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Solutions

Manual for College Physics

2nd Edition by Urone, Hinrichs

ISBN: 9781711470832

CHAPTER 1: INTRODUCTION: THE NATURE OF SCIENCE AND PHYSICS

1.2 PHYSICAL QUANTITIES AND UNITS

1. 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?

Solution (a) 100km h × 1000m 1km × 1h 3600s = 2777m/s= 278 m/s

(b) 100km h × 1mi 1609km = 62mi/h

2. A car is traveling at a speed of 33m/s. (a) What is its speed in kilometers per hour? (b) Is it exceeding the 90km/h speed limit?

Solution (a) 33m s × 1km 1000m × 3600s 1h = 1188km/h= 12×102 km/h (b) At 120 km/h, the car is travelling faster than the speed limit.

3. Show that 1.0 ��/�� =3.6��/ℎ. Hint: Show the explicit steps involved in converting 1.0 ��/�� =3.6��/ℎ.

Solution 10m s = 10m s × 3600s hr × 1km 1000m = 3.6km/h

4. American football is played on a 100-yd-long field, excluding the end zones. How long is the field in meters? (Assume that 1 meter equals 3.281 feet.)

Solution

100yd× 3ft 1yd × 1m 3281ft = 91.44m = 91.4m

5. Soccer fields vary in size. A large soccer field is 115 m long and 85 m wide. What are its dimensions in feet and inches? (Assume that 1 meter equals 3.281 feet.)

Solution

115m× 1ft

03048m = 3773ft= 377ftlong

3773ft × 12in 1.0ft =4528in =453×103 in long

85m × 1ft

03048m =278.9ft=2.8×102 ft wide

2789ft × 12in. 1.0ft =3346in =33×103 in wide

6. What is the height in meters of a person who is 6 ft 1.0 in. tall? (Assume that 1 meter equals 39.37 in.)

Solution 6ft, 10in =(6ft× 12in ft )+10in =730in ; 730in× 1m 3937in =185m

7. Mount Everest, at 29,028 feet, is the tallest mountain on the Earth. What is its height in kilometers? (Assume that 1 kilometer equals 3,281 feet.)

Solution

29,028ft× 1km 3281ft =8.847km

8. The speed of sound is measured to be 342��/�� on a certain day. What is this in km/h?

Solution 342m s × 1km 1000m × 3600s 1h =123×103 km/h

9. Tectonic plates are large segments of the Earth’s crust that move slowly. Suppose that one such plate has an average speed of 4.0 cm/year. (a) What distance does it move in 1.0 s at this speed? (b) What is its speed in kilometers per million years?

10. (a) Refer to Table 1.3 to determine the average distance between the Earth and the Sun. Then calculate the average speed of the Earth in its orbit in kilometers per second. (b) What is this in meters per second?

1.3 ACCURACY, PRECISION, AND SIGNIFICANT FIGURES

11. Suppose that your bathroom scale reads your mass as 65 kg with a 3% uncertainty. What is the uncertainty in your mass (in kilograms)? Solution ��m= 3% 100%×65kg = 2kg

12. A good-quality measuring tape can be off by 0.50 cm over a distance of 20 m. What is its percent uncertainty?

13. (a) A car speedometer has a 5.0% uncertainty. What is the range of possible speeds when it reads 90km/h? (b) Convert this range to miles per hour.(1 �� = 06214 ��)

Solution (a) �� = 50% 100% ×900km/h=45km/h

Thus,therange = 90.0 ± 5km/h = 85to95km/h.

(b) 85.5km h ×0.6214mi 1km =53.1mi/h; 94.5km 1h ×0.6214mi 1km = 58.7mi/h

Sotherangeis531to587mi/h

14. An infant’s pulse rate is measured to be 130±5 beats/min. What is the percent uncertainty in this measurement?

Solution %�� = �� ×100% = 5beats/min 130beats/min×100%=384%=4%

15. (a) Suppose that a person has an average heart rate of 72.0 beats/min. How many beats do they have in 2.0 y? (b) In 2.00 y? (c) In 2.000 y?

Solution 72.0beats 1min ×60.0min 100h ×24.0h 100d× 365.25d 100y ×2.0y= 7.5738 ×107 beats

(a) 76×107beats (limited by 2.0 y)

(b) 7.57×107beats (limited by 2.00 y)

16. A can contains 375 mL of soda. How much is left after 308 mL is removed?

Solution 375mL 308mL = 67mL (uncertainty in the 1’s column)

17. State how many significant figures are proper in the results of the following calculations: (a) (106.7)(98.2)/(46.210)(1.01) (b)(18.7)2 (c) (1.60× 10 19)(3712).

Solution (a) 3 (limited by 98.2 and 1.01)

(b) 3 (limited by 18.7)

18. (a) How many significant figures are in the numbers 99 and 100? (b) If the uncertainty in each number is 1, what is the percent uncertainty in each? (c) Which is a more meaningful way to express the accuracy of these two numbers, significant figures or percent uncertainties?

Solution (a) 99 has 2 sig. figs. ; 100 has 3 sig. figs. at most (b) 1 99 ×100= 101%=10%;

×100=100%(if all zeros are significant)

19. (a) If your speedometer has an uncertainty of 20km/h at a speed of 90km/h, what is the percent uncertainty? (b) If it has the same percent uncertainty when it reads 60km/h, what is the range of speeds you could be going?

Solution (a) %�� = 20km/h 90km/h ×100% = 22%

(b) �� = 22% 100% ×60km/h = 1km/h

So the range is 60±1km/h or 59to61km/h

20. (a) A person’s blood pressure is measured to be 120±2��. What is its percent uncertainty? (b) Assuming the same percent uncertainty, what is the uncertainty in a blood pressure measurement of 80��?

Solution (a) %�� = 2mmHg 120mmHg ×100% = 1.7% = 2%(1sig.figbecauseof2mmHg) (b) ��bp= 17% 100% ×80mmHg = 1.3mmHg = 1mmHg(1sig.figbecauseof2mmHg)

21. A person measures their heart rate by counting the number of beats in 30s. If 40±1 beats are counted in 30.0±0.5s, what is the heart rate and its uncertainty in beats per minute?

Solution beats minute = 40beats 300s × 600 s 100min=80beats/min. %�� = 1beat 40beats×100%+ 05s 300s ×100%=2.5%+1.7%=4.2% =4% ��= %�� 100% ×�� = 4.2% 100%×80beats/min=33beats/min=3beats/min

The heart rate is 80 ± 3beats/min.

22. What is the area of a circle 3102cm in diameter? Solution �� =��2 =��(��2)2 =��(3.102cm 2 )2 =7.557cm2

23. If a marathon runner averages 9.5 mi/h, how long does it take him or her to run a 26.22-mi marathon? Solution 2622mi 95mi/h =2.8h

24. A marathon runner completes a 42.188 km course in 2 h, 30 min, and 12 s. There is an uncertainty of 25�� in the distance traveled and an uncertainty of 1 s in the elapsed time. (a) Calculate the percent uncertainty in the distance. (b) Calculate the uncertainty in the elapsed time. (c) What is the average speed in meters per second? (d) What is the uncertainty in the average speed?

Solution (a) % ��distance= 25m 42188km × 1km 1000m ×100% =0.0593%=0.059%

(b) % ��time= 1s 9012s ×100%=0.0111%=0.01%

(d) %��speed = %��distance + %��time = 00593% + 00111%= 00704% = 007%

δspeed= 0.07% 100% ×4681m/s = 0003m/s

25. The sides of a small rectangular box are measured to be 1.80±0.01cm, 2.05± 002cm , and 31±01cm long. Calculate its volume and uncertainty in cubic centimeters.

Solution V = 180cm × 205cm × 31cm = 114cm3 Use the methods of adding percents.

1.80±0.01cm→ 0.01cm 180cm ×100%=0.556%

205±002cm→ 0.02cm 2.05cm×100%=0976%

31±01cm→ 01cm 31cm ×100%=3226%

Adding these values and rounding to 1 sig. fig., the percent uncertainty of the volume is 5%. The uncertainty in the volume is therefore (11.4)(0.05) = 0.6. The volume is thus 114±06cm3

26. When non-metric units were used in the United Kingdom, a unit of mass called the pound-mass (lbm) was employed, where 1��=0.4539 ��. (a) If there is an uncertainty of 00001 �� in the pound-mass unit, what is its percent uncertainty? (b) Based on that percent uncertainty, what mass in pound-mass has an uncertainty of 1 kg when converted to kilograms?

Solution (a) %��lbm = 00001kg 0.4539kg ×100%=0.022%=0.02%

(b) lbm = ��lbm %��lbm ×100%= 1kg 002% × 1lbm 04539kg ×100%= 1×104 lbm,or10,000lbm.

27. The length and width of a rectangular room are measured to be 3955±0005m and 3050±0005m. Calculate the area of the room and its uncertainty in square meters.

Solution The area is 3.995m × 3.050m = 12.06m2 . Now use method of adding percents to get uncertainty in the area.

%��width = 0.005m 3050m×100%=0.16%

%��length = 0.005m 3.955m×100%=0.13%

%��area = 0.13% +0.16%=0.29%=0.3%

��area = 0.29% 100% ×12.06m2 =0.035m2 =0.04m2

So the area is 1206 ± 004m2

28. A car engine moves a piston with a circular cross section of 7500±0002 �� diameter a distance of 3.250±0.001 �� to compress the gas in the cylinder. (a) By what amount is the gas decreased in volume in cubic centimeters? (b) Find the uncertainty in this volume.

Solution (a) �� =��2ℎ=��[(0.5)(7.500cm)]2 ×3.250cm=143.6cm3

(b) Now use method of adding percents to get uncertainty in the volume

%�� = 0002cm 7500cm×100%=0.0267%

%��ℎ = 0.001cm 3.250cm×100%=00308%

%�� =2(0.0267%)+0.0308%=0.0842%; �� = 00842% 100% ×143.6cm3 =0.121cm3 =0.1cm3

1.4 APPROXIMATION

29. How many heartbeats are there in a lifetime?

Solution 1lifetime× 109 s 05lifetime × 1heartbeat 1s =2×109 heartbeats

30. A generation is about one-third of a lifetime. Approximately how many generations have passed since the year 0 AD?

Solution history× 1011 s history × 1generation 1/3lifetime × 05lifetime 109 s =150generations

31. How many times longer than the mean life of an extremely unstable atomic nucleus is the lifetime of a human? (Hint: The lifetime of an unstable atomic nucleus is on the order of 10 22 ��.)

Solution Th Tn = 2×109 s 10 22 s =2×1031 times

32. Calculate the approximate number of atoms in a bacterium. Assume that the average mass of an atom in the bacterium is ten times the mass of a hydrogen atom. (Hint: The mass of a hydrogen atom is on the order of 10 27 �� and the mass of a bacterium is on the order of 10 15��.)

33. Approximately how many atoms thick is a cell membrane, assuming all atoms there average about twice the size of a hydrogen atom?

34. (a) What fraction of Earth’s diameter is the greatest ocean depth? (b) The greatest mountain height?

Solution (a) 104 m(greatestoceandepth) 107 m(earth′sdiameter) =1⁄1000 (b) Take the highest mountain to be roughly 104 m. Then,

=1⁄1000

35. (a) Calculate the number of cells in a hummingbird assuming the mass of an average cell is ten times the mass of a bacterium. (b) Making the same assumption, how many cells are there in a human?

Solution (a) 10 2 kg/hummingbird 10×10 15 kg/cell =1012 cells/hummingbird (b) 102 kg/person 10×10 15 kg/cell =1016 cells/person

36. Assuming one nerve impulse must end before another can begin, what is the maximum firing rate of a nerve in impulses per second?

Solution

1nerveimpulse 10 3 s =103 nerveimpulses/s