Measurement

Using conversions and data in the chapter, determine the number of hydrogen atoms required to obtain${}^{\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{kg}}$ of hydrogen. A hydrogen atom has a mass of ${}^{\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{u}}$.

A cord is a volume of cut wood equal to a stack 8 ft long, 4 ft wide, and 4 ft high. How many cords are in${}^{\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{ }{\mathbf{m}}^{\mathbf{3}}}$?

One molecule of water${}^{\left(H_{2}O\right)}$ contains two atoms of hydrogen and one atom of oxygen. A hydrogen atom has a mass of${}^{\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{u}}$ and an atom of oxygen has a mass of ${}^{\mathbf{16}\mathbf{u}}$ , approximately. (a) What is the mass in kilograms of one molecule of water? (b) How many molecules of water are in the world’s oceans, which have an estimated total mass of ${}^{1.4\times {10}^{21}\mathbf{ }\mathbf{kg}}$ ?

A person on a diet might lose ${}^{\mathbf{2}\mathbf{.}\mathbf{3}\mathbf{ }\mathbf{kg}\mathbf{ }\mathbf{per}\mathbf{ }\mathbf{week}}$. Express the mass loss rate in milligrams per second, as if the dieter could sense the second-by-second loss.

A unit of time sometimes used in microscopic physics is the shake. One shake equals ${}^{{\mathbf{10}}^{\mathbf{-}\mathbf{8}}}$ s. Are there more shakes in a second than there are seconds in a year? (b) Humans have existed for about ${}^{{\mathbf{10}}^{\mathbf{6}}}$ years, whereas the universe is about ${}^{{\mathbf{10}}^{\mathbf{10}}}$ years old. If the age of the universe is defined as${}^{\mathbf{1}}$ “universe day,” where a universe day consists of “universe seconds” as a normal day consists of normal seconds, how many universe seconds have humans existed?

A unit of area often used in measuring land areas is the hectare, defined as${\mathbf{10}}^{\mathbf{4}}{\mathbf{m}}^{\mathbf{2}}$ . An open-pit coal mine consumes $\mathbf{75}$ hectares of land, down to a depth of $\mathbf{26}\mathbf{ }\mathbf{m}$ , each year. What volume of earth, in cubic kilometers, is removed in this time?

An astronomical unit (AU) is the average distance between Earth and the Sun, approximately ${}^{\mathbf{1}\mathbf{.}\mathbf{50}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\mathbf{ }\mathbf{km}}$ . The speed of light is about ${}^{\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}}$. Express the speed of light in astronomical units per minute.

The common Eastern mole, a mammal, typically has a mass of ${}^{\mathbf{75}\mathbf{ }\mathbf{g}}$ , which corresponds to about ${}^{\mathbf{7}\mathbf{.}\mathbf{5}\mathbf{ }\mathbf{moles}}$ of atoms. (A mole of atoms is${}^{\mathbf{6}\mathbf{.}\mathbf{02}\mathbf{\times }{\mathbf{10}}^{\mathbf{23}}\mathbf{ }\mathbf{atoms}}$.) In atomic mass units (u), what is the average mass of the atoms in the common Eastern mole?

A traditional unit of length in Japan is the ken (${}^{1 \mathrm{ken}=1.97 \mathrm{m}}$ ). What are the ratios of (a) square kens to square meters and (b) cubic kens to cubic meters? What is the volume of a cylindrical water tank of height ${}^{5.50 \mathrm{kens}}$ and radius ${}^{3.00 \mathrm{kens}}$ in (c) cubic kens and (d) cubic meters?

You receive orders to sail due east for $24.5\mathbf{ }\mathbf{mi}$  to put your salvage ship directly over a sunken pirate ship. However, when your divers probe the ocean floor at that location and find no evidence of a ship, you radio back to your source of information, only to discover that the sailing distance was supposed to be $24.5\mathbf{ }\mathbf{nautical}\mathbf{ }\mathbf{miles}$ , not regular miles. Use the Length table in Appendix D to calculate how far horizontally you are from the pirate ship in kilometers.

The cubit is an ancient unit of length based on the distance between the elbow and the tip of the middle finger of the measurer. Assume that the distance ranged from ${}^{43 \mathrm{to} 53 \mathrm{cm}}$ , and suppose that ancient drawings indicate that a cylindrical pillar was to have a length of ${}^{9 \mathrm{cubits}}$ and a diameter of . For the stated range, what are the lower value and the upper value, respectively${}^{2 \mathrm{cubits}}$ , for (a) the cylinder’s length in meters, (b) the cylinder’s length in millimeters, and (c) the cylinder’s volume in cubic meters?

As a contrast between the old and the modern and between the large and the small, consider the following: In old rural England 1 hide (between 100 and 120 acres) was the area of land needed to sustain one family with a single plough for one year. (An area of 1 acre is equal to 4047 m2.) Also, 1 wapentake was the area of land needed by 100 such families. In quantum physics, the cross-sectional area of a nucleus (defined in terms of the chance of a particle hitting and being absorbed by it) is measured in units of barns, where 1 barn is ${}^{\mathbf{1}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{28}}\mathbf{ }{\mathbf{m}}^{\mathbf{2}}}$ . (In nuclear physics jargon, if a nucleus is “large,” then shooting a particle at it is like shooting a bullet at a barn door, which can hardly be missed.) What is the ratio of ${}^{\mathbf{15}}$ wapentakes to ${}^{\mathbf{11}}$ barns?

An astronomical unit (AU) is equal to the average distance from Earth to the Sun, about $92.9\times {10}^6 \mathrm{mi}$. A parsec (pc) is the distance at which a length of 1 AU would subtend an angle of exactly 1 second of arc (Fig. 1-8). A light-year (ly) is the distance that light, traveling through a vacuum with a speed of $186000 \mathrm{mi}/\mathrm{s}$ , would cover in 1.0 year . Express the Earth – Sun distance in (a) parsecs and (b) light-years.

The description for a certain brand of house paint claims a coverage of ${}^{460 {\mathrm{ft}}^2 /\mathrm{gal}}$. (a) Express this quantity in square meters per liter. (b) Express this quantity in an SI unit (see Appendices A and D). (c) What is the inverse of the original quantity, and (d) what is its physical significance?

Strangely, the wine for a large wedding reception is to be served in a stunning cut-glass receptacle with the interior dimensions of ${}^{40\times 40\times 30}$ (height). The receptacle is to be initially filled to the top. The wine can be purchased in bottles of the sizes given in the following table. Purchasing a larger bottle instead of multiple smaller bottles decreases the overall cost of the wine. To minimize the cost, (a) which bottle sizes should be purchased and how many of each should be purchased and, once the receptacle is filled, how much wine is left over in terms of (b) standard bottles and (c) liters?

1 standard bottle

1 magnum = 2 standard bottles

1 jeroboam = 4 standard bottles

1 rehoboam = 6 standard bottles

1 methuselah = 8 standard bottles

1 salmanazar = 12 standard bottles

1 balthazar = 16 standard bottles = 11.356 L

1 nebuchadnezzar =20 standard bottles

The corn–hog ratio is a financial term used in the pig market and presumably is related to the cost of feeding a pig until it is large enough for market. It is defined as the ratio of the market price of a pig with a mass of 3.108 slugs to the market price of a U.S. bushel of corn. (The word “slug” is derived from an old German word that means “to hit”; we have the same meaning for “slug” as a verb in modern English.) A U.S. bushel is equal to 35.238 L. If the corn–hog ratio is listed as 5.7 on the market exchange, what is it in the metric units of,

                                               $\frac{\mathrm{price} \mathrm{of} 1 \mathrm{kilogram} \mathrm{of} \mathrm{pig}}{\mathrm{price} \mathrm{of} 1 \mathrm{liter} \mathrm{of} \mathrm{corn}}?$

(Hint: See the Mass table in Appendix D.)

You are to fix dinners for 400 people at a convention of Mexican food fans. Your recipe calls for 2 jalapeño peppers per serving (one serving per person). However, you have only habanero peppers on hand. The spiciness of peppers is measured in terms of the scoville heat unit (SHU). On average, one jalapeño pepper has a spiciness of 4000 SHU and one habanero pepper has a spiciness of 300000 SHU. To get the desired spiciness, how many habanero peppers should you substitute for the jalapeño peppers in the recipe for the 400 dinners?

A standard interior staircase has steps each with a rise (height) of 19 cm and a run (horizontal depth) of 23 cm. Research suggests that the stairs would be safer for descent if the run were, instead, 28 cm. For a particular staircase of total height 4.57 m, how much farther into the room would the staircase extend if this change in run were made?

In purchasing food for a political rally, you erroneously order shucked medium-size Pacific oysters (which come 8 to 12 per U.S. pint) instead of shucked medium-size Atlantic oysters (which come 26 to 38 per U.S. pint). The filled oyster container shipped to you has the interior measure of$\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{m}\mathbf{\times }\mathbf{12}\mathbf{cm}\mathbf{\times }\mathbf{20}\mathbf{cm}$, and a U.S. pint is equivalent to 0.4732 liter . By how many oysters is the order short of your anticipated count?

An old English cookbook carries this recipe for cream of nettle soup: “Boil stock of the following amount: 1 breakfastcup plus 1 teacup plus 6 tablespoons plus 1 dessertspoon. Using gloves, separate nettle tops until you have 0.5 quart; add the tops to the boiling stock. Add 1 tablespoon of cooked rice and 1 saltspoon of salt. Simmer for 15 min.” The following table gives some of the conversions among old (premetric) British measures and among common (still premetric) U.S. measures. (These measures just scream for metrication.) For liquid measures, 1 British teaspoon = 1 U.S. teaspoon. For dry measures, 1 British teaspoon = 2 U.S. teaspoons and 1 British quart =1 U.S. quart. In U.S. measures, how much (a) stock, (b) nettle tops, (c) rice, and (d) salt are required in the recipe?