Basics and Fluids - Mastering Physics Assignment

Converting between Different Units Unit conversion problems can seem tedious and unnecessary at times. However, different systems of units are used in different parts of the world, so when dealing with international transactions, documents, software, etc., unit conversions are often necessary. Here is a simple example. The inhabitants of a small island begin exporting beautiful cloth made from a rare plant that grows only on their island. Seeing how popular the small quantity that they export has been, they steadily raise their prices. A clothing maker from New York, thinking that he can save money by "cutting out the middleman," decides to travel to the small island and buy the cloth himself. Ignorant of the local custom of offering strangers outrageous prices and then negotiating down, the clothing maker accepts (much to everyone's surpise) the initial price of 400 tepizes / m 2. The price of this cloth in New York is 120 dollars / yard 2. Part A If the clothing maker bought 500 m 2 of this fabric, how much money did he lose? Use 1tepiz = 0.625dollar and 0.9144m = 1yard. Express your answer in dollars using two significant figures. Hint 1. How to approach the problem To find how much money the clothing maker loses, you must find how much money he spent and how much he would have spent in New York. Furthermore, since the problem asks how much he lost in dollars, you need to determine both in dollars. This will require unit conversions. Hint 2. Find how much he paid If the clothing maker bought 500 m 2 at a cost of 400 tepizes / m 2, then simple multiplication will give how much he spent in tepizes. Once you've found that, convert to dollars. How much did the clothing maker spend in dollars? Express your answer in dollars to three significant figures. Hint 1. Find how much he paid in tepizes If the clothing maker bought 500 m 2 at a cost of 400 tepizes / m 2, then how much did he pay in total, in tepizes? Express your answer in tepizes. ANSWER: 2.00×105 tepizes ANSWER: 1.25×105 dollars Hint 3. Find the price in New York You know that the price of the fabric in New York is 120 dollars / yard 2. Thus, you need only to find the number of square yards that the clothing maker purchased and then multiply to find the price in New York. What would it have cost him to buy the fabric in New York? Express your answer in dollars to three significant figures. Hint 1. Determine how much cloth he bought in yard 2 You are given that 0.9144m = 1yard. Squaring both sides, you would get that 0.8361m 2 = 1yard 2. How much is 500m2? Express your answer in yard2 to three significant figures. ANSWER: 598 yard 2 ANSWER: 7.18×104 dollars ANSWER: 5.3×104 dollars Correct Still think that unit conversion isn't important? Here is a widely publicized, true story about how failing to convert units resulted in a huge loss. In 1998, the Mars Climate Orbiter probe crashed into the surface of Mars, instead of entering orbit. The resulting inquiry revealed that NASA navigators had been making minor course corrections in SI units, whereas the software written by the probe's makers implicitly used British units. In the United States, most scientists use SI units, whereas most engineers use the British, or Imperial, system of units. (Interestingly, British units are not used in Britain.) For these two groups to be able to communicate to one another, unit conversions are necessary. The unit of force in the SI system is the newton (N), which is defined in terms of basic SI units as 1N = 1kg m / s 2. The unit of force in the British system is the pound (lb), which is defined in terms of the slug (British unit of mass), foot (ft), and second (s) as 1lb = 1slug ft / s 2. Part B Find the value of 15.0 N in pounds. Use the conversions 1slug = 14.59kg and 1ft = 0.3048m. Express your answer in pounds to three significant figures. Hint 1. How to approach the problem When doing a unit conversion, you should begin by comparing the units you are starting with and the units you need to finish with. In this problem, we have the following: Starting units Final units kg m slug ft s2 s2 Notice that both have seconds squared in the denominator. You will only have to change the units in the numerator. Match up the units that measure the same quantity (e.g., kilograms and slugs both measure mass). Once you've done this, create a fraction (e.g., 1hour / 60minutes) based on conversion factors such that the old unit is canceled out of the expression and the new unit appears in the position (i.e., numerator or denominator) of the old unit. In this problem, there are two pairs within the starting and final units that must be converted in this way (i.e., kilograms/slugs and meters/feet). Hint 2. Calculate the first conversion The first step is to eliminate kilograms from the expression for newtons in favor of slugs. What is the value of 15 kg m / s 2 in slug m / s 2? Express your answer in slug-meters per second squared to four significant figures. ANSWER: 1.028 ANSWER: 15.0 N = 3.37 lb Correct Thus, if the NASA navigators believed that they were entering a force value of 15 N (3.37 lb), they were actually entering a value nearly four and a half times higher, 15 lb. Though these errors were only in tiny course corrections, they added up during the trip of many millions of kilometers. In the end, the blame for the loss of the 125-million-dollar probe was placed on the lack of communication between people at NASA that allowed the units mismatch to go unnoticed. Nonetheless, this story makes apparent how important it is to carefully label the units used to measure a number. Problem 1.47 An average family of four uses roughly 1200 L (about 300 gallons) of water per day ( 1 L = 1000 cm3). Part A How much depth would a lake lose per year if it uniformly covered an area of 40 km2 and supplied a local town with a population of 70000 people? Consider only population uses, and neglect evaporation and so on. Express your answer using one significant figure. ANSWER: d = 20 cm / y Correct 18-Karat Gold vs. 14-Karat Gold Gold in its pure form is too soft to be used for most jewelry. Therefore, the gold is mixed with other metals to produce an alloy. The composition of gold alloys are always calculated by mass, using the karat (kt) as a unit of measure. A karat represents a proportion by mass of one part in twenty-four. The higher the karat value, the higher the proportion of gold in relation to the total metal content. Pure gold is therefore 24 karat, while an 18-karat gold alloy contains (at least) 18 parts (by mass) of gold out of 24 parts total. Part A In a sample of 18-karat gold, 75 percent of the total mass is pure gold, while the rest is typically 16 percent silver and 9 percent copper. If the density of pure gold is ρ gold = 19.3g / cm3, while the densitites of silver and copper are respectively ρ silver = 10.5g / cm3 and ρ copper = 8.90g / cm3, what is the overall density ρ 18ktrho_18kt of this alloy of 18-karat gold? Express your answer in grams per cubic centimeter to three significant figures. Hint 1. How to approach the problem To compute the density of a material, you need to know its mass per unit volume, or alternatively, the volume of a given mass of that material. In this problem you are given information based on percentages by weight of each metal content of 18-karat gold. Therefore, in order to determine the density of the alloy, it is most convenient to find the total volume of a given mass of 18-karat gold, say 1 gram. The total volume is simply the sum of the volumes of each metal component in the alloy; however, before evaluating the volume of each metal, first you need to determine each metal's mass in one gram of alloy. Hint 2. Find the volume of pure gold in 1 gram of alloy What is the volume V goldV_gold of pure gold in 1 gram of 18-karat gold? Express your answer numerically in cubic centimeters. Hint 1. How are mass, volume, and density related? Consider a piece of a homogeneous material of mass mm whose volume is VV. The density ρrho of that material is given by m ρ = V. If the mass mm and the density ρrho of a material are known, you can then determine the volume VV. Write an expression for volume in terms of mass and density. Express your answer in terms of ρrho and mm. ANSWER: VV = m ρ Hint 2. Find the mass of pure gold in 1 gram of alloy Consider a sample of 18-karat gold; 75 percent of the total mass is pure gold. What is the mass m goldm_gold of pure gold contained in 1 gram of 18-karat gold? Express your answer numerically in grams. ANSWER: m goldm_gold = 0.75 g ANSWER: V goldV_gold = 3.89×10−2 cm 3 ANSWER: ρ 18ktrho_18kt = 15.6 g / cm 3 Correct Part B The percentage composition by mass of the less expensive 14-karat gold is typically 58.5 percent gold, 4.0 percent silver, 31.2 percent copper, and 6.3 percent zinc. Find the percentage composition by volume of 14-karat gold. The density of zinc is ρ zinc = 7.14g / cm 3. Enter four answers as percentages, separated by commas. Round each to the nearest integer. Hint 1. How to approach the problem Find the volume of each metal present in a gram of alloy and add the volumes to obtain the total volume. The percentage composition by volume of a metal is just the volume of that metal in the alloy divided by the total volume of the alloy. Hint 2. Find the volume of pure gold in 1 gram of alloy What is the volume V goldV_gold of pure gold in 1 gram of 14-karat gold? Express your answer numerically in cubic centimeters, to three significant figures. Hint 1. How are mass, volume, and density related? Consider a piece of a homogeneous material of mass mm whose volume is VV. The density ρrho of that material is given by m ρ = V. If the mass mm and the density ρrho of a material are known, you can then determine the volume VV. Write an expression for volume in terms of mass and density. Express your answer in terms of ρrho and mm. ANSWER: VV = m ρ Hint 2. Find the mass of pure gold in 1 gram of alloy Consider a sample of 14-karat gold; 58.5 percent of the total weight is pure gold. What is the mass m goldm_gold of pure gold contained in 1 gram of 14-karat gold? Express your answer numerically in grams. ANSWER: m goldm_gold = 0.585 g ANSWER: V goldV_gold = 3.03×10−2 cm 3 ANSWER: V %gold, V %silver, V %copper, V %zinc = 39,5,45,11 % Correct In a sample of 14-karat gold, less than 50 percent of the volume is pure gold!