Chapter 1 1. THINK In this problem we’re given the radius of Earth, and asked to compute its circumference, surface area and volume. EXPRESS Assuming Earth to be a sphere of radius R  6.37106m103km m  6.37103 km, E the corresponding circumference, surface area and volume are: 4 C 2R , A4R2, V  R3. E E 3 E The geometric formulas are given in Appendix E. ANALYZE (a) Using the formulas given above, we find the circumference to be C 2R 2(6.37103 km)4.00104 km. E (b) Similarly, the surface area of Earth is  2 A4R2  4 6.37 103 km  5.10108 km2, E (c) and its volume is 4 4 3 V  R3  6.37 103 km 1.081012 km3. 3 E 3 LEARN From the formulas given, we see that C R , A R2, and V R3. The ratios E E E of volume to surface area, and surface area to circumference are V /AR /3 and E A/C 2R . E 2. The conversion factors are: 1 gry1/10 line, 1 line1/12 inchand 1 point = 1/72 inch. The factors imply that 1 gry = (1/10)(1/12)(72 points) = 0.60 point. Thus, 1 gry2 = (0.60 point)2 = 0.36 point2, which means that 0.50 gry2= 0.18 point2. 3. The metric prefixes (micro, pico, nano, …) are given for ready reference on the inside front cover of the textbook (see also Table 1–2). 1 2 CHAPTER 1 (a) Since 1 km = 1  103 m and 1 m = 1  106 m,    1km 103m  103m 106 m m 109 m. The given measurement is 1.0 km (two significant figures), which implies our result should be written as 1.0  109 m. (b) We calculate the number of microns in 1 centimeter. Since 1 cm = 102 m, 1cm =102m = 102m106m m 104 m. We conclude that the fraction of one centimeter equal to 1.0 m is 1.0  104. (c) Since 1 yd = (3 ft)(0.3048 m/ft) = 0.9144 m, 1.0yd = 0.91m106m m  9.1105 m. 4. (a) Using the conversion factors 1 inch = 2.54 cm exactly and 6 picas = 1 inch, we obtain  1inch 6 picas 0.80 cm = 0.80 cm    1.9 picas. 2.54 cm 1inch  (b) With 12 points = 1 pica, we have  1inch 6 picas12 points 0.80 cm = 0.80 cm      23 points. 2.54 cm 1inch  1pica  5. THINK This problem deals with conversion of furlongs to rods and chains, all of which are units for distance. EXPRESS Given that 1 furlong  201.168 m,1rod5.0292 m and 1chain20.117 m, the relevant conversion factors are 1 rod 1.0 furlong 201.168 m(201.168 m) 40 rods, 5.0292 m and 1 chain 1.0 furlong 201.168 m(201.168 m) 10 chains. 20.117 m Note the cancellation of m (meters), the unwanted unit. ANALYZE Using the above conversion factors, we find 40 rods (a) the distance d in rods to be d  4.0 furlongs 4.0 furlongs 160 rods, 1 furlong 3 10 chains (b) and in chains to be d  4.0 furlongs 4.0 furlongs  40 chains. 1 furlong LEARN Since 4 furlongs is about 800 m, this distance is approximately equal to 160 rods (1 rod  5 m) and 40 chains (1 chain  20 m). So our results make sense. 6. We make use of Table 1-6. (a) We look at the first (“cahiz”) column: 1 fanega is equivalent to what amount of cahiz? We note from the already completed part of the table that 1 cahiz equals a dozen fanega. Thus, 1 fanega = 1 cahiz, or 8.33  102 cahiz. Similarly, “1 cahiz = 48 cuartilla” (in the 12 already completed part) implies that 1 cuartilla = 1 cahiz, or 2.08  102 cahiz. 48 Continuing in this way, the remaining entries in the first column are 6.94  103 and 3.47103. (b) In the second (“fanega”) column, we find 0.250, 8.33  102, and 4.17  102 for the last three entries. (c) In the third (“cuartilla”) column, we obtain 0.333 and 0.167 for the last two entries. 1 (d) Finally, in the fourth (“almude”) column, we get = 0.500 for the last entry. 2 (e) Since the conversion table indicates that 1 almude is equivalent to 2 medios, our amount of 7.00 almudes must be equal to 14.0 medios. (f) Using the value (1 almude = 6.94  103 cahiz) found in part (a), we conclude that 7.00 almudes is equivalent to 4.86  102 cahiz. (g) Since each decimeter is 0.1 meter, then 55.501 cubic decimeters is equal to 0.055501 m3 or 55501 cm3. Thus, 7.00 almudes = 7.00 fanega = 7.00 (55501 cm3) = 3.24  104 cm3. 12 12 7. We use the conversion factors found in Appendix D. 1 acreft = (43,560 ft2)ft = 43,560 ft3 Since 2 in. = (1/6) ft, the volume of water that fell during the storm is V (26 km2)(1/6 ft)(26 km2)(3281ft/km)2(1/6 ft)  4.66107 ft3. Thus, 4.66  107 ft3 V   1.1103 acre ft. 4.3560  104 ft3 acreft 4 CHAPTER 1 8. From Fig. 1-4, we see that 212 S is equivalent to 258 W and 212 – 32 = 180 S is equivalent to 216 – 60 = 156 Z. The information allows us to convert S to W or Z. (a) In units of W, we have 258 W 50.0 S  50.0 S    60.8 W  212 S  (b) In units of Z, we have 156 Z 50.0 S  50.0 S    43.3 Z 180 S 9. The volume of ice is given by the product of the semicircular surface area and the thickness. The area of the semicircle is A = r2/2, where r is the radius. Therefore, the volume is  V  r2 z 2 where z is the ice thickness. Since there are 103 m in 1 km and 102 cm in 1 m, we have 103m 102cm r  2000km      2000105 cm.  1km   1m  In these units, the thickness becomes 102cm z3000m 3000m    3000102 cm  1m    2   which yields V  2000105 cm 3000102cm 1.91022 cm3. 2 10. Since a change of longitude equal to 360corresponds to a 24 hour change, then one expects to change longitude by360/2415 before resetting one's watch by 1.0 h. 11. (a) Presuming that a French decimal day is equivalent to a regular day, then the ratio of weeks is simply 10/7 or (to 3 significant figures) 1.43. (b) In a regular day, there are 86400 seconds, but in the French system described in the problem, there would be 105 seconds. The ratio is therefore 0.864. 12. A day is equivalent to 86400 seconds and a meter is equivalent to a million micrometers, so 5 b gc h 3.7m 106m m b gb g  3.1m s. 14day 86400s day 13. The time on any of these clocks is a straight-line function of that on another, with slopes  1 and y-intercepts  0. From the data in the figure we deduce 2 594 33 662 t  t  , t  t  . C 7 B 7 B 40 A 5 These are used in obtaining the following results. (a) We find 33 t  t  t  t   495 s B B 40 A A when t'  t = 600 s. A A 2 b g 2 b g (b) We obtain t  t  t  t  495  141s. C C 7 B B 7 (c) Clock B reads t = (33/40)(400) (662/5)  198 s when clock A reads t = 400 s. B A (d) From t = 15 = (2/7)t + (594/7), we get t  245 s. C B B 14. The metric prefixes (micro (), pico, nano, …) are given for ready reference on the inside front cover of the textbook (also Table 1–2). (a) 1century  106 century  100 y  365 day  24 h  60 min 52.6 min. 1century  1 y  1day  1h  (b) The percent difference is therefore 52.6 min  50 min  4.9%. 52.6 min 15. A week is 7 days, each of which has 24 hours, and an hour is equivalent to 3600 seconds. Thus, two weeks (a fortnight) is 1209600 s. By definition of the micro prefix, this is roughly 1.21  1012 s. 16. We denote the pulsar rotation rate f (for frequency). 1rotation f  1.55780644887275103 s 6 CHAPTER 1 (a) Multiplying f by the time-interval t = 7.00 days (which is equivalent to 604800 s, if we ignore significant figure considerations for a moment), we obtain the number of rotations:  1rotation  N    604800 s  388238218.4 1.55780644887275103 s which should now be rounded to 3.88  108 rotations since the time-interval was specified in the problem to three significant figures. (b) We note that the problem specifies the exact number of pulsar revolutions (one million). In this case, our unknown is t, and an equation similar to the one we set up in part (a) takes the form N = ft, or  1rotation  1106    t 1.55780644887275103 s which yields the result t = 1557.80644887275 s (though students who do this calculation on their calculator might not obtain those last several digits). (c) Careful reading of the problem shows that the time-uncertainty per revolution is 31017s . We therefore expect that as a result of one million revolutions, the uncertainty should be (31017)(1106)= 31011 s. 17. THINK In this problem we are asked to rank 5 clocks, based on their performance as timekeepers. EXPRESS We first note that none of the clocks advance by exactly 24 h in a 24-h period but this is not the most important criterion for judging their quality for measuring time intervals. What is important here is that the clock advance by the same (or nearly the same) amount in each 24-h period. The clock reading can then easily be adjusted to give the correct interval. ANALYZE The chart below gives the corrections (in seconds) that must be applied to the reading on each clock for each 24-h period. The entries were determined by subtracting the clock reading at the end of the interval from the clock reading at the beginning. Clocks C and D are both good timekeepers in the sense that each is consistent in its daily drift (relative to WWF time); thus, C and D are easily made “perfect” with simple and predictable corrections. The correction for clock C is less than the correction for clock D, so we judge clock C to be the best and clock D to be the next best. The correction that must be applied to clock A is in the range from 15 s to 17s. For clock B it is the range from 5 s to +10 s, for clock E it is in the range from 70 s to 2 s. After C and D, A has 7 the smallest range of correction, B has the next smallest range, and E has the greatest range. From best to worst, the ranking of the clocks is C, D, A, B, E. CLOCK Sun. Mon. Tues. Wed. Thurs. Fri. -Mon. -Tues. -Wed. -Thurs. -Fri. -Sat. A 16 16 15 17 15 15 B 3 +5 10 +5 +6 7 C 58 58 58 58 58 58 D +67 +67 +67 +67 +67 +67 E +70 +55 +2 +20 +10 +10 LEARN Of the five clocks, the readings in clocks A, B and E jump around from one 24- h period to another, making it difficult to correct them. 18. The last day of the 20 centuries is longer than the first day by 20 century 0.001s century  0.02 s. The average day during the 20 centuries is (0 + 0.02)/2 = 0.01 s longer than the first day. Since the increase occurs uniformly, the cumulative effect T is T  average increase in length of a daynumber of days 0.01s 365.25 day      2000 y  day   y   7305 s or roughly two hours. 19. When the Sun first disappears while lying down, your line of sight to the top of the Sun is tangent to the Earth’s surface at point A shown in the figure. As you stand, elevating your eyes by a height h, the line of sight to the Sun is tangent to the Earth’s surface at point B. Let d be the distance from point B to your eyes. From the Pythagorean theorem, we have d2 r2 (rh)2 r2 2rhh2 8 CHAPTER 1 or d2 2rhh2,where r is the radius of the Earth. Since r h, the second term can be dropped, leading to d2 2rh. Now the angle between the two radii to the two tangent points A and B is , which is also the angle through which the Sun moves about Earth during the time interval t = 11.1 s. The value of  can be obtained by using  t  . 360 24 h This yields (360)(11.1 s)  0.04625. (24 h)(60 min/h)(60 s/min) Using d rtan, we have d2 r2tan22rh, or 2h r  tan2 Using the above value for  and h = 1.7 m, we have r 5.2106 m. 20. (a) We find the volume in cubic centimeters 3 231in3  2.54cm 193gal = 193gal      7.31105cm3  1gal   1in  and subtract this from 1  106 cm3 to obtain 2.69  105 cm3. The conversion gal  in3 is given in Appendix D (immediately below the table of Volume conversions). (b) The volume found in part (a) is converted (by dividing by (100 cm/m)3) to 0.731 m3, which corresponds to a mass of c h c h 1000 kg m3 0.731 m2 = 731 kg using the density given in the problem statement. At a rate of 0.0018 kg/min, this can be filled in 731kg  4.06105min = 0.77 y 0.0018 kg min after dividing by the number of minutes in a year (365 days)(24 h/day) (60 min/h). 21. If M is the mass of Earth, m is the average mass of an atom in Earth, and N is the E number of atoms, then M = Nm or N = M /m. We convert mass m to kilograms using E E Appendix D (1 u = 1.661  1027 kg). Thus, 9 M 5.98  1024 kg N  E  b g c h  9.0  1049. m 40 u 1.661 1027 kg u 22. The density of gold is m 19.32g   19.32g/cm3. V 1 cm3 (a) We take the volume of the leaf to be its area A multiplied by its thickness z. With density  = 19.32 g/cm3 and mass m = 27.63 g, the volume of the leaf is found to be m V   1.430 cm3.  We convert the volume to SI units: 3 V 1.430cm3  1m   1.430 106 m3. 100 cm Since V = Az with z = 1  10-6 m (metric prefixes can be found in Table 1–2), we obtain 1.430  106 m3 A   1.430 m2. 1 106 m (b) The volume of a cylinder of length  is V  A where the cross-section area is that of a circle: A = r2. Therefore, with r = 2.500  106 m and V = 1.430  106 m3, we obtain V  7.284104 m72.84 km. r2 23. THINK This problem consists of two parts: in the first part, we are asked to find the mass of water, given its volume and density; the second part deals with the mass flow rate of water, which is expressed as kg/s in SI units. EXPRESS From the definition of density:  m/V , we see that mass can be calculated as mV, the product of the volume of water and its density. With 1 g = 1  103 kg and 1 cm3 = (1  102m)3 = 1  106m3, the density of water in SI units (kg/m3) is  1g  103 kg  cm3  1g/cm3        1103 kg m3. cm3   g  106 m3  To obtain the flow rate, we simply divide the total mass of the water by the time taken to drain it. ANALYZE (a) Using mV, the mass of a cubic meter of water is