text{km}}^{2}$$<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1168328063269\">Unit conversions may not seem very interesting, but not doing them can be costly. One famous example of this situation was seen with the <span class=\"no-emphasis\"><em>Mars Climate Orbiter<\/em><\/span>. This probe was launched by NASA on December 11, 1998. On September 23, 1999, while attempting to guide the probe into its planned orbit around Mars, NASA lost contact with it. Subsequent investigations showed a piece of software called SM_FORCES (or \u201csmall forces\u201d) was recording thruster performance data in the English units of pound-seconds (lb-s). However, other pieces of software that used these values for course corrections expected them to be recorded in the SI units of newton-seconds (N-s), as dictated in the software interface protocols. This error caused the probe to follow a very different trajectory from what NASA thought it was following, which most likely caused the probe either to burn up in the Martian atmosphere or to shoot out into space. This failure to pay attention to unit conversions cost hundreds of millions of dollars, not to mention all the time invested by the scientists and engineers who worked on the project.<\/p>\r\n\r\n<div id=\"fs-id1168327869372\" class=\"textbox exercises check-understanding\">\r\n<h3>Check Your Understanding<\/h3>\r\n<div id=\"fs-id1168328059712\" class=\"problem textbox\">\r\n<div id=\"fs-id1168328224586\">\r\n<p id=\"fs-id1168327873248\">Given that 1 lb (pound) is 4.45 N, were the numbers being output by SM_FORCES too big or too small?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1168328289312\" class=\"solution\">\r\n\r\n[reveal-answer q=\"fs-id1168328289312\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"fs-id1168328289312\"]\r\n<p id=\"fs-id1168328242737\">The numbers were too small, by a factor of 4.45.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168328267596\" class=\"textbox key-takeaways\">\r\n<h3>Summary<\/h3>\r\n<ul id=\"fs-id1168328066342\">\r\n \t<li>To convert a quantity from one unit to another, multiply by conversions factors in such a way that you cancel the units you want to get rid of and introduce the units you want to end up with.<\/li>\r\n \t<li>Be careful with areas and volumes. Units obey the rules of algebra so, for example, if a unit is squared we need two factors to cancel it.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1168327956216\" class=\"review-problems textbox exercises\">\r\n<h3>Problems<\/h3>\r\n<div id=\"fs-id1168327939270\" class=\"problem textbox\">\r\n<div id=\"fs-id1168327926028\">\r\n<p id=\"fs-id1168328193378\">The volume of Earth is on the order of 10<sup>21<\/sup> m<sup>3<\/sup>. (a) What is this in cubic kilometers (km<sup>3<\/sup>)? (b) What is it in cubic miles (mi<sup>3<\/sup>)? (c) What is it in cubic centimeters (cm<sup>3<\/sup>)?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168327854111\" class=\"problem textbox\">\r\n<div id=\"fs-id1168327874834\">\r\n<p id=\"fs-id1168328177870\">The speed limit on some interstate highways is roughly 100 km\/h. (a) What is this in meters per second? (b) How many miles per hour is this?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1168328294280\" class=\"solution\">\r\n\r\n[reveal-answer q=\"fs-id1168328294280\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"fs-id1168328294280\"]\r\n<p id=\"fs-id1168328269910\">a. 27.8 m\/s; b. 62 mi\/h<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168328285914\" class=\"problem textbox\">\r\n<div id=\"fs-id1168328071458\">\r\n<p id=\"fs-id1168327876205\">A car is traveling at a speed of 33 m\/s. (a) What is its speed in kilometers per hour? (b) Is it exceeding the 90 km\/h speed limit?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168328059510\" class=\"problem textbox\">\r\n<div id=\"fs-id1168328229974\">\r\n<p id=\"fs-id1168327951931\">In SI units, speeds are measured in meters per second (m\/s). But, depending on where you live, you\u2019re probably more comfortable of thinking of speeds in terms of either kilometers per hour (km\/h) or miles per hour (mi\/h). In this problem, you will see that 1 m\/s is roughly 4 km\/h or 2 mi\/h, which is handy to use when developing your physical intuition. More precisely, show that (a) $$ 1.0\\,\\text{m\/s}=3.6\\,\\text{km\/h} $$ and (b) $$ 1.0\\,\\text{m\/s}=2.2\\,\\text{mi\/h}.$$<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1168328204121\" class=\"solution\">\r\n\r\n[reveal-answer q=\"fs-id1168328204121\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"fs-id1168328204121\"]\r\n<p id=\"fs-id1168327958773\">a. 3.6 km\/h; b. 2.2 mi\/h<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168327924198\" class=\"problem textbox\">\r\n<div id=\"fs-id1168328170085\">\r\n<p id=\"fs-id1168328293595\">American football is played on a 100-yd-long field, excluding the end zones. How long is the field in meters? (Assume that 1 m = 3.281 ft.)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168328297906\" class=\"problem textbox\">\r\n<div id=\"fs-id1168327931904\">\r\n<p id=\"fs-id1168327933995\">Soccer fields vary in size. A large soccer field is 115 m long and 85.0 m wide. What is its area in square feet? (Assume that 1 m = 3.281 ft.)<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1168328059139\" class=\"solution\">\r\n\r\n[reveal-answer q=\"fs-id1168328059139\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"fs-id1168328059139\"]\r\n<p id=\"fs-id1168328169332\">$$1.05\\,\u00d7\\,{10}^{5}{\\,\\text{ft}}^{2}$$<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168328222458\" class=\"problem textbox\">\r\n<div id=\"fs-id1168327943116\">\r\n<p id=\"fs-id1168327956404\">What is the height in meters of a person who is 6 ft 1.0 in. tall?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168327939869\" class=\"problem textbox\">\r\n<div id=\"fs-id1168327922500\">\r\n<p id=\"fs-id1168328295075\">Mount Everest, at 29,028 ft, is the tallest mountain on Earth. What is its height in kilometers? (Assume that 1 m = 3.281 ft.)<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1168328025151\" class=\"solution\">\r\n\r\n[reveal-answer q=\"fs-id1168328025151\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"fs-id1168328025151\"]\r\n<p id=\"fs-id1168328059572\">8.847 km<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168328222906\" class=\"problem textbox\">\r\n<div id=\"fs-id1168328189735\">\r\n<p id=\"fs-id1168328301961\">The speed of sound is measured to be 342 m\/s on a certain day. What is this measurement in kilometers per hour?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168328171083\" class=\"problem textbox\">\r\n<div id=\"fs-id1168328247343\">\r\n<p id=\"fs-id1168328222416\">Tectonic plates are large segments of Earth\u2019s crust that move slowly. Suppose one such plate has an average speed of 4.0 cm\/yr. (a) What distance does it move in 1.0 s at this speed? (b) What is its speed in kilometers per million years?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1168327911343\" class=\"solution\">\r\n\r\n[reveal-answer q=\"fs-id1168327911343\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"fs-id1168327911343\"]\r\n<p id=\"fs-id1168327940797\">a. $$ 1.3\\,\u00d7\\,{10}^{-9}\\text{m}; $$ b. 40 km\/My<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168328294774\" class=\"problem textbox\">\r\n<div id=\"fs-id1168328222883\">\r\n<p id=\"fs-id1168327872186\">The average distance between Earth and the Sun is $$ 1.5\\,\u00d7\\,{10}^{11}\\text{m.} $$ (a) Calculate the average speed of Earth in its orbit (assumed to be circular) in meters per second. (b) What is this speed in miles per hour?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168328311640\" class=\"problem textbox\">\r\n<div id=\"fs-id1168328364226\">\r\n<p id=\"fs-id1168328025058\">The density of nuclear matter is about 10<sup>18<\/sup> kg\/m<sup>3<\/sup>. Given that 1 mL is equal in volume to cm<sup>3<\/sup>, what is the density of nuclear matter in megagrams per microliter (that is, $$ \\text{Mg\/}\\mu \\text{L}$$)?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1168328240950\" class=\"solution\">\r\n\r\n[reveal-answer q=\"fs-id1168328240950\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"fs-id1168328240950\"]\r\n<p id=\"fs-id1168328329673\">$${10}^{6}\\text{Mg\/}\\mu \\text{L}$$<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168328064287\" class=\"problem textbox\">\r\n<div id=\"fs-id1168328201081\">\r\n<p id=\"fs-id1168328175768\">The density of aluminum is 2.7 g\/cm<sup>3<\/sup>. What is the density in kilograms per cubic meter?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168328170090\" class=\"problem textbox\">\r\n<div id=\"fs-id1168327961174\">\r\n<p id=\"fs-id1168327923356\">A commonly used unit of mass in the English system is the pound-mass, abbreviated lbm, where 1 lbm = 0.454 kg. What is the density of water in pound-mass per cubic foot?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1168327931285\" class=\"solution\">\r\n\r\n[reveal-answer q=\"fs-id1168327931285\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"fs-id1168327931285\"]\r\n<p id=\"fs-id1168327879321\">62.4 lbm\/ft<sup>3<\/sup><\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168327871170\" class=\"problem textbox\">\r\n<div id=\"fs-id1168327935365\">\r\n<p id=\"fs-id1168327873821\">A furlong is 220 yd. A fortnight is 2 weeks. Convert a speed of one furlong per fortnight to millimeters per second.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168327941954\" class=\"problem textbox\">\r\n<div id=\"fs-id1168327918284\">\r\n<p id=\"fs-id1168328181668\">It takes $$ 2\\pi $$ radians (rad) to get around a circle, which is the same as 360\u00b0. How many radians are in 1\u00b0?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1168327854847\" class=\"solution\">\r\n\r\n[reveal-answer q=\"fs-id1168327854847\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"fs-id1168327854847\"]\r\n<p id=\"fs-id1168328190635\">0.017 rad<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168328293255\" class=\"problem textbox\">\r\n<div id=\"fs-id1168327953647\">\r\n<p id=\"fs-id1168328185095\">Light travels a distance of about $$ 3\\,\u00d7\\,{10}^{8}\\text{m\/s.} $$ A light-minute is the distance light travels in 1 min. If the Sun is $$ 1.5\\,\u00d7\\,{10}^{11}\\text{m} $$ from Earth, how far away is it in light-minutes?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168328303932\" class=\"problem textbox\">\r\n<div id=\"fs-id1168327927709\">\r\n<p id=\"fs-id1168327983780\">A light-nanosecond is the distance light travels in 1 ns. Convert 1 ft to light-nanoseconds.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1168327943341\" class=\"solution\">\r\n\r\n[reveal-answer q=\"fs-id1168327943341\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"fs-id1168327943341\"]\r\n<p id=\"fs-id1168328329635\">1 light-nanosecond<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168327934441\" class=\"problem textbox\">\r\n<div id=\"fs-id1168327982880\">\r\n<p id=\"fs-id1168327943847\">An electron has a mass of $$ 9.11\\,\u00d7\\,{10}^{-31}\\text{kg.} $$ A proton has a mass of $$ 1.67\\,\u00d7\\,{10}^{-27}\\text{kg}\\text{.} $$ What is the mass of a proton in electron-masses?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1168328195795\" class=\"problem textbox\">\r\n<div id=\"fs-id1168328286055\">\r\n<p id=\"fs-id1168328289390\">A fluid ounce is about 30 mL. What is the volume of a 12 fl-oz can of soda pop in cubic meters?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1168328280449\" class=\"solution\">\r\n\r\n[reveal-answer q=\"fs-id1168328280449\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"fs-id1168328280449\"]\r\n<p id=\"fs-id1168327870505\">$$3.6\\,\u00d7\\,{10}^{-4}{\\text{m}}^{3}$$<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Glossary<\/h3>\r\n<dl id=\"fs-id1168327920641\">\r\n \t<dt>conversion factor<\/dt>\r\n \t<dd id=\"fs-id1168328329236\">a ratio that expresses how many of one unit are equal to another unit<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Use conversion factors to express the value of a given quantity in different units.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1168327934938\">It is often necessary to convert from one unit to another. For example, if you are reading a European cookbook, some quantities may be expressed in units of liters and you need to convert them to cups. Or perhaps you are reading walking directions from one location to another and you are interested in how many miles you will be walking. In this case, you may need to convert units of feet or meters to miles.<\/p>\n<p id=\"fs-id1168328205340\">Let\u2019s consider a simple example of how to convert units. Suppose we want to convert 80 m to kilometers. The first thing to do is to list the units you have and the units to which you want to convert. In this case, we have units in <em>meters<\/em> and we want to convert to <em>kilometers<\/em>. Next, we need to determine a conversion factor relating meters to kilometers. A<strong> conversion factor<\/strong> is a ratio that expresses how many of one unit are equal to another unit. For example, there are 12 in. in 1 ft, 1609 m in 1 mi, 100 cm in 1 m, 60 s in 1 min, and so on. Refer to <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/back-matter\/conversion-factors\/\">Appendix B<\/a> for a more complete list of conversion factors. In this case, we know that there are 1000 m in 1 km. Now we can set up our unit conversion. We write the units we have and then multiply them by the conversion factor so the units cancel out, as shown:<\/p>\n<div id=\"fs-id1168328201234\" class=\"unnumbered\">$$80\\,\\overline{)\\text{m}}\\,\u00d7\\,\\frac{1\\,\\text{km}}{1000\\,\\overline{)\\text{m}}}=0.080\\,\\text{km}.$$<\/div>\n<p id=\"fs-id1168327941758\">Note that the unwanted meter unit cancels, leaving only the desired kilometer unit. You can use this method to convert between any type of unit. Now, the conversion of 80 m to kilometers is simply the use of a metric prefix, as we saw in the preceding section, so we can get the same answer just as easily by noting that<\/p>\n<div id=\"fs-id1168328191070\" class=\"unnumbered\">$$80\\,\\text{m}=8.0\\,\u00d7\\,{10}^{1}\\text{m}=8.0\\,\u00d7\\,{10}^{-2}\\text{km}=0.080\\,\\text{km,}$$<\/div>\n<p id=\"fs-id1168328144040\">since \u201ckilo-\u201d means 10<sup>3<\/sup> (see <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/chapter\/1-2-units-and-standards#T1.2\">(Figure)<\/a>) and $$ 1=-2+3. $$ However, using conversion factors is handy when converting between units that are not metric or when converting between derived units, as the following examples illustrate.<\/p>\n<div id=\"fs-id1168328168078\" class=\"textbox examples\">\n<h3>Example<\/h3>\n<h4>Converting Nonmetric Units to Metric<\/h4>\n<p>The distance from the university to home is 10 mi and it usually takes 20 min to drive this distance. Calculate the average speed in meters per second (m\/s). (<em>Note:<\/em> Average speed is distance traveled divided by time of travel.)<\/p>\n<h4>Strategy<\/h4>\n<p>First we calculate the average speed using the given units, then we can get the average speed into the desired units by picking the correct conversion factors and multiplying by them. The correct conversion factors are those that cancel the unwanted units and leave the desired units in their place. In this case, we want to convert miles to meters, so we need to know the fact that there are 1609 m in 1 mi. We also want to convert minutes to seconds, so we use the conversion of 60 s in 1 min.<\/p>\n<h4>Solution<\/h4>\n<ol id=\"fs-id1168327949701\" type=\"1\">\n<li>Calculate average speed. Average speed is distance traveled divided by time of travel. (Take this definition as a given for now. Average speed and other motion concepts are covered in later chapters.) In equation form,\n<div id=\"fs-id1168328070162\" class=\"unnumbered\">$$\\text{Average speed}=\\frac{\\text{Distance}}{\\text{Time}}.$$<\/div>\n<\/li>\n<li>Substitute the given values for distance and time:\n<div id=\"fs-id1168327942214\" class=\"unnumbered\">$$\\text{Average speed}=\\frac{10\\,\\text{mi}}{20\\,\\text{min}}=0.50\\,\\frac{\\text{mi}}{\\text{min}}.$$<\/div>\n<\/li>\n<li>Convert miles per minute to meters per second by multiplying by the conversion factor that cancels miles and leave meters, and also by the conversion factor that cancels minutes and leave seconds:\n<div id=\"fs-id1168327891322\" class=\"unnumbered\">$$0.50\\,\\frac{\\overline{)\\text{mile}}}{\\overline{)\\text{min}}}\\,\u00d7\\,\\frac{1609\\,\\text{m}}{1\\,\\overline{)\\text{mile}}}\\,\u00d7\\,\\frac{1\\,\\overline{)\\text{min}}}{60\\,\\text{s}}=\\frac{(0.50)(1609)}{60}\\,\\text{m\/s}=13\\,\\text{m\/s}.$$<\/div>\n<\/li>\n<\/ol>\n<h4>Significance<\/h4>\n<p>Check the answer in the following ways:<\/p>\n<ol id=\"fs-id1168328187129\" type=\"1\">\n<li>Be sure the units in the unit conversion cancel correctly. If the unit conversion factor was written upside down, the units do not cancel correctly in the equation. We see the \u201cmiles\u201d in the numerator in 0.50 mi\/min cancels the \u201cmile\u201d in the denominator in the first conversion factor. Also, the \u201cmin\u201d in the denominator in 0.50 mi\/min cancels the \u201cmin\u201d in the numerator in the second conversion factor.<\/li>\n<li>Check that the units of the final answer are the desired units. The problem asked us to solve for average speed in units of meters per second and, after the cancellations, the only units left are a meter (m) in the numerator and a second (s) in the denominator, so we have indeed obtained these units.<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1168327874278\" class=\"textbox exercises check-understanding\">\n<h3>Check Your Understanding<\/h3>\n<div id=\"fs-id1168328327747\" class=\"problem textbox\">\n<div id=\"fs-id1168327988106\">\n<p id=\"fs-id1168328242532\">Light travels about 9 Pm in a year. Given that a year is about $$ 3\\,\u00d7\\,{10}^{7}\\text{s}, $$ what is the speed of light in meters per second?<\/p>\n<\/div>\n<div id=\"fs-id1168328199684\" class=\"solution\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1168328199684\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1168328199684\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1168327986853\">$$3\\,\u00d7\\,{10}^{8}\\text{m\/s}$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168328289500\" class=\"textbox examples\">\n<h3>Example<\/h3>\n<h4>Converting between Metric Units<\/h4>\n<p>The density of iron is $$ 7.86\\,{\\text{g\/cm}}^{3} $$ under standard conditions. Convert this to kg\/m<sup>3<\/sup>.<\/p>\n<h4>Strategy<\/h4>\n<p>We need to convert grams to kilograms and cubic centimeters to cubic meters. The conversion factors we need are $$ 1\\,\\text{kg}={10}^{3}\\text{g} $$ and $$ 1\\,\\text{cm}={10}^{-2}\\text{m}\\text{.} $$ However, we are dealing with cubic centimeters $$ {\\text{(cm}}^{3}=\\text{cm}\\,\u00d7\\,\\text{cm}\\,\u00d7\\,\\text{cm),} $$ so we have to use the second conversion factor three times (that is, we need to cube it). The idea is still to multiply by the conversion factors in such a way that they cancel the units we want to get rid of and introduce the units we want to keep.<\/p>\n<h4>Solution<\/h4>\n<div id=\"fs-id1168328194417\" class=\"unnumbered\">$$7.86\\,\\frac{\\overline{)\\text{g}}}{{\\overline{)\\text{cm}}}^{3}}\\,\u00d7\\,\\frac{\\text{kg}}{{10}^{3}\\overline{)\\text{g}}}\\,\u00d7\\,{(\\frac{\\overline{)\\text{cm}}}{{10}^{-2}\\text{m}})}^{3}=\\frac{7.86}{({10}^{3})({10}^{-6})}\\,{\\text{kg\/m}}^{3}=7.86\\,\u00d7\\,{10}^{3}{\\,\\text{kg\/m}}^{3}$$<\/div>\n<h4>Significance<\/h4>\n<p>Remember, it\u2019s always important to check the answer.<\/p>\n<ol id=\"fs-id1168327984172\" type=\"1\">\n<li>Be sure to cancel the units in the unit conversion correctly. We see that the gram (\u201cg\u201d) in the numerator in 7.86 g\/cm<sup>3<\/sup> cancels the \u201cg\u201d in the denominator in the first conversion factor. Also, the three factors of \u201ccm\u201d in the denominator in 7.86 g\/cm<sup>3<\/sup> cancel with the three factors of \u201ccm\u201d in the numerator that we get by cubing the second conversion factor.<\/li>\n<li>Check that the units of the final answer are the desired units. The problem asked for us to convert to kilograms per cubic meter. After the cancellations just described, we see the only units we have left are \u201ckg\u201d in the numerator and three factors of \u201cm\u201d in the denominator (that is, one factor of \u201cm\u201d cubed, or \u201cm<sup>3<\/sup>\u201d). Therefore, the units on the final answer are correct.<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1168327874671\" class=\"textbox exercises check-understanding\">\n<h3>Check Your Understanding<\/h3>\n<div id=\"fs-id1168327876191\" class=\"problem textbox\">\n<div id=\"fs-id1168328204361\">\n<p id=\"fs-id1168327929521\">We know from <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/chapter\/1-1-the-scope-and-scale-of-physics#1.4\">(Figure)<\/a> that the diameter of Earth is on the order of 10<sup>7<\/sup> m, so the order of magnitude of its surface area is 10<sup>14<\/sup> m<sup>2<\/sup>. What is that in square kilometers (that is, km<sup>2<\/sup>)? (Try doing this both by converting 10<sup>7<\/sup> m to km and then squaring it and then by converting 10<sup>14<\/sup> m<sup>2<\/sup> directly to square kilometers. You should get the same answer both ways.)<\/p>\n<\/div>\n<div id=\"fs-id1168327864339\" class=\"solution\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1168327864339\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1168327864339\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1168328228670\">$$1{0}^{8}{\\,\\text{km}}^{2}$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168328063269\">Unit conversions may not seem very interesting, but not doing them can be costly. One famous example of this situation was seen with the <span class=\"no-emphasis\"><em>Mars Climate Orbiter<\/em><\/span>. This probe was launched by NASA on December 11, 1998. On September 23, 1999, while attempting to guide the probe into its planned orbit around Mars, NASA lost contact with it. Subsequent investigations showed a piece of software called SM_FORCES (or \u201csmall forces\u201d) was recording thruster performance data in the English units of pound-seconds (lb-s). However, other pieces of software that used these values for course corrections expected them to be recorded in the SI units of newton-seconds (N-s), as dictated in the software interface protocols. This error caused the probe to follow a very different trajectory from what NASA thought it was following, which most likely caused the probe either to burn up in the Martian atmosphere or to shoot out into space. This failure to pay attention to unit conversions cost hundreds of millions of dollars, not to mention all the time invested by the scientists and engineers who worked on the project.<\/p>\n<div id=\"fs-id1168327869372\" class=\"textbox exercises check-understanding\">\n<h3>Check Your Understanding<\/h3>\n<div id=\"fs-id1168328059712\" class=\"problem textbox\">\n<div id=\"fs-id1168328224586\">\n<p id=\"fs-id1168327873248\">Given that 1 lb (pound) is 4.45 N, were the numbers being output by SM_FORCES too big or too small?<\/p>\n<\/div>\n<div id=\"fs-id1168328289312\" class=\"solution\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1168328289312\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1168328289312\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1168328242737\">The numbers were too small, by a factor of 4.45.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168328267596\" class=\"textbox key-takeaways\">\n<h3>Summary<\/h3>\n<ul id=\"fs-id1168328066342\">\n<li>To convert a quantity from one unit to another, multiply by conversions factors in such a way that you cancel the units you want to get rid of and introduce the units you want to end up with.<\/li>\n<li>Be careful with areas and volumes. Units obey the rules of algebra so, for example, if a unit is squared we need two factors to cancel it.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1168327956216\" class=\"review-problems textbox exercises\">\n<h3>Problems<\/h3>\n<div id=\"fs-id1168327939270\" class=\"problem textbox\">\n<div id=\"fs-id1168327926028\">\n<p id=\"fs-id1168328193378\">The volume of Earth is on the order of 10<sup>21<\/sup> m<sup>3<\/sup>. (a) What is this in cubic kilometers (km<sup>3<\/sup>)? (b) What is it in cubic miles (mi<sup>3<\/sup>)? (c) What is it in cubic centimeters (cm<sup>3<\/sup>)?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1168327854111\" class=\"problem textbox\">\n<div id=\"fs-id1168327874834\">\n<p id=\"fs-id1168328177870\">The speed limit on some interstate highways is roughly 100 km\/h. (a) What is this in meters per second? (b) How many miles per hour is this?<\/p>\n<\/div>\n<div id=\"fs-id1168328294280\" class=\"solution\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1168328294280\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1168328294280\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1168328269910\">a. 27.8 m\/s; b. 62 mi\/h<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168328285914\" class=\"problem textbox\">\n<div id=\"fs-id1168328071458\">\n<p id=\"fs-id1168327876205\">A car is traveling at a speed of 33 m\/s. (a) What is its speed in kilometers per hour? (b) Is it exceeding the 90 km\/h speed limit?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1168328059510\" class=\"problem textbox\">\n<div id=\"fs-id1168328229974\">\n<p id=\"fs-id1168327951931\">In SI units, speeds are measured in meters per second (m\/s). But, depending on where you live, you\u2019re probably more comfortable of thinking of speeds in terms of either kilometers per hour (km\/h) or miles per hour (mi\/h). In this problem, you will see that 1 m\/s is roughly 4 km\/h or 2 mi\/h, which is handy to use when developing your physical intuition. More precisely, show that (a) $$ 1.0\\,\\text{m\/s}=3.6\\,\\text{km\/h} $$ and (b) $$ 1.0\\,\\text{m\/s}=2.2\\,\\text{mi\/h}.$$<\/p>\n<\/div>\n<div id=\"fs-id1168328204121\" class=\"solution\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1168328204121\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1168328204121\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1168327958773\">a. 3.6 km\/h; b. 2.2 mi\/h<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168327924198\" class=\"problem textbox\">\n<div id=\"fs-id1168328170085\">\n<p id=\"fs-id1168328293595\">American football is played on a 100-yd-long field, excluding the end zones. How long is the field in meters? (Assume that 1 m = 3.281 ft.)<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1168328297906\" class=\"problem textbox\">\n<div id=\"fs-id1168327931904\">\n<p id=\"fs-id1168327933995\">Soccer fields vary in size. A large soccer field is 115 m long and 85.0 m wide. What is its area in square feet? (Assume that 1 m = 3.281 ft.)<\/p>\n<\/div>\n<div id=\"fs-id1168328059139\" class=\"solution\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1168328059139\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1168328059139\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1168328169332\">$$1.05\\,\u00d7\\,{10}^{5}{\\,\\text{ft}}^{2}$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168328222458\" class=\"problem textbox\">\n<div id=\"fs-id1168327943116\">\n<p id=\"fs-id1168327956404\">What is the height in meters of a person who is 6 ft 1.0 in. tall?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1168327939869\" class=\"problem textbox\">\n<div id=\"fs-id1168327922500\">\n<p id=\"fs-id1168328295075\">Mount Everest, at 29,028 ft, is the tallest mountain on Earth. What is its height in kilometers? (Assume that 1 m = 3.281 ft.)<\/p>\n<\/div>\n<div id=\"fs-id1168328025151\" class=\"solution\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1168328025151\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1168328025151\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1168328059572\">8.847 km<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168328222906\" class=\"problem textbox\">\n<div id=\"fs-id1168328189735\">\n<p id=\"fs-id1168328301961\">The speed of sound is measured to be 342 m\/s on a certain day. What is this measurement in kilometers per hour?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1168328171083\" class=\"problem textbox\">\n<div id=\"fs-id1168328247343\">\n<p id=\"fs-id1168328222416\">Tectonic plates are large segments of Earth\u2019s crust that move slowly. Suppose one such plate has an average speed of 4.0 cm\/yr. (a) What distance does it move in 1.0 s at this speed? (b) What is its speed in kilometers per million years?<\/p>\n<\/div>\n<div id=\"fs-id1168327911343\" class=\"solution\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1168327911343\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1168327911343\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1168327940797\">a. $$ 1.3\\,\u00d7\\,{10}^{-9}\\text{m}; $$ b. 40 km\/My<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168328294774\" class=\"problem textbox\">\n<div id=\"fs-id1168328222883\">\n<p id=\"fs-id1168327872186\">The average distance between Earth and the Sun is $$ 1.5\\,\u00d7\\,{10}^{11}\\text{m.} $$ (a) Calculate the average speed of Earth in its orbit (assumed to be circular) in meters per second. (b) What is this speed in miles per hour?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1168328311640\" class=\"problem textbox\">\n<div id=\"fs-id1168328364226\">\n<p id=\"fs-id1168328025058\">The density of nuclear matter is about 10<sup>18<\/sup> kg\/m<sup>3<\/sup>. Given that 1 mL is equal in volume to cm<sup>3<\/sup>, what is the density of nuclear matter in megagrams per microliter (that is, $$ \\text{Mg\/}\\mu \\text{L}$$)?<\/p>\n<\/div>\n<div id=\"fs-id1168328240950\" class=\"solution\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1168328240950\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1168328240950\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1168328329673\">$${10}^{6}\\text{Mg\/}\\mu \\text{L}$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168328064287\" class=\"problem textbox\">\n<div id=\"fs-id1168328201081\">\n<p id=\"fs-id1168328175768\">The density of aluminum is 2.7 g\/cm<sup>3<\/sup>. What is the density in kilograms per cubic meter?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1168328170090\" class=\"problem textbox\">\n<div id=\"fs-id1168327961174\">\n<p id=\"fs-id1168327923356\">A commonly used unit of mass in the English system is the pound-mass, abbreviated lbm, where 1 lbm = 0.454 kg. What is the density of water in pound-mass per cubic foot?<\/p>\n<\/div>\n<div id=\"fs-id1168327931285\" class=\"solution\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1168327931285\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1168327931285\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1168327879321\">62.4 lbm\/ft<sup>3<\/sup><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168327871170\" class=\"problem textbox\">\n<div id=\"fs-id1168327935365\">\n<p id=\"fs-id1168327873821\">A furlong is 220 yd. A fortnight is 2 weeks. Convert a speed of one furlong per fortnight to millimeters per second.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1168327941954\" class=\"problem textbox\">\n<div id=\"fs-id1168327918284\">\n<p id=\"fs-id1168328181668\">It takes $$ 2\\pi $$ radians (rad) to get around a circle, which is the same as 360\u00b0. How many radians are in 1\u00b0?<\/p>\n<\/div>\n<div id=\"fs-id1168327854847\" class=\"solution\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1168327854847\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1168327854847\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1168328190635\">0.017 rad<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168328293255\" class=\"problem textbox\">\n<div id=\"fs-id1168327953647\">\n<p id=\"fs-id1168328185095\">Light travels a distance of about $$ 3\\,\u00d7\\,{10}^{8}\\text{m\/s.} $$ A light-minute is the distance light travels in 1 min. If the Sun is $$ 1.5\\,\u00d7\\,{10}^{11}\\text{m} $$ from Earth, how far away is it in light-minutes?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1168328303932\" class=\"problem textbox\">\n<div id=\"fs-id1168327927709\">\n<p id=\"fs-id1168327983780\">A light-nanosecond is the distance light travels in 1 ns. Convert 1 ft to light-nanoseconds.<\/p>\n<\/div>\n<div id=\"fs-id1168327943341\" class=\"solution\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1168327943341\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1168327943341\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1168328329635\">1 light-nanosecond<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168327934441\" class=\"problem textbox\">\n<div id=\"fs-id1168327982880\">\n<p id=\"fs-id1168327943847\">An electron has a mass of $$ 9.11\\,\u00d7\\,{10}^{-31}\\text{kg.} $$ A proton has a mass of $$ 1.67\\,\u00d7\\,{10}^{-27}\\text{kg}\\text{.} $$ What is the mass of a proton in electron-masses?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1168328195795\" class=\"problem textbox\">\n<div id=\"fs-id1168328286055\">\n<p id=\"fs-id1168328289390\">A fluid ounce is about 30 mL. What is the volume of a 12 fl-oz can of soda pop in cubic meters?<\/p>\n<\/div>\n<div id=\"fs-id1168328280449\" class=\"solution\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1168328280449\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1168328280449\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1168327870505\">$$3.6\\,\u00d7\\,{10}^{-4}{\\text{m}}^{3}$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1168327920641\">\n<dt>conversion factor<\/dt>\n<dd id=\"fs-id1168328329236\">a ratio that expresses how many of one unit are equal to another unit<\/dd>\n<\/dl>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-150\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>OpenStax University Physics. <strong>Authored by<\/strong>: OpenStax CNX. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/cnx.org\/contents\/1Q9uMg_a@10.16:Gofkr9Oy@15\">https:\/\/cnx.org\/contents\/1Q9uMg_a@10.16:Gofkr9Oy@15<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@10.16<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":311,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"OpenStax University Physics\",\"author\":\"OpenStax CNX\",\"organization\":\"\",\"url\":\"https:\/\/cnx.org\/contents\/1Q9uMg_a@10.16:Gofkr9Oy@15\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@10.16\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-150","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":142,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/pressbooks\/v2\/chapters\/150","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":5,"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/pressbooks\/v2\/chapters\/150\/revisions"}],"predecessor-version":[{"id":2264,"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/pressbooks\/v2\/chapters\/150\/revisions\/2264"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/pressbooks\/v2\/parts\/142"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/pressbooks\/v2\/chapters\/150\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/wp\/v2\/media?parent=150"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/pressbooks\/v2\/chapter-type?post=150"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/wp\/v2\/contributor?post=150"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/wp\/v2\/license?post=150"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}