Measurement

Question: The sport with the fastest moving ball is jai alai, where measured speeds have reached  303 km/hr. If a professional jai alai player faces a ball at that speed and involuntarily blinks, he blacks out the scene for 100 ms. How far does the ball move during the blackout?

Light containing a mixture of two wavelengths, $\mathbf{500}$  and $\mathbf{600}\mathbf{ }\mathit{n}\mathit{m}$ , is incident normally on a diffraction grating. It is desired (1) that the first and second maxima for each wavelength appear at$\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathbf{⩽}{\mathbf{30}}^{\mathbf{°}}$  , (2) that the dispersion be as high as possible, and (3) that the third order for the $\mathbf{600}\mathbf{ }\mathit{n}\mathit{m}$  light be a missing order. 

(a) What should be the slit separation? 

(b) What is the smallest individual slit width that can be used? 

(c) For the values calculated in (a) and (b) and the light of wavelength$\mathbf{600}\mathbf{ }\mathit{n}\mathit{m}$ , what is the largest order of maxima produced by the grating?

Question: The head of a rattlesnake can accelerate at 50 m/s²  in striking a victim. If a car could do as well, how long would it take to reach a speed of 100 km from rest?

Question: A jumbo jet must reach a speed of 360 km/hr on the runway for takeoff. What is the lowest constant acceleration needed for takeoff from a 1.80 km runway?

Question: An automobile driver increases the speed at a constant rate from 25 km.hr to 55 km/hr in 0.50 min. A bicycle rider speeds up at a constant rate from rest to 30 km/hr in 0.50 min. What are the magnitudes of (a) the driver’s acceleration and (b) the rider’s acceleration?

A point object is 10 cm away from a plane mirror, and the eye of an observer

(with pupil diameter 5.0 mm ) is 20 cm away. Assuming the eye and the object

to be on the same line perpendicular to the mirror surface, find the area of the

mirror used in observing the reflection of the point.

A football player punts the football so that it will have a “hang time” (time of flight) of 4.5 s and land 46 m away. If the ball leaves the player’s foot 150 cm above the ground, what must be the (a) magnitude and (b) angle (relative to the horizontal) of the ball’s initial velocity?

An airport terminal has a moving sidewalk to speed passengers through a long corridor. Larry does not use the moving sidewalk; he takes 150 s to walk through the corridor. Curly, who simply stands on the moving sidewalk, covers the same distance in 70 s. Moe boards the sidewalk and walks along it. How long does Moe take to move through the corridor? Assume that Larry and Moe walk at the same speed.

Figure 29-26 shows four arrangements in which long parallel wires carry equal currents directly into or out of the page at the corners of identical squares. Rank the arrangements according to the magnitude of the net magnetic field at the center of the square, greatest first.

Figure 29-25 represents a snapshot of the velocity vectors of four electrons near a wire carrying current i. The four velocities have the same magnitude; velocity is directed into the page. Electrons 1 and 2 are at the same distance from the wire, as are electrons 3 and 4. Rank the electrons according to the magnitudes of the magnetic forces on them due to current i, greatest first.

Question: The system in Fig. 12-28 is in equilibrium, with the string in the center exactly horizontal. Block A weighs 40 N , block B weighs  50 N , and angle $\mathit{\varphi }$ is 35⁰ . Find (a) tension T₁ , (b) tension  T₂ , (c) tension T₃ , and (d) angle $\mathit{\theta }$ .

Question: Earth is approximately a sphere of radius $6.37\times {10}^{6}m$ . What are (a) its circumference in kilometers, (b) its surface area in square kilometers, and (c) its volume in cubic kilometers?

Question: A gry is an old English measure for length, defined as 1/10  of a line, where line is another old English measure for length, defined as 1/12  inch. A common measure for length in the publishing business is a point, defined as 1/72   inch. What is an area of 0.50gry² in points squared (points²)

Question: The micrometer $\left(1\mu m\right)$   is often called the micron. (a) How many microns make up 1km ? (b) What fraction of a centimeter equals  $\left(1\mu m\right)$? (c) How many microns are in 1.0 yd ?

Question: Spacing in this book was generally done in units of points and picas: 12 Points = 1 pica  , and  6 picas =1 inch. If a figure was misplaced in the page proofs by 0.80cm , what was the misplacement in (a) picas and (b) points?

Question: Horses are to race over a certain English meadow for a distance of . What is the race distance in (a) rods and (b) chains? ( 1 furlong = 201.168m, 1 rod = 5.0292 m and 1 chain =20.117 m)

You can easily convert common units and measures electronically, but you still should be able to use a conversion table, such as those in Appendix D. Table 1-6 is part of a conversion table for a system of volume measures once common in Spain; a volume of 1 fanega is equivalent to 55.501 dm3 (cubic decimeters). To complete the table, what numbers (to three significant figures) should be entered in (a) the cahiz column, (b) the fanega column, (c) the cuartilla column, and (d) the almude column, starting with the top blank? Express 7.00 almudes in (e) medios, (f) cahizes, and (g) cubic centimeters (cm3)

Question: Hydraulic engineers in the United States often use, as a unit of volume of water, the acre-foot, defined as the volume of water that will cover 1 acre of land to a depth of 1 ft. A severe thunderstorm dumped 2.0 in. of rain in 30 min on a town of area 26 km². What volume of water, in acre-feet, fell on the town?

Question: Harvard Bridge, which connects MIT with its fraternities across the Charles River, has a length of 364.4 Smoots plus one ear. The unit of one Smoot is based on the length of Oliver Reed Smoot, Jr., class of 1962, who was carried or dragged length by length across the bridge so that other pledge members of the Lambda Chi Alpha fraternity could mark off (with paint) 1-Smoot lengths along the bridge. The marks have been repainted biannually by fraternity pledges since the initial measurement, usually during times of traffic congestion so that the police cannot easily interfere. (Presumably, the police were originally upset because the Smoot is not an SI base unit, but these days they seem to have accepted the unit.) Figure 1-4 shows three parallel paths, measured in Smoots (S), Willies (W), and Zeldas (Z). What is the length of 50.0 Smoots in (a) Willies and (b) Zeldas?

Figure 1-4 Problem 8

Question: Antarctica is roughly semicircular, with a radius of 2000 km (Fig. 1-5). The average thickness of its ice cover is 3000 m. How many cubic centimeters of ice does Antarctica contain? (Ignore the curvature of Earth.)

Figure 1-5 Problem 9

Until 1883, every city and town in the United States kept its own local time. Today, travelers reset their watches only when the time change equals 1.0 h. How far, on the average, must you travel in degrees of longitude between the time-zone boundaries at which your watch must be reset by 1.0 h? (Hint: Earth rotates 360° in about 24 h.)

Question: For about 10 years after the French Revolution, the French government attempted to base measures of time on multiples of ten: One week consisted of 10 days, one day consisted of 10 hours, one hour consisted of 100 minutes, and one minute consisted of 100 seconds. What are the ratios of (a) the French decimal week to the standard week and (b) the French decimal second to the standard second?

Question: The fastest growing plant on record is a Hesperoyucca whipplei that grew 3.7 m in 14 days. What was its growth rate in micrometers per second?

Question: Three digital clocks A, B, and C run at different rates and do not have simultaneous readings of zero. Figure 1-6 shows simultaneous readings on pairs of the clocks for four occasions. (At the earliest occasion, for example, B reads 25.0 s and C reads 92.0 s.) If two events are 600 s apart on clock A, how far apart are they on (a) clock B and (b) clock C? (c) When clock A reads 400 s, what does clock B read? (d) When clock C reads 15.0 s, what does clock B read? (Assume negative readings for prezero times.)

Figure 1-6 Problem 13

Question: A lecture period (50 min) is close to 1 microcentury. (a) How long is a microcentury in minutes? (b) Using 

, $\mathit{p}\mathit{e}\mathit{r}\mathit{c}\mathit{e}\mathit{n}\mathit{t}\mathit{a}\mathit{g}\mathit{e}\mathbf{ }\mathit{d}\mathit{i}\mathit{f}\mathit{f}\mathit{e}\mathit{r}\mathit{e}\mathit{n}\mathit{c}\mathit{e}\mathbf{=}\left(\frac{actual -approximation}{actual}\right)\mathbf{\times }\mathbf{100}$

find the percentage difference from the approximation.

A fortnight is a charming English measure of time equal to 2.0 weeks (the word is a contraction of “fourteen nights”). That is a nice amount of time in pleasant company but perhaps a painful string of microseconds in unpleasant company. How many microseconds are in a fortnight?

Time standards are now based on atomic clocks. A promising second standard is based on pulsars, which are rotating neutron stars (highly compact stars consisting only of neutrons). Some rotate at a rate that is highly stable, sending out a radio beacon that sweeps briefly across Earth once with each rotation, like a lighthouse beacon. Pulsar PSR 1937 + 21 is an example; it rotates once every 1.557 806 448 872 75 $\pm$ 3 ms, where the trailing $\pm$3 indicates the uncertainty in the last decimal place (it does not mean $\pm$3 ms). (a) How many rotations does PSR 1937 + 21 make in 7.00 days? (b) How much time does the pulsar take to rotate exactly one million times and (c) what is the associated uncertainty?

Question: Earth has a mass of $5.98\times {10}^{24}kg$ . The average mass of the atoms that make up Earth is 40 u. How many atoms are there in Earth?

Five clocks are being tested in a laboratory. Exactly at noon, as determined by the WWV time signal, on successive days of a week the clocks read as in the following table. Rank the five clocks according to their relative value as good timekeepers, best to worst. Justify your choice

Because Earth’s rotation is gradually slowing, the length of each day increases: The day at the end of 1.0 century is 1.0 ms longer than the day at the start of the century. In 20 centuries, what is the total of the daily increases in time?

Suppose that, while lying on a beach near the equator watching the Sunset over a calm ocean, you start a stopwatch just as the top of the Sun disappears. You then stand, elevating your eyes by a height H = 1.70 m, and stop the watch when the top of the Sun again disappears. If the elapsed time is t = 11.1 s, what is the radius r of Earth?

Question: The record for the largest glass bottle was set in 1992 by a team in Millville, New Jersey—they blew a bottle with a volume of 193 U.S. fluid gallons. (a) How much short of 1.0 million cubic centimeters is that? (b) If the bottle were filled with water at the leisurely rate of 1.8 g/min, how long would the filling take? Water has a density of 1000 kg/m³_.

Gold, which has a density of $\mathbf{19}\mathbf{.}\mathbf{32}\mathbf{ }\mathbf{g}\mathbf{/}{\mathbf{cm}}^{\mathbf{3}}$, is the most ductile metal and can be pressed into a thin leaf or drawn out into a long fiber. (a) If a sample of gold, with a mass of 27.63 g, is pressed into a leaf of 1.000 µm thickness, what is the area of the leaf? (b) If, instead, the gold is drawn out into a cylindrical fiber of radius 2.500 µm, what is the length of the fiber?

Question: (a) Assuming that water has a density of exactly 1 gm/cm³, find the mass of one cubic meter of water in kilograms. (b) Suppose that it takes 10.0 h to drain a container of 5700 m³ of water. What is the “mass flow rate,” in kilograms per second, of water from the container?

Grains of fine California beach sand are approximately spheres with an average radius of $50 \mu m$and are made of silicon dioxide, which has a density of $\mathbf{2600}\mathbf{kg}/{\mathbf{m}}^{\mathbf{3}}$.What mass of sand grains would have a total surface area (the total area of all the individual spheres) equal to the surface area of a cube 1.00 m on an edge?

During heavy rain, a section of a mountainside measuring 2.5 km horizontally, 0.80 km up along the slope, and 2.0 m deep slips into a valley in a mud slide. Assume that the mud ends up uniformly distributed over a surface area of the valley measuring 0.40 km × 0.40 km and that mud has a density of . What is the mass of the mud sitting above a 4.0 m² area of the valley floor?

One cubic centimeter of a typical cumulus cloud contains 50 to 500 water drops, which have a typical radius of 10 µm. For that range, give the lower value and the higher value, respectively, for the following. (a) How many cubic meters of water are in a cylindrical cumulus cloud of height 3.0 km and radius 1.0 km? (b) How many 1-liter pop bottles would that water fill? (c) Water has a density of $\mathbf{1000}\mathbf{ }\mathbf{kg}\mathbf{ }\mathbf{/}{\mathbf{m}}^{\mathbf{2}}$. How much mass does the water in the cloud have?

Iron has a density of $\mathbf{7}\mathbf{.}\mathbf{87}\mathbf{ }\mathbf{g}\mathbf{/}{\mathbf{cm}}^{\mathbf{3}}$ , and the mass of an iron atom is   $\mathbf{9}\mathbf{.}\mathbf{27}\mathbf{\times }{\mathbf{10}}^{\mathbf{‐}\mathbf{26}}\mathbf{ }\mathbf{kg}$ . If the atoms are spherical and tightly packed, (a) what is the volume of an iron atom, and (b) what is the distance between the centers of adjacent atoms?

A mole of atoms is $\mathbf{6}\mathbf{.}\mathbf{20}\mathbf{\times }{\mathbf{10}}^{\mathbf{23}}$atoms. To the nearest order of magnitude, how many moles of atoms are in a large domestic cat? The masses of a hydrogen atom, an oxygen atom, and a carbon atom are 1.0 u, 16 u, and 12 u, respectively. (Hint: Cats are sometimes known to kill a mole.)

 On a spending spree in Malaysia, you buy an ox with a weight of 28.9 piculs in the local unit of weights: 1 picul $\mathbf{=}$100 gins, 1 gin $\mathbf{=}$ 16 tahils, 1 tahil $\mathbf{=}$ 10 chees, and 1 chee $\mathbf{=}$ 10 hoons. The weight of 1 hoon corresponds to a mass of 0.3779 g. When you arrange to ship the ox home to your astonished family, how much mass in kilograms must you declare on the shipping manifest? (Hint: Set up multiple chain-link conversions.)

 Water is poured into a container that has a small leak. The mass m of the water is given as a function of time t by $\mathbf{m}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{5}\mathbf{.}\mathbf{00}{\mathbf{t}}^{\mathbf{0}\mathbf{.}\mathbf{8}\mathbf{ }}\mathbf{‐}\mathbf{ }\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{t}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{20}\mathbf{.}\mathbf{00}$, with $t \ge 0$, m in grams, and t in seconds. (a) At what time is the water mass greatest, and (b) what is that greatest mass? In kilograms per minute, what is the rate of mass change at (c) $\mathbf{t}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{s}$and (d) $\mathbf{t}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{s}$ ?

A vertical container with base area measuring $\mathbf{14}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{cm}\mathbf{ }\mathbf{by}\mathbf{ }\mathbf{17}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{cm}\mathbf{ }$ is being filled with identical pieces of candy, each with a volume of $\mathbf{50}\mathbf{.}\mathbf{0}\mathbf{ }{\mathbf{mm}}^{\mathbf{3}}$  and a mass of $\mathbf{0}\mathbf{.}\mathbf{0200}\mathbf{ }\mathbf{g}$ . Assume that the volume of the empty spaces between the candies is negligible. If the height of the candies in the container increases at the rate of $\mathbf{0}\mathbf{.}\mathbf{250}\mathbf{ }\mathbf{cm}\mathbf{/}\mathbf{s}$ , at what rate (kilograms per minute) does the mass of the candies in the container increase?

In the United States, a doll house has the scale of $\mathbf{1}\mathbf{:}\mathbf{12}$  of a real house (that is, each length of the doll house is $\frac{\mathbf{1}}{\mathbf{12}}$  that of the real house) and a miniature house (a doll house to fit within a doll house) has the scale of $\mathbf{1}\mathbf{:}\mathbf{144}$ of a real house. Suppose a real house (Fig. 1-7) has a front length of 20 m, a depth of 12 m, a height of 6.0 m, and a standard sloped roof (vertical triangular faces on the ends) of height 3.0 m. In cubic meters, what are the volumes of the corresponding (a) doll house and (b) miniature house?

Figure 1-7 Problem 32

A ton is a measure of volume frequently used in shipping, but that use requires some care because there are at least three types of tons: A displacement ton is equal to 7 barrels bulk, a freight ton is equal to 8 barrels bulk, and a register ton is equal to 20 barrels bulk. A barrel bulk is another measure of volume: $\mathbf{1}\mathbf{ }\mathbf{barrel}\mathbf{ }\mathbf{bulk}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{0}\mathbf{.}\mathbf{1415}\mathbf{ }{\mathbf{m}}^{\mathbf{3}}$. Suppose you spot a shipping order for “73 tons” of  $\mathbf{M}\mathbf{\&}\mathbf{M}$ candies, and you are certain that the client who sent the order intended “ton” to refer to volume (instead of weight or mass, as discussed in Chapter 5). If the client actually meant displacement tons, how many extra U.S. bushels of the candies will you erroneously ship if you interpret the order as (a) 73 freight tons and (b) 73 register tons? ($\mathbf{1}{\mathbf{m}}^{\mathbf{3}}\mathbf{=}\mathbf{28}\mathbf{.}\mathbf{378}$U.S. bushels.)

Two types of barrel units were in use in the 1920s in the United States. The apple barrel had a legally set volume of 7056 cubic inches; the cranberry barrel, 5826 cubic inches. If a merchant sells 20 cranberry barrels of goods to a customer who thinks he is receiving apple barrels, what is the discrepancy in the shipment volume in liters?

An old English children’s rhyme states, “Little Miss Muffet sat on a tuffet, eating her curds and whey, when along came a spiders who sat down beside her. . . .” The spider sat down not because of the curds and whey but because Miss Muffet had a stash of 11 tuffets of dried flies. The volume measure of a tuffet is given by 1 tuffet ${}^{\mathbf{=}}$ 2 pecks ${}^{\mathbf{=}}$ 0.50 Imperial bushel, where 1 Imperial bushel ${}^{\mathbf{=}}$ 36.3687 liters (L). What was Miss Muffet’s stash in (a) pecks, (b) Imperial bushels, and (c) liters?

Table 1-7 shows some old measures of liquid volume. To complete the table, what numbers (to three significant figures) should be entered in (a) the wey column, (b) the chaldron column, (c) the bag column, (d) the pottle column, and (e) the gill column, starting from the top down? (f) The volume of 1 bag is equal to 0.1091 m3. If an old story has a witch cooking up some vile liquid in a cauldron of volume 1.5 chaldrons, what is the volume in cubic meters?

A typical sugar cube has an edge length of 1 cm. If you had a cubical box that contained a mole of sugar cubes, what would its edge length be? (${}^{\mathbf{One}\mathbf{ }\mathbf{mole}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{02}\mathbf{\times }{\mathbf{10}}^{\mathbf{23}\mathbf{ }\mathbf{units}}}$ .)

An old manuscript reveals that a landowner in the time of King Arthur held 3.00 acres of plowed land plus a livestock area of 25.0 perches by 4.00 perches. What was the total area in (a) the old unit of roods and (b) the more modern unit of square meters? Here, 1 acre is an area of 40 perches by 4 perches, 1 rood is an area of 40 perches by 1 perch, and 1 perch is the length 16.5 ft.

A tourist purchases a car in England and ships it home to the United States. The car sticker advertised that the car’s fuel consumption was at the rate of 40 miles per gallon on the open road. The tourist does not realize that the U.K. gallon differs from the U.S. gallon: 

    $\mathbf{1}\mathbf{U}\mathbf{.}\mathbf{K}\mathbf{.}\mathbf{gallon}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{546}\mathbf{ }\mathbf{090}\mathbf{ }\mathbf{0}\mathbf{ }\mathbf{liters}$

    $\mathbf{1}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{.}\mathbf{gallon}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{785}\mathbf{ }\mathbf{411}\mathbf{ }\mathbf{8}\mathbf{ }\mathbf{liters}$

For a trip of 750 miles (in the United States), how many gallons of fuel does (a) the mistaken tourist believe she needs and (b) the car actually require?