#### Hugh D. Young, College Physics, 9th edition, Addison Wesley 2011

Chapter 12

Multiple-Choice Problems

1.  A hiker sees a lightning flash; 15 s later he hears the sound of the thunder. Recalling from his study of physics that the speed of sound in air is approximately he estimates that the distance to where the lightning flash occurred is approximately
A. 5 km.          B. 10 km.          C. 15 km.         D. 45 km.

2.  A segment A of wire stretched tightly between two posts a distance L apart vibrates in its fundamental mode with frequency A segment B of an identical wire is stretched with the same tension, but between two different posts. You observe that the frequency of the second harmonic of wire B is the same as the fundamental frequency of wire A. The length of wire B must be
A. L.               B. 2L.             C. 4L.

3. Two pulses of exactly the same size and shape are traveling toward each other along a stretched rope. They differ only in that one is upright while the other is inverted. Superposition tells us that when the pulses meet each other, they will cancel each other exactly at that instant and the rope will show no evidence of a pulse. What happens afterwards?
A. Each pulse continues as though it had never met the other one.
B. The rope remains straight, since the pulses have cancelled each other.
C. The pulses rebound from each other, each going back in the direction from which it came.

4. An organ pipe open at one end, but closed at the other, is vibrating in its fundamental mode, producing sound of frequency 1000 Hz. If you now open the closed end, the new fundamental frequency will be
A. 2000 Hz.      B. 1000 Hz.      C. 500 Hz.       D. 250 Hz.

5. A person listening to a siren from a stationary police car observes the frequency and wavelength of that sound. This person now moves rapidly toward the police car.
A. The wavelength of the sound the person observes is shorter than it was, but the frequency does not change.
B. The frequency of the sound the person observes is higher than it was, but the wavelength does not change.
C. The wavelength of the sound the person observes is shorter than it was, and the frequency is higher than it was.

6. A person listening to a siren from a stationary police car observes the frequency and wavelength of that sound. The car now drives rapidly toward the person.
A. The wavelength of the sound the person observes will be shorter than it was, but the frequency will not be changed.
B. The frequency of the sound the person observes will be higher than it was, but the wavelength will  not be changed.
C. The wavelength of the sound the person observes will be shorter than it was, and the frequency will also be higher than it was.

7. A string of length 0.600 m is vibrating at 100.0 Hz in its second harmonic and producing sound that moves at 340 m/s. What is true about the frequency and wavelength of this sound?
A. The wavelength of the sound is 0.600 m.
B. The wavelength of the sound is 3.40 m.
C. The frequency of the sound is 100.0 Hz.
D. The frequency of the sound is 567 Hz.

8. When a 15 kg mass is hung vertically from a thin, light wire, pulses take time  t to  travel  the  full  length of the wire. If an additional 15 kg mass is added to the first one without changing the length of the wire, the time taken for pulses to travel the length of the wire will be
A. 2t.              B. √2t             C. t/2               D. t/√2

9. An omnidirectional loudspeaker produces sound waves uniformly in all directions. The total power received by a sphere of radius 2.0 m centered on the speaker is 100 W. A sphere of radius 4.0 m will receive a total power of
A. 100 W         B. 50 W         C.  25 W

10. An organ pipe open at both ends is resonating in its fundamental mode at frequency    If you close both ends, the fundamental frequency will now be
A. 2f               B. f                 C. f/2             D. f/4

11. You are standing between two stereo speakers that are emitting sound of wavelength 10.0 cm in step with each other. They are very far apart compared with that wavelength. If you start in the middle and walk a distance x directly toward one speaker, you observe that the sound from these speakers first cancels when x is equal to
A. 20.0 cm.      B. 10.0 cm.       C. 5.0 cm.        D. 2.5 cm.

12. On a cold day, a siren emits sound waves with a wavelength of 17 cm. On a hot day, the wavelength of the sound produced by the same siren oscillating at the same frequency will be
A. greater than 17 cm.
B. less than 17 cm.
C. 17 cm.

13.  Traffic noise on Beethoven Boulevard has an intensity level of 80 dB; the traffic noise on Mozart Alley is only 60 dB. Compared to the sound intensity on Beethoven Boulevard, the sound intensity on Mozart Alley is
A. 25% lower                         B. 20 times lower                   C. 100 times lower     D. 20 W/m2 lower

14. A thin, light string supports a weight W hanging from the ceiling. In this situation, the string produces a note of frequency f when vibrating in its fundamental mode. In order to cause this string to produce a note one octave higher in its fundamental mode without stretching it, we must change the weight to
A. 4W             B. 2W             C. √2W           D. W/2                        E. W/4

15. String A weighs twice as much as string B. Both strings are thin and light and have the same length. If you hang equal weights at the bottom of each of these strings, the ratio of the speed of waves on string A to the speed of waves on string B will be
A. vA/vB = 2                B. vA/vB = √2              C. vA/vB =1/√2            D. vA/vB = ½

Problems

1. (a) Audible wavelengths. The range of audible frequencies is from about 20 Hz to 20,000 Hz. What is range of the wavelengths of audible sound in air? (b) Visible light. The range of visible light extends from 400 nm to 700 nm. What is the range of visible frequencies of light? (c) Brain surgery. Surgeons can remove brain tumors by using a cavitron ultrasonic surgical aspirator, which produces sound waves of frequency 23 kHz. What is the wavelength of these waves in air?  (d) Sound in the body. What would be the wavelength of the sound in part (c) in bodily fluids in which the speed of sound is 1480 m/s, but the frequency is unchanged?

2. The electromagnetic spectrum. Electromagnetic waves, which include light, consist of vibrations of electric and magnetic fields, and they all travel at the speed of light. (a) FM radio. Find the wavelength of an FM radio station signal broadcasting at a frequency of 104.5 MHz. (b) X rays. X rays have a wavelength of about 0.10 nm. What is their frequency? (c) The Big Bang. Microwaves with a wavelength of 1.1 mm, left over from soon after the Big Bang, have been detected. What is their frequency? (d) Sunburn. Sunburn (and skin cancer) are caused by ultraviolet light waves having a frequency of around What is their wavelength? (e) SETI. It has been suggested that extraterrestrial civilizations (if they exist) might  try  to  communicate  by  using  electromagnetic  waves having the same frequency as that given off by the spin flip of the electron in hydrogen, which is 1.43 GHz. To what wavelength should we tune our telescopes in order to search for such signals? (f)  Microwave ovens. Microwave ovens cook food with electromagnetic waves of frequency around 2.45 GHz. What wavelength do these waves have?

3. If an earthquake wave having a wavelength of 13 km causes the ground to vibrate 10.0 times each minute, what is the speed of the wave?

4. A fisherman notices that his boat is moving up and down periodically, owing to waves on the surface of the water. It takes 2.5 s for the boat to travel from its highest point to its lowest, a total distance of 0.62 m. The fisherman sees that the wave crests are spaced 6.0 m apart. (a) How fast are the waves traveling? (b) What is the amplitude of each wave? (c) If the total vertical distance traveled by the boat were 0.30 m, but the other data remained the same, how would the answers to parts (a) and (b) be affected?

5. A steel wire 4.00 m long has a mass of 0.0600 kg and is stretched with a tension of 1000 N. What is the speed of propagation of a transverse wave on the wire?

6. With what tension must a rope with length 2.50 m and mass 0.120 kg be stretched for transverse waves of frequency 40.0 Hz to have a wavelength of 0.750 m?

7. One end of a horizontal rope is attached to a prong of an electrically driven tuning fork that vibrates at 120 Hz. The other end passes over a pulley and supports a 1.50 kg mass. The linear mass density of the rope is 0.0550 kg/m. (a) What is the speed of a transverse wave on the rope? (b) What is the wavelength? (c) How would your answers to parts (a) and (b) change if the mass were increased to 3.00 kg?

8. (a) If the amplitude in a sound wave is doubled, by what factor does the intensity of the wave increase? (b) By what factor must the amplitude of a sound wave be increased in order to increase the intensity by a factor of 9?

9. When a mass M hangs from a vertical wire of length L, waves travel on this wire with a speed V. What will be the speed of these waves (in terms of V) if (a) we double M without stretching the wire? (b) we replace the wire with an identical one, except twice as long? (c) we replace the wire with one of the same length,  but  three times as heavy? (d) we stretch the wire to twice its original length? (e) we increase M by a factor of 10, which stretches the wire to double its original length?

10. A certain transverse wave is described by the equation
y(x,t)=(6.50 mm) sin2π(t/(0.0360 s) – x/(0.280 m))
Determine this wave’s (a) amplitude, (b) wavelength, (c) frequency, (d) speed of propagation, and (e) direction of propagation.

11. Transverse waves on a string have wave speed 8.00 m/s, amplitude 0.0700 m, and wavelength 0.320 m. These waves travel in the x direction, and at t = 0 the x = 0 end of the string is at y = 0 and moving downward. (a) Find the frequency, period, and wave number of these waves. (b) Write the equation for y(x,t) describing these waves. (c) Find the transverse displacement of a point on the string at x = 0.360 m at time t = 0.150 s.

12. The equation describing a transverse wave on a string is
y(x,t) = (1.50 mm) sin[(157 s–1)t –(41.9 m–1)x]
Find (a) the wavelength,  frequency,  and  amplitude  of  this wave, (b) the speed and direction of motion of the wave, and (c) the transverse displacement of a point on the string when t = 0.100 s and at a position x = 0.135 m.

13. Transverse waves are traveling on a long string that is under a tension of 4.00 N. The equation describing these waves is
y(x,t) = (1.25 cm) sin[(415 s–1)t –(44.9 m–1)x]
Find the linear mass density of this string.

14. Mapping the ocean floor. The ocean floor is mapped by sending sound waves (sonar) downward and measuring the time it takes for their echo to return. From this information, the ocean depth can be calculated if one knows that sound travels at 1531 m/s in seawater. If a ship sends out sonar pulses and records  their echo 3.27 s later, how deep is the ocean floor at that point, assuming that the speed of sound is the same at all depths?

15. In Figure 12.39, each pulse is traveling on a string at 1 cm/s and each square represents 1 cm.  Draw the shape of the string at the end of 6 s, 7 s, and 8 s.

16. A 1.50-m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 48.0 m/s. What are the wavelength and frequency of (a) the fundamental tone? (b) the second overtone? (c) the fourth harmonic?

17. A piano tuner stretches a steel piano wire with a tension of 800 N. The wire is 0.400 m long and has a mass of 3.00 g. (a) What is the frequency of its fundamental mode of vibration?  (b) What is the number of the highest harmonic that could be heard by a person who is capable of hearing frequencies up to 10,000 Hz?

18. A wire with mass 40.0 g is stretched so that its ends are tied down at points 80.0 cm apart. The wire vibrates in its fundamental mode with frequency 60.0 Hz and with an amplitude of 0.300 cm at the antinodes. (a) What is the speed of propagation of transverse waves in the wire? (b) Compute the tension in the wire.

19. The portion of string between  the  bridge  and  upper end of the fingerboard (the part of  the  string  that  is  free  to vibrate) of a certain musical instrument is 60.0 cm long and has a mass of 2.00 g. The string sounds an A4 note (440 Hz) when played. (a) Where must the player put a finger (at what distance x from the bridge) to play a D5 note (587 Hz)? For both notes, the string vibrates in its fundamental mode. (b) Without retuning, is it possible to play a G4 note (392 Hz) on this string? Why or why not?

21. Voiceprints. Suppose a singer singing F# (370 Hz, the fundamental frequency) has one overtone of frequency 740 Hz with half the amplitude of the fundamental and another overtone of frequency 1110 Hz having one-third the amplitude of the fundamental. Using the previous problem as a guide, graph the superposition of these three waves to show the complex sound wave produced by this singer.

22. Guitar string. One of the 63.5-cm-long strings of an ordinary guitar is tuned to produce the note B3 (frequency 245 Hz) when vibrating in its fundamental mode. (a) Find the speed of transverse waves on this string. (b) If the tension in this string is increased by 1.0%, what will be the new fundamental frequency of the string? (c) If the speed of sound in the surrounding air is 344 m/s, find the frequency and wavelength of the sound wave produced in the air by the vibration of the B3 string. How do these compare to the frequency and wavelength of the standing wave on the string?

23. Standing sound waves are produced in a pipe that is 1.20 m long.  For the fundamental frequency and the first two overtones, determine the locations along the pipe (measured from the left end) of the displacement nodes if (a) the pipe is open at both ends; (b) the pipe is closed at the left end and open at the right end.

24. Find the fundamental frequency and the frequency of the first three overtones of a pipe 45.0 cm long (a) if the pipe is open at both ends; (b) if the pipe is closed at one end. (c) For each of the preceding cases, what is the number of the highest harmonic that may be heard by a person who can hear frequencies from 20 Hz to 20,000 Hz?

25. The longest pipe found in most medium-size pipe organs is 4.88 m (16 ft) long. What is the frequency of the note corresponding to the fundamental mode if the pipe is (a) open at both ends, (b) open at one end and closed at the other?

26. The fundamental frequency of a pipe that is open at both ends is 594 Hz. (a) How long is this pipe? If one end is now closed, find (b) the wavelength and (c) the frequency of the new fundamental.

27. The role of the mouth in sound. The production of sound during speech or singing is a complicated process. Let’s concentrate on the mouth. A typical depth for the human mouth is about 8.0 cm, although this number can vary. We can model the mouth as an organ pipe that is open at the back of the throat. What are the wavelengths and frequencies of the first four harmonics you can produce if your mouth is (a) open, (b) closed? Use v = 354 m/s.

28. A certain pipe produces a fundamental frequency of 262 Hz in air at 20°C. (a) If the pipe is filled with helium at the same temperature, what fundamental frequency does it produce? (b) Does your answer to part (a) depend on whether the pipe is open or stopped? Why or why not?

29. The vocal tract. Many opera singers (and some pop singers) have a range of about 2.5 octaves or even greater. Suppose a soprano’s range extends from A below middle C (frequency 220 Hz) up to Eb above high C (frequency 1244 Hz). Although the vocal tract is quite complicated, we can model it as a resonating air column, like an organ pipe, that is open at the top and closed at the bottom. The column extends from the mouth down to the diaphragm in the chest cavity, and we can also assume that the lowest note is the fundamental. How long is this column of air if v = 354 m/s. Does your result seem reasonable, on the basis of observations of your own body?

30. Singing in the shower! We all sound like great singers in the shower, due to standing waves. Assume that your shower is 2.45 m (about 8 ft) tall and can be modeled as an organ pipe. (a) What will we have at the floor and ceiling, displacement nodes or antinodes? (b) What are the wavelength and frequency of the fundamental harmonic for standing waves in this shower? (c) What are the wavelength and frequency of the first two overtones for this shower?

31. French horn. The French horn, one of the most beautiful sounding instruments in the orchestra, consists of about 3.7 m (roughly 12 ft) of thin tubing, rolled into a spiral shape (although sizes do vary). The player blows into the mouthpiece, which can be treated as a closed end, and places his hand in the opposite end, which has a large flared opening. In brass instruments, the fundamental note is not normally playable. Instead the first overtone is the lowest playable note. (a) If the player’s hand keeps the large end open, what is the frequency of the lowest playable note? (b) If the player now closes the large end with his hand, what is the frequency of the lowest playable note?

32. You blow across the open mouth of an empty test tube and produce the fundamental standing wave of the air column inside the test tube. The speed of sound in air is 344 m/s and the test tube acts as a stopped pipe. (a) If the length of the air column in the test tube is 14.0 cm, what is the frequency of this standing wave?  (b) What is the frequency of the fundamental standing wave in the air column if the test tube is half filled with water?

33. Two small speakers A and B are driven in step at 725 Hz by the same audio oscillator. These speakers both start out 4.50 m from the listener, but speaker A is slowly moved away. (a) At what distance d will the sound from the speakers first produce destructive interference at the location of the listener? (b) If A keeps moving, at what distance d will the speakers next produce destructive interference at the listener? (c) After A starts moving away, at what distance will the speakers first produce constructive interference at the listener?

34. In a certain home sound system, two small speakers are located so that one is 45.0 cm closer to the listener than the other. For what frequencies of audible sound will these speakers produce (a) destructive interference at the listener, (b) constructive interference at the listener? In each case, find only the three lowest audible frequencies.

35. Two small stereo speakers are driven in step by the same variable-frequency oscillator. Their sound is picked up by a microphone arranged as shown in Figure 12.43. For what frequencies does their sound at the speakers produce (a) constructive interference, (b) destructive interference?

36. Human hearing. The human outer ear contains a more-or-less cylindrical cavity called the auditory canal that behaves like a resonant tube to aid in the hearing process. One end terminates at the eardrum   (tympanic membrane), while the other opens to the outside. Typically, this canal is approximately 2.4 cm long. (a) At what frequencies would it resonate in its first two harmonics? (b) What are the corresponding sound wavelengths in part (a)?

38. A 75.0 cm wire of mass 5.625 g is tied at both ends and adjusted to a tension of 35.0 N. When it is vibrating in its second overtone, find (a) the frequency and wavelength at which it  is  vibrating  and  (b)  the  frequency  and  wavelength  of  the sound waves it is producing.

39. A small omnidirectional stereo speaker produces waves in all directions that have an intensity of 6.50 W/m2 at a distance of 2.50 m from the speaker. (a) At what rate does this speaker produce energy? (b) What is the intensity of this sound 7.00 m from the speaker? (c) What is the total amount of energy received each second by the walls (including windows and doors) of the room in which this speaker is located?

40. Find the intensity (in W/m2) of (a) a 55.0 dB sound, (b) a 92.0 dB sound, (c) a –2.0 dB sound.

41. Find the noise level (in dB) of a sound having an intensity of (a) 0.000127 W/m2 (b) 6.53 × 10–10 W/cm2 (c) 1.5 × 10–14 W/m2

42. (a) By what factor must the sound intensity be increased to raise the sound intensity level by 13.0  dB? (b) Explain why you don’t need to know the original sound intensity.

43. Eavesdropping! You are trying to overhear a juicy conversation, but from your distance of 15.0 m, it sounds like only an average whisper of 20.0 dB. So you decide to move closer to give the conversation a sound level of 60.0 dB instead. How close should you come?

44. Energy delivered to the ear. Sound is detected when a sound wave causes the tympanic membrane (the eardrum) to vibrate.  Typically, the diameter of this membrane is about 8.4 mm in humans. (a) How much energy is delivered to the eardrum each second when someone whispers (20 dB) a secret in your ear? (b) To comprehend how sensitive the ear is to very small amounts of energy, calculate how fast a typical 2.0 mg mosquito would have to fly (in mm/s) to have this amount of kinetic energy.

45. Human hearing. A fan at a rock concert is 30 m from the stage, and at this point the sound intensity level is 110 dB. (a) How much energy is transferred to her eardrums each second? (b) How fast would a 2.0 mg mosquito have to fly to have this much kinetic energy? Compare the mosquito’s speed with that found for the whisper in part (a) of the previous problem.

46. The intensity due to a number of independent sound sources is the sum of the individual intensities. (a) When four quadruplets cry simultaneously, how many decibels greater is the sound intensity level than when a single one cries? (b) To increase the sound intensity level again by the same number of decibels as in part (a), how many more crying babies are required?

47. What is the sound intensity level in a car when the sound intensity is 0.500 μW/m2 (b) What is the sound intensity in the air near a jackhammer when the sound intensity level is 103 dB?

48. A trumpet player is tuning his instrument by playing an A note simultaneously with the first-chair trumpeter, who has perfect pitch. The first-chair player’s note is exactly 440 Hz, and 2.8 beats per second are heard. What are the two possible frequencies of the other player’s note?

49. Two tuning forks are producing sounds of wavelength 34.40 cm and 33.94 cm simultaneously. How many beats do you hear each second?

50. Two guitarists attempt to play the same note of wavelength 6.50 cm at the same time, but one of the instruments is slightly out of tune and plays a note of wavelength 6.52 cm instead. What is the frequency of the beat these musicians hear when they play together?

51. Tuning a violin. A violinist is tuning her instrument to concert A (440 Hz). She plays the note while listening to an electronically generated tone of exactly that frequency and hears a beat of frequency 3 Hz, which increases to 4 Hz when she tightens her violin string slightly. (a) What was the frequency of her violin when she heard the 3-Hz beat? (b) To get her violin perfectly tuned to concert A, should she tighten or loosen her string from what it was when she heard the 3-Hz beat?

52. A railroad train is traveling at 25.0 m/s in still air. The frequency of the note emitted by the locomotive whistle is 400 Hz. What is the wavelength of the sound waves (a) in front of the locomotive? (b) behind the locomotive? What is the frequency of the sound heard by a stationary listener (c) in front of the locomotive? (d) behind the locomotive?

53. Two train whistles, A and B, each have a frequency of 392 Hz. A is stationary and B is moving toward the right (away from A) at a speed of 35.0 m/s. A listener is between the two whistles and is moving toward the right with a speed of 15.0 m/s. (See Figure 12.45.) (a) What is the frequency from A as heard by the listener? (b) What is the frequency from B as heard by the listener? (c) What is the beat frequency detected by the listener?

54. On the planet Arrakis, a male ornithoid is flying toward his stationary  mate  at 25.0 m/s while  singing at a frequency of 1200 Hz. If the female hears a tone of 1240 Hz, what is the speed of sound in the atmosphere of Arrakis?

55. A car alarm is emitting sound waves of frequency 520 Hz. You are on a motorcycle, traveling directly away from the car. How fast must you be traveling if you detect a frequency of 490 Hz?

56. A railroad train is traveling at 30.0 m/s in still air. The frequency of the note emitted by the train whistle is 262 Hz. What frequency is heard by a passenger on a train moving in the opposite direction to the first at 18.0 m/s and (a) approaching the first; and (b) receding from the first?

57. The siren of a fire engine that is driving northward at 30.0 m/s emits a sound of frequency 2000 Hz. A truck in front of this fire engine is moving northward at 20.0 m/s (a) What is the frequency of the siren’s sound that the fire engine’s driver hears reflected from the back of the truck? (b) What wavelength would this driver measure for these reflected sound waves?

58. A stationary police car emits a sound of frequency 1200 Hz that bounces off of a car on the highway and returns with a frequency of 1250 Hz. The police car is right next to the highway, so the moving car is traveling directly toward or away from it. (a) How fast was the moving car going? Was it moving towards or away from the police car? (b) What frequency would the police car have received if it had been traveling toward the other car at 20.0 m/s.

59. A container ship is traveling westward at a speed of 5.00 m/s. The waves on the surface of the ocean have a wavelength of 40.0 m and are traveling eastward at a speed of 16.5 m/s. (a) At  what  time intervals does the ship encounter the crest of a wave? (b) At what time intervals will the ship encounter wave crests if it turns around and heads eastward?

60. While sitting in your car by the side of a country road, you see your friend, who happens to have an identical car with an identical horn, approaching you. You blow your horn, which has a frequency of 260 Hz; your friend begins to blow his horn as well, and you hear a beat frequency of 6.0 Hz. How fast is your friend approaching you?

61. Moving source vs. moving listener. (a) A sound source producing 1.00 kHz waves moves toward a stationary listener at one-half the speed of sound. What frequency will the listener hear? (b) Suppose instead that the source is stationary and the listener moves toward the source at one-half the speed of sound. What frequency does the listener hear? How does your answer compare with that in part (a)? Did you expect to get the same answer in both cases? Explain on physical grounds why the two answers differ.

62. How fast (as a percentage of light speed) would a star have to be moving so that the frequency of the light we receive from it is 10.0% higher than the frequency of the light it is emitting? Would it be moving away from us or toward us? (Assume it is moving either directly away from us or directly toward us.)

63. One end of a 14.0-m-long wire having a total mass of 0.800 kg is fastened to a fixed support in the ceiling, and a 7.50 kg object is hung from the other end. If the wire is struck a transverse blow at one end, how much time does the pulse take to reach the other end? Neglect the variation in tension along the length of the wire.

64. Ultrasound in medicine. A 2.00 MHz sound wave travels through a pregnant woman’s abdomen and is reflected from the fetal heart wall of her unborn baby. The heart wall is moving toward the sound receiver as the heart beats. The reflected sound is then mixed with the transmitted sound, and 85 beats per second are detected. The speed of sound in body tissue is 1500 m/s. Calculate the speed of  the fetal heart wall at the instant this measurement is made.

65. A very noisy chain saw operated by a tree surgeon emits a total acoustic power of 20.0 W uniformly in all directions. At what distance from the source is the  sound level equal to (a) 100 dB, (b) 60 dB?

66. Tuning a cello. A cellist tunes the C-string of her instrument to a fundamental frequency of 65.4 Hz. The vibrating portion of the string is 0.600 m long and has a mass of 14.4 g. (a) With what tension must she stretch that portion of the string? (b) What percentage increase in tension is needed to increase the frequency from 65.4 Hz to 73.4 Hz, corresponding to a rise in pitch from C to D?

67. A person is playing a small flute 10.75 cm long, open at one end and closed at the other, near a taut string having a fundamental frequency of 600.0 Hz. If the speed of sound is 344.0 m/s, for which harmonics of the flute will the string resonate? In each case, which harmonic of the string is in resonance?

68. A bat flies toward a wall, emitting a steady sound of frequency 2000 Hz. The bat hears its own sound, plus the sound reflected by the wall. How fast should the bat fly in order to hear a beat frequency of 10.0 Hz?

69. You’re standing between two speakers that are driven by the same amplifier and are emitting sound waves with frequency 229 Hz. The two speakers are facing each other, 15 meters apart. (a) You begin walking away from one speaker toward the other one, and as you walk, you hear what sounds like beats, with a frequency of 2.50 Hz. How fast are you walking?  (b)  If the frequency of the sound emitted by the speakers increases to 573 Hz and you continue to walk at the same speed, what frequency of beats will you hear?

70. The sound source of a ship’s sonar system operates at a frequency of 22.0 kHz. The speed of sound in water (assumed to be at a uniform 20 °C) is 1482 m/s. (a) What is the wavelength of the waves emitted by the source? (b) What is the difference in frequency between the directly radiated waves and the waves reflected from a whale traveling straight toward the ship at 4.95 m/s. The ship is at rest in the water.

71. The range of human hearing. A young person with normal hearing can hear sounds ranging from 20 Hz to 20 kHz. How many octaves can such a person hear? (Recall that if two tones differ by an octave, the higher frequency is twice the lower frequency.)

72. A person leaning over a 125-m-deep well accidentally drops a siren emitting sound of frequency 2500 Hz. Just before this siren hits the bottom of the well, find the frequency and wavelength of the sound the person hears (a) coming directly from the siren, (b) reflected off the bottom of the well. (c) What beat frequency does this person perceive?

73. A police siren of frequency fsiren is attached to a vibrating platform. The platform and siren oscillate up and down in simple harmonic motion with amplitude Ap and frequency fp (a) Find the maximum and minimum sound frequencies that you would hear at a position directly above the siren. (b) At what point in the motion of the platform is the maximum frequency heard? The minimum frequency? Explain.

74. A flexible stick can oscillate in a standing-wave pattern, much as a string does. (a) Are the ends of the stick displacement nodes or antinodes? Why? (If in doubt, use a handy stick at home or in the laboratory and try it out.) (b) A flexible stick 3.00 m long is free to wiggle. Using the conditions from part (a), find the wavelengths of the first five harmonics for this stick.

75. A turntable 1.50 m in diameter rotates at 75 rpm. Two speakers, each giving off sound of wavelength 31.3 cm, are attached to the rim of the table at opposite ends of a diameter. A listener stands in front of the turntable. (a) What is the greatest beat frequency the listener will receive from this system? (b) Will the listener be able to distinguish individual beats?

76. Musical scale. The frequency ratio of a semitone interval on the equally tempered scale is 21/12. (a) Show that this ratio is 1.059. (b) Find the speed of an automobile passing a listener at rest in still air if the pitch of the car’s horn drops a semitone between the times when the car is coming directly toward him and when it is moving directly away from him.

77. Two organ pipes, open at one end but closed at the other, are each 1.14 m long. One is now lengthened by 2.00 cm. Find the frequency of the beat they produce when playing together in their fundamental.

Chapter 13

Multiple-Choice Problems

1. Which has a greater buoyant force on it, a 25 cm3 piece of wood floating with part of its volume above water or a 25 cm3 piece of submerged iron?
A. The floating wood.
B. The submerged iron.
C. They have the same buoyant force on them.
D. It is impossible to tell without knowing their weights.

2. As a rigid (i.e., constant volume), lighter-than-air balloon leaves the ground and rises, the buoyant force on it
A. increases.               B. decreases.          C. stays the same.

3. A mass of sunken lead is resting against the bottom in a glass of water. You take this lead, put it in a small boat of negligible mass, and float the boat in the water. Which of the following statements are true? (There may be more than one correct choice.)
A. The sunken lead displaces a volume of water equal to the lead’s own volume.
B. The floating lead displaces a volume of water equal to the lead’s own volume.
C. The sunken lead displaces a volume of water whose weight equals the lead’s weight.
D. The floating lead displaces a volume of water whose weight equals the lead’s weight.

4. Two equal-sized buckets are filled to the brim with water, but one of them has a piece of wood floating in it. Which bucket of water weighs more?
A. The bucket with the wood.
B. The bucket without the wood.
C. They weigh the same amount.

5. Two equal mass pieces of metal are sitting side by side at the bottom of a deep lake. One piece is aluminum and the other is lead. Which piece has the greater buoyant force acting on it?
A. The aluminum.
C. They both have the same buoyant force acting on them.

6. If a 5 lb force is required to keep a block of wood 1 ft beneath the surface of water, the force needed to keep it 2 ft beneath the surface is
A. 2.5 lb          B. 5 lb            C. 10 lb.

7. A horizontal pipe with water flowing through it has a circular cross section that varies in diameter. The diameter at the wide section is 3 times that of the diameter at the narrow section. If the rate of flow of the water in the narrow section is 9.0 L/min the rate of flow of the water in the wide section is
A. 1.0 L/min               B. 3.0 L/min               C. 9.0 L/min               D. 18 L/min                E. 36.0 L/min

8. A horizontal cylindrical pipe has a part with a diameter half that of the rest of the pipe. If V is the speed of the fluid in the wider section of pipe, then the speed of the fluid in the narrower section is
A. V/4             B. V/2             C. 2V             D. 4V

9. If the absolute pressure at a depth d in a lake is P, the absolute pressure at a depth 2d will be
A. 2P.             B. P                 C. Greater than 2P.                D. Less than 2P.

10. If the gauge pressure at a depth d in a lake is P, the gauge pressure at a depth 2d will be
A. 2P.             B. P                 C. Greater than 2P.                D. Less than 2P.

11. There is a great deal of ice floating on the oceans near the North Pole. If this ice were to melt due to global warming, what would happen to the level of the oceans?
A.  The level would rise.
B.  The level would fall.
C.  The level would stay the same.

12. A rigid metal object is dropped into a lake and sinks to the bottom. The  density  of  the  water  in  this  lake  is  the  same  everywhere. As the object sinks deeper and deeper below the surface, the buoyant force on it
A. gets greater and greater.
B. gets less and less.
C. does not change.
D. is equal to the weight of this object.

13. A spherical object has a density ρ. If it is compressed under high pressure to half of its original diameter, its density will now be
A. ρ/8              B. ρ/4             C. 2ρ              D. 4ρ               E. 8ρ

14. Identical-size cubes of lead and aluminum are suspended at different depths by two wires in a tank of water. Which wire will have a greater tension?
A. The wire holding the lead cube
B. The wire holding the aluminum cube
C. The tensions in the two wires will be equal.

15. Two small holes are drilled in the side of a barrel filled with water. One hole is twice as far below the surface as the other. If the speed of the water flowing from the upper hole is V, the speed of the water flowing from the lower hole is
A. V                B. V/2             C. 2V              D. √2V

Problems

1. You purchase a rectangular piece of metal that has dimensions 5.0 mm × 15.0 mm × 30.0 mm and mass 0.0158 kg. The seller tells you that the metal is gold. To check this, you compute the average density of the piece. What value do you get? Were you cheated?

2. A kidnapper demands a 40.0 kg cube of platinum as a ransom. What is the length of a side?

3. Calculate the weight of air at 20 °C in a room that measures 5.00 × 4.50 × 3.25 m. Give your answer in newtons and in pounds.

4. By how many newtons do you increase the weight of your car when you fill up your 11.5 gal gas tank with gasoline? A gallon is equal to 3.788 L and the density of gasoline is 737 kg/m3.

5. How big is a million dollars? At the time this problem was written, the price of gold was about \$1239 per ounce, while that of platinum was about \$1508 an ounce. The “ounce” in this case is the troy ounce, which is equal to 31.1035 g. (The more familiar avoirdupois ounce is equal to 28.35 g.) The density of gold is 19.3 g/cm3 and that of platinum is 21.4 g/cm3 (a) If you find a spherical gold nugget worth 1.00 million dollars, what would be its diameter? (b) How much would a platinum nugget of this size be worth?

6. A cube 5.0 cm on each side is made of a metal alloy. After you drill a cylindrical hole 2.0 cm in diameter all the way through and perpendicular to one face, you find that the cube weighs 7.50 N. (a) What is the density of this metal? (b) What did the cube weigh before you drilled the hole in it?

7. A cube of compressible material (such as Styrofoam™ or balsa wood) has a density ρ and sides of length L. (a) If you keep its mass the same, but compress each side to half its length, what will be its new density, in terms of ρ (b) If you keep the mass and shape the same, what would the length of each side have to be (in terms of L) so that the density of the cube was three times its original value?

8. A hollow cylindrical copper pipe is 1.50 m long and has an outside diameter of 3.50 cm and an inside diameter of 2.50 cm. How much does it weigh?

9. A uniform lead sphere and a uniform aluminum sphere have the same mass. What is the ratio of the radius of the aluminum sphere to the radius of the lead sphere?

10. Blood pressure. Systemic blood pressure is defined as the ratio of two pressures, both expressed in millimeters of mercury. Normal blood pressure is (120 mm)/(80 mm) about which is usually just stated as 120/80 (See also Problem 24.) What would normal systemic blood pressure be if, instead of millimeters of mercury, we expressed pressure in each of the following units, but continued to use the same ratio format? (a) atmospheres, (b) torr, (c) Pa, (d) N/m2 (e) psi.

11. Blood. (a) Mass of blood. The human body typically contains 5 L of blood of density 1060 kg/m3. How many kilograms of blood are in the body? (b) The average blood pressure is 13,000 Pa at the heart. What average force does the blood exert on each square centimeter of the heart? (c) Red blood cells. Red blood cells have a specific gravity of 5.0 and a diameter of about 7.5 μm.  If they are spherical in shape (which is not quite true), what is the mass of such a cell?

12. Landing on Venus. One of the great difficulties in landing on Venus is dealing with the crushing pressure of the atmosphere, which is 92 times the earth’s atmospheric pressure. (a) If you are designing a lander for Venus in the shape of a hemisphere 2.5 m in diameter, how many newtons of inward force must it be prepared to withstand due to the Venusian atmosphere? (Don’t forget about the bottom!) (b) How much force would the lander have to withstand on the earth?

13. You are designing a diving bell to withstand the pressure of seawater at a depth of 250 m. (a) What is the gauge pressure at this depth? (You can ignore the small changes in the density of the water with depth.) (b) At the 250 m depth, what is the net force due to the water outside and the air inside the bell on a circular glass window 30.0 cm in diameter if the pressure inside the diving bell equals the pressure at the surface of the water?

14. Glaucoma. Under normal circumstances, the vitreous humor, a jelly-like substance in the main part of the eye, exerts a pressure of up to 24 mm of mercury that maintains the shape of the eye. If blockage of the drainage duct for aqueous humor causes this pressure to increase to about 50 mm of mercury, the condition is called glaucoma. What is the increase in the total force (in newtons) on the walls of the eye if the pressure increases from 24 mm to 50 mm of mercury? We can quite accurately model the eye as a sphere 2.5 cm in diameter.

15. By means of physiological adaptations that are still not very well understood, sperm whales are thought to be able to hunt for their food at depths of between 400 m and 3000 m. (a) What range of gauge pressures (in Pa and atm) do the whales withstand at these depths? (b) Estimate the total inward force of water pressure on the surface of a sperm whale at a depth of 3000 m, modeling the whale as a cylinder 16 m long and 4 m in diameter.

16. What gauge pressure must a pump produce to pump water from the bottom of the Grand Canyon (elevation 730 m) to Indian  Gardens (elevation 1370 m)? Express your result in pascals and in atmospheres.

17. Intravenous feeding. A hospital patient is being fed intravenously with a liquid of density 1060 kg/m3. (See Figure 13.40.) The container of liquid is raised 1.20 m above the patient’s arm where the fluid enters his veins. What is the pressure this fluid exerts on his veins, expressed in millimeters of mercury?

18. A 975-kg car has its tires each inflated to “32.0 pounds.” (a) What are the absolute and gauge pressures in these tires in lb/in2, Pa, and atm? (b) If the tires were perfectly round, could the tire pressure exert any force on the pavement? (Assume that the tire walls are flexible so that the pressure exerted by the tire on the pavement equals the air pressure inside the tire.) (c) If you examine a car’s tires, it is obvious that there is some flattening at the bottom. What is the total contact area for all four tires of the flattened part of the tires at the pavement?

19. An electrical short cuts off all power to a submersible diving vehicle when it is 30 m below the surface of the ocean. The crew must push out a hatch of area 0.75 m2 and weight 300 N on the bottom to escape. If the pressure inside is 1.0 atm, what downward force must the crew exert on the hatch to open it?

20. Standing on your head. (a) When you stand on your head, what is the difference in pressure of the blood in your brain compared with the pressure when you stand on your feet if you are 1.85 m tall? The density of blood is 1060 kg/m3 (b) What effect does the increased pressure have on the blood vessels in your brain?

21. You are designing a machine for a space exploration vehicle. It contains an enclosed column of oil that is 1.50 m tall, and you need the pressure difference between the top and the bottom of this column to be 0.125 atm. (a) What must be the density of the oil? (b) If the vehicle is taken to Mars, where the acceleration due to gravity is 0.379g, what will be the pressure difference (in earth atmospheres) between the top and bottom of the oil column?

22. Ear damage from diving. If the force on the tympanic membrane (eardrum) increases by about 1.5 N above the force from atmospheric pressure, the membrane can be damaged. When you go scuba diving in the ocean, below what depth could damage to your eardrum start to occur? The eardrum is typically 8.2 mm in diameter.

23. A barrel contains a 0.120 m layer of oil of density 600 kg/m3 floating on water that is 0.250 m deep. (a) What is the gauge pressure at the oil–water interface? (b) What is the gauge pressure at the bottom of the barrel?

24. Blood pressure. Systemic blood pressure is expressed as the ratio of the systolic pressure (when the heart first ejects blood into the arteries) to the diastolic pressure (when the heart is relaxed). Both pressures are measured at the level of the heart and are expressed in millimeters of mercury (or torr), although the units are not written. Normal systemic blood pressure is 120/80 (a) What are the maximum and minimum forces (in newtons) that the blood exerts against each square centimeter of the heart for a person with normal blood pressure? (b) As pointed out in the text, blood pressure is normally measured on the upper arm at the same height as the heart. Due to therapy for an injury, a patient’s upper arm is extended 30.0 cm above his heart. In that position, what should be his systemic blood pressure reading, expressed in the standard way, if he has normal blood pressure? The density of blood is 1060 kg/m3.

25. Blood pressure on the moon. When we eventually establish lunar colonies, people living there will need to have their blood pressure taken. Assume that we continue to express the systemic blood pressure as we now do on earth (see previous problem) and that the density of blood does not change. Suppose also that normal blood pressure on the moon is still 120/80 (which may not actually be true). If a lunar colonizer has her blood pressure taken at her upper arm when it is raised 25 cm above her heart, what will be her systemic blood pressure reading, expressed in the standard way, if she has normal blood pressure? The acceleration due to gravity on the moon is 1.67 m/s2.

26. The piston of a hydraulic automobile lift is 0.30 m in diameter. What gauge pressure, in pascals, is required to lift a car with a mass of 1200 kg? Now express this pressure in atmospheres.

27. Hydraulic lift. You are designing a hydraulic lift for an automobile garage. It will consist of two oil-filled cylindrical pipes of different diameters. A worker pushes down on a piston at one end, raising the car on a platform at the other end. To handle a full range of jobs, you must be able to lift cars up to 3000 kg, plus the 500 kg platform on which they are parked. To avoid injury to your workers, the maximum amount of force a worker should need to exert is 100 N. (a) What should be the diameter of the pipe under the platform? (b) If the worker pushes down with a stroke 50 cm long, by how much will he raise the car at the other end?

28. There is a maximum depth at which a diver can breathe through a snorkel tube, because as the depth increases, so does the pressure difference, which tends to collapse the diver’s lungs. Since the snorkel connects the air in the lungs to the atmosphere at the surface, the pressure inside the lungs is atmospheric pressure.  What is the external–internal pressure difference when the diver’s lungs are at a depth of 6.1 m (about 20 ft)? Assume that the diver is in freshwater.

29. A solid aluminum ingot weighs 89 N in air. (a) What is its volume? (b) The ingot is suspended from a rope and totally immersed in water. What is the tension in the rope (the apparent weight of the ingot in water)?

30. Fish navigation. (a) As you can tell by watching them in an aquarium, fish are able to remain at any depth in water with no effort. What does this ability tell you about their density? (b) Fish are able to inflate themselves using a sac (called the swim bladder) located under their spinal column. These sacs can be filled with an oxygen–nitrogen mixture that comes from the blood. If a 2.75 kg fish in fresh water inflates itself and increases its volume by 10%, find the net force that the water exerts on it. (c) What is the net external force on it? Does the fish go up or down when it inflates itself?

31. When an open-faced boat has a mass of 5750 kg, including its cargo and passengers, it floats with the water just up to the top of its gunwales (sides) on a freshwater lake. (a) What is the volume of this boat? (b) The captain decides that it is too dangerous to float with his boat on the verge of sinking, so he decides to throw some cargo overboard so that 20% of the boat’s volume will be above water. How much mass should he throw out?

32. An ore sample weighs 17.50 N in air. When the sample is suspended by a light cord and totally immersed in water, the tension in the cord is 11.20 N. Find the total volume and the density of the sample.

33. A slab of ice floats on a freshwater lake. What minimum volume must the slab have for a 45.0 kg woman to be able to stand on it without getting her feet wet?

34. Using data from Appendix E, calculate the average density of the planet Saturn. How does your answer compare to the density of water, and what does this imply about the buoyancy of Saturn, if you could find an ocean big enough to drop it into?

35. A hollow plastic sphere is held below the surface of a freshwater lake by a cord anchored to the bottom of the lake. The sphere has a volume of 0.650 m3 and the tension in the cord is 900 N. (a) Calculate the buoyant force exerted by the water on the sphere. (b) What is the mass of the sphere? (c) The cord breaks and the sphere rises to the surface. When the sphere comes to rest, what fraction of its volume will be submerged?

36. (a) Calculate the buoyant force of air (density 1.20 kg/m3) on a spherical party balloon that has a radius of 15.0 cm. (b) If the rubber of the balloon itself has a mass of 2.00 g and the balloon is filled with helium (density 0.166 kg/m3), calculate the net upward force (the “lift”) that acts on it in air.

37. The tip of the iceberg. Icebergs consist of freshwater ice and float in the ocean with only about 10% of their volume above water (the “tip of the iceberg,” so to speak). This percentage can vary, depending on the condition of the ice. Assume that the ice has the density given in Table 13.1, although, in reality, this can vary considerably, depending on the condition of the ice and the amount of impurities in it. (a) What does this 10% observation tell us is the density of seawater? (b) What percentage of the icebergs’ volume would be above water if they were floating in a large freshwater lake such as Lake Superior?

38. At 20° C, the surface tension of water is 72.8 dynes/cm.  Find the excess pressure inside of (a) an ordinary-size water drop of radius 1.50 mm and (b) a fog droplet of radius 0.0100 mm.

39. Find the gauge pressure in pascals inside a soap bubble 7.00 cm in diameter. The surface tension of this soap is 25 dynes/cm.

40. What radius must a water drop have for the difference between the inside and outside pressures to be 0.0200 atm? The surface tension of water is 72.8 dynes/cm.

41. At 20° C, the surface tension of water is 72.8 dynes/cm and that of carbon tetrachloride (CCl4) is 26.8 dynes/cm. If the gauge pressure is the same in two drops of these liquids, what is the ratio of the volume of the water drop to that of the CCl4 drop?

42. At a point where an irrigation canal having a rectangular cross section is 18.5 m wide and 3.75 m deep, the water flows at 2.50 cm/s. At a point downstream, but on the same level, the canal is 16.5 m wide, but the water flows at 11.0 cm/s. How deep is the canal at this point?

43. Water is flowing in a pipe with a varying cross-sectional area, and at all points the water completely fills the pipe. At point 1, the cross-sectional area of the pipe is 0.070 m2 and the magnitude of the fluid velocity is 3.50 m/s. What is the fluid speed at points in the pipe where the cross-sectional area is (a) 0.105 m2 (b) 0.047 m2?

44. Water is flowing in a cylindrical pipe of varying circular cross-sectional area, and at all points the water completely fills the pipe. (a) At one point in the pipe, the radius is 0.150 m. What is the speed of the water at this point if the volume flow rate in the pipe is 1.20 m3/s (b) At a second point in the pipe, the water speed is 3.80 m/s. What is the radius of the pipe at this point?

45. A shower head has 20 circular openings, each with radius 1.0 mm. The shower head is connected to a pipe with radius 0.80 cm. If the speed of water in the pipe is 3.00 m/s, what is its speed as it exits the shower-head openings?

46. You’re holding a hose at waist height and spraying water horizontally with it. The hose nozzle has a diameter of 1.80 cm, and the water splashes on the ground a distance of 0.950 m horizontally from the nozzle. Suppose you now constrict the nozzle to a diameter of 0.750 cm; how far horizontally from
the  nozzle  will  the  water  travel  before  hitting  the  ground?

47. A small circular hole 6.00 mm in diameter is cut in the side of a large water tank, 14.0 m below the water level in the tank. The top of the tank is open to the air. Find the speed at which the water shoots out of the tank.

48. A sealed tank containing seawater to a height of 11.0 m also contains air above the water at a gauge pressure of 3.00 atm. Water flows out from the bottom through a small hole. Calculate the speed with which the water comes out of the tank.

49. What gauge pressure is required in the city water mains for a stream from a fire hose connected to the mains to reach a vertical height of 15.0 m? (Assume that the mains have a much larger diameter than the fire hose.)

50. At one point in a pipeline, the water’s speed is 3.00 m/s and the gauge pressure is 4.00 × 104 Pa.  Find the gauge pressure at a second point in the line 11.0 m lower than the first if the pipe diameter at the second point is twice that at the first.

51. Lift on an airplane. Air streams horizontally past a small airplane’s wings such that the speed is 70.0 m/s over the top surface and 60.0 m/s past the bottom surface. If the plane has a mass of 1340 kg and a wing area of 16.2 m2, what is the net vertical force (including the effects of gravity) on the airplane? The density of the air is 1.20 kg/m3.

52. A golf course sprinkler system discharges water from a horizontal pipe at the rate of 7200 cm3/s. At one point in the pipe, where the radius is 4.00 cm, the water’s absolute pressure is 2.40 × 105 Pa. At a second point in the pipe, the water passes through a constriction where the radius is 2.00 cm. What is the water’s absolute pressure as it flows through this constriction?

53. Water discharges from a horizontal cylindrical pipe at the rate of 465 cm3/s. At a point in the pipe where the radius is 2.05 cm, the absolute pressure is 1.60 × 105 Pa. What is the pipe’s radius at a constriction if the pressure there is reduced to 1.20 × 105 Pa?

54. Artery blockage. A medical technician is trying to determine what percentage of a patient’s artery is blocked by plaque. To do this, she measures the blood pressure just before the region of blockage and finds that it is 1.20 × 104 Pa while in the region of blockage it is 1.15 × 104 Pa. Furthermore, she knows that blood flowing through the normal artery just before the point of blockage is traveling at 30.0 cm/s and the specific gravity of this patient’s blood is 1.06. What percentage of the cross-sectional area of the patient’s artery is blocked by the plaque?

55. At a certain point in a horizontal pipeline, the water’s speed is 2.50 m/s and the gauge pressure is 1.80 × 104 Pa. Find the gauge pressure at a second point in the line if the cross-sectional area at the second point is twice that at the first.

56. With what terminal speed would a steel ball bearing 2.00 mm in diameter fall in a liquid of viscosity 0.150 N·s/m2 if we could neglect buoyancy?

57. What speed must a gold sphere of radius 3.00 mm have in castor oil for the viscous drag force to be one-fourth of the weight of the sphere? The density of gold is 19300 kg/m3 and the viscosity of the oil is 0.986 N·s/m2.

58. A copper sphere with a mass of 0.20 g and a density of 8900 kg/m3 is observed to fall with a terminal speed of 6.0 cm/s in an unknown liquid. Find the viscosity of the unknown liquid if its buoyancy can be neglected.

59. Clogged artery. Viscous blood is flowing through an artery partially clogged by cholesterol. A surgeon wants to remove enough of the cholesterol to double the flow rate of blood through this artery. If the original diameter of the artery is D, what should be the new diameter (in terms of D) to accomplish this for the same pressure gradient?

60. Advertisements for a certain small car claim that it floats in water. (a) If the car’s mass is 900 kg and its interior volume is 3.0 m3 what fraction of the car is immersed when it floats? You can ignore the volume of steel and other materials. (b) Water gradually leaks in and displaces the air in the car. What fraction of the interior volume is filled with water when the car sinks?

61. A U-shaped tube open to the air at both ends contains some mercury. A quantity of water is carefully poured into the left arm of the U-shaped tube until the vertical height of the water column is 15.0 cm. (a) What is the gauge pressure at the water-mercury interface? (b) Calculate the vertical distance h from the top of  the  mercury in the right-hand arm of the tube to the top of the water in the left-hand arm.

62. An open barge has the dimensions shown in Figure 13.44. If the barge is made out of 4.0-cm-thick steel plate on each of its four sides and its bottom, what mass of coal can the barge carry in fresh water without sinking? Is there enough room in the barge to hold this amount of coal?  (The density of coal is about 1500 kg/m3.

63. A piece of wood is 0.600 m long, 0.250 m wide, and 0.080 m thick. Its density is 600 kg/m3. What volume of lead must be fastened underneath it to sink the wood in calm water so that its top is just even with the water level? What is the mass of this volume of lead?

64. A hot-air balloon has a volume of 2200 m3. The balloon fabric (the envelope) weighs 900 N. The basket with gear and full propane tanks weighs 1700 N. If the balloon can barely lift an additional 3200 N of passengers, breakfast, and champagne when the outside air density is 1.23 kg/m3, what is the average density of the heated gases in the envelope?

65. In seawater, a life preserver with a volume of 0.0400 m3 will support a 75.0 kg person (average density 980 kg/m3) with 20% of the person’s volume above water when the life preserver is fully submerged. What is the density of the material composing the life preserver?

66. Block A in Figure 13.45 hangs by a cord from spring balance D and is submerged in a liquid C contained in beaker B. The mass of the beaker is 1.00 kg; the mass of the liquid is 1.80 kg. Balance D reads 3.50 kg and balance E reads 7.50 kg. The volume of block A is 3.80 × 10–3 m3 (a) What is the density of the liquid? (b) What will each balance read if block A is pulled up out of the liquid?

67. A hunk of aluminum is completely covered with a gold shell to form an ingot of weight 45.0 N. When you suspend the ingot from a spring balance and submerge the ingot in water, the balance reads 39.0 N. What is the weight of the gold in the shell?

68. A liquid is used to make a mercury-type barometer, as described in Section 13.2. The barometer is intended for spacefaring astronauts. At the surface of the earth, the column of liquid rises to a height of 2185 mm, but on the surface of Planet X, where the acceleration due to gravity is one-fourth of its value on earth, the column rises to only 725 mm. Find (a) the density of the liquid and (b) the atmospheric pressure at the surface of Planet X.

69. An  open  cylindrical  tank  of  acid  rests  at  the  edge  of  a table 1.4 m above the floor of the chemistry lab. If this tank springs a small hole in the side at its base, how far from the foot of the table will the acid hit the floor if the acid in the tank is 75 cm deep?

70. Water stands at a depth H in a large, open tank whose side walls are vertical (Fig. 13.46). A hole is made in one of the walls at a depth h below the water surface. (a) At what distance R from the foot of the wall does the emerging stream strike the floor? (b) How far above the bottom of the tank could a second hole be cut so that the stream emerging from it could have the same range as for the first hole?

71. Exploring Europa’s oceans. Europa, a satellite of Jupiter, appears to have an ocean beneath its icy surface. Proposals have been made to send a robotic submarine to Europa to see if there might be life there. There is no atmosphere on Europa, and we shall assume that the surface ice is thin enough that we can neglect its weight and that the oceans are fresh water having the same density as on the earth. The mass and diameter of Europa have been measured to be 4.78 × 1022 kg and 3130 km, respectively. (a) If the submarine intends to submerge to a depth of 100 m, what pressure must it be designed to withstand? (b) If you wanted to test this submarine before sending it to Europa, how deep would it have to go in our oceans to experience the same pressure as the pressure at a depth of 100 m on Europa?

72. The horizontal pipe shown in Figure 13.47 has a cross-sectional area of 40.0 cm2 at the wider portions and 10.0 cm2 at the constriction. Water is flowing in the pipe, and the discharge from the pipe is 6.00 × 10–3 m3/s.  Find (a) the flow speeds at the wide and the narrow portions; (b) the pressure difference between these portions; (c) the difference in height between the mercury columns in the U-shaped tube.

73. Venturi meter. The Venturi meter is a device used to measure the speed of a fluid traveling through a pipe. Two cylinders are inserted in small holes in the pipes, as shown in Figure 13.48. Since the cross-sectional area is different at the two places, the speed and pressure will be different there also. The difference in the heights of the two columns can easily be measured, as can the cross-sectional areas A1 and A2. Notice that points 1 and 2 in the figure are both at the same vertical height. (a) Show that Δp = ρgh, where ρ is the density of the fluid and p is the pressure difference between points 1 and 2. (b) Apply Bernoulli’s equation and the continuity condition to show that the speed at point 1 is given by the equation v1 = √(2gh/((A1/A2)2 – 1) (c) How would you find the speed at point 2?

74. Compressible fluids. Throughout this chapter, we have dealt only with incompressible fluids. But under very high pressure, fluids do, in fact, compress. (a) Show that the continuity condition for compressible fluids is ρ1A1v1 = ρ2A2v2, where ρ is the density of the fluid. (b) Show that your result reduces to the familiar result for incompressible fluids.

Chapter 14

Multiple-Choice Problems

1. If heat Q is required to increase the temperature of a metal object from 4 °C to 6 °C the amount of heat necessary to increase its temperature from 6 °C to 12 °C is most likely
A. Q.               B. 2Q.                         C. 3Q.                         D. 4Q.

2. A metal bar expands by 1.0 mm when its temperature is increased by 1.0 °C. Which of the following statements about this bar must be true?
A. It will expand by more than 1.0 mm if its temperature is increased by 1.0 K.
B. It will expand by less than 1.0 mm if its temperature is increased by 1.0 F°
C. It will expand by 2.00 mm if its temperature is increased by 2.0 K.
D. It will expand by more than 2.0 mm if its temperature is increased by 2.0 F°

3.  If an amount of heat Q is needed to increase the temperature of a solid metal sphere of diameter D  from 4 °C to 7 °C the amount of heat needed to increase the temperature of a solid sphere of  diameter 2D of the same metal from 4 °C to 7 °C is
A. Q                B. 2Q              C. 4Q              D. 8Q

4.  If you mix 100 g of ice at 0 °C with 100 g of boiling water at 100 °C in a perfectly insulated container, the final stabilized temperature will be
A. 0 °C            B. between 0 °C and 50 °C                 C. 50 °C          D.  between 50 °C and 100 °C

5. A sphere radiates energy at a rate of 1.00 J/s when its temperature is 100 °C. At what rate will it radiate energy if its temperature is increased to 200 °C? (Neglect any heat transferred back into the sphere.)
A. 1.27 J/s       B. 2.00 J/s       D. 2.59 J/s       D. 16.0 J/s

6.  Two identical closed, insulated containers hold equal amounts of water at a temperature of 50 °C. One kilogram of ice at 0 °C is added to the water in container A; one kilogram of steam at 100 °C is added to container B. After all the ice has melted and all the steam has condensed, which container’s temperature will have changed more?
A. Container A
B. Container B
C. Their temperatures change by the same amount.

7. The wall of a furnace conducts 1000 cal/min of heat out of the furnace and into the factory. If the wall’s thickness is doubled, but nothing else changes (including the inner and outer surface temperatures), the amount of heat it conducts into the factory will be
A. 4000 cal/min           B. 2000 cal/min           C. 500 cal/min             D. 250 cal/min

8. A thin metal rod expands 1.5 mm when its temperature is increased by 2.0 °C. If an identical rod of the same material, except twice as long, also has its temperature increased by it will expand
A. 0.75 mm.     B. 1.0 mm.        C. 3.0 mm.       D. 6.0 mm.

9.  Water has a specific heat capacity nearly 9 times that of iron. Suppose a 50-g pellet of iron at a temperature of 200 °C is dropped into 50 g of water at a temperature of 20 °C.  When the system reaches thermal equilibrium, its temperature will be
A. closer to 20 °C
B. closer to 200 °C
C. halfway between the two initial temperatures.

10. A cylindrical metal bar conducts heat at a rate R from a hot reservoir to a cold reservoir. If both its length and diameter are doubled, it will conduct heat at a rate
A. R                B. 2R               C. 4R              D. 8R.

11. Two rods P and Q of identical shape and size, but made of different metals, are joined end to end. The left end of rod P is kept at 100 °C while the right end of rod Q is at 0 °C causing heat to flow down the rods. The graph in Figure 14.25 shows the temperature distribution along the rods from the hot end to the cold end. Which rod has the higher thermal conductivity?
A. rod P.
B. rod Q.
C. We cannot tell from the available data.

12. The thermal conductivity of concrete is 0.80 W/(m·K) and the thermal conductivity of a certain wood is 0.10 W/(m·K). How thick would a solid concrete wall have to be to have the same rate of heat flow as an 8.0-cm-thick solid wall of this wood? Both walls have the same surface area and the same
interior and exterior temperatures.
A. 1.0 cm.        B. 16 cm.          C. 32 cm.         D. 64 cm.

13. The graph in Figure 14.26 shows the temperature as a function of time for a sample of material being heated at a constant rate. What segment or segments of this graph show the sample existing in two phases (or states)?
A.  OA.
B.  AB.
C.  OA and BC.

14. For the sample in the previous question, what segment of the graph shows the specific heat capacity of the sample to be greatest?
A. OA.            B. AB.             C. BC.

15. You enter a cold room containing a wooden table and a steel table; both have been there for a long time. Why does the steel table feel much colder than the wooden one?
A. Steel is a much better conductor of heat than wood is.
B. The steel stores much more cold than the wood does.
C. Steel has a higher specific heat capacity than wood has.
D. The temperature of the steel table is lower than that of the wooden table.

Problems

5. At what temperature do the Fahrenheit and Celsius scales coincide? (b) Is there any temperature at which the Kelvin and Celsius scales coincide?

6. Convert the following Kelvin temperatures to the Celsius and Fahrenheit scales: (a) the midday temperature at the surface of the moon (400 K); (b) the temperature at the tops of the clouds in the atmosphere of Saturn (95 K); (c) the temperature at the center of the sun (1.55 × 107 K)

7. The Eiffel Tower in Paris is 984 ft tall and is made mostly of steel.  If  this  is  its  height  in  winter  when  its  temperature  is –8.00 °C, how much additional vertical distance must you cover if you  decide to climb it during a summer heat wave when its temperature is 40.0 °C (b) Express the coefficient of linear expansion of steel in terms of Fahrenheit degrees.

8. A steel bridge is built in the summer when its temperature is 35.0 °C. At the time of construction, its length is 80.00 m. What is the length of the bridge on a cold winter day when its temperature is –12.0°C?

9. A metal rod is 40.125 cm long at 20.0 °C and 40.148 cm long at 45.0 °C. Calculate the average coefficient of linear expansion of the rod’s material for this temperature range.

10. (a) Steel train rails are laid in 12.0-m-long segments placed end to end. The rails are laid on a winter day when their temperature is –2.00 °C. How much space must be left between adjacent rails if they are just to touch on a summer day when their temperature is 33.0 °C? (b) If the rails are mistakenly laid in contact with each other, what is the stress in them on a summer day when their temperature is 33.0 °C?

11. An underground tank with a capacity of 1700 L is completely filled with ethanol that has an initial temperature of 19.0 °C. After the ethanol has cooled off to the temperature of the tank and ground, which is 10.0 °C, how much air space will there be above the ethanol in the tank?

12. A copper cylinder is initially at 20.0 °C. At what temperature will its volume be 0.150% larger than it is at 20.0 °C?

13. A geodesic dome constructed with an aluminum framework is a nearly perfect hemisphere; its diameter measures 55.0 m on a winter day at a temperature of –15 °C. How much more interior space does the dome have in the summer, when the temperature is 35 °C?

14. The outer diameter of a glass jar and the inner diameter of its iron lid are both 725 mm at room temperature (20.0 °C). What will be the size of the mismatch between the lid and the jar if the lid is briefly held under hot water until its temperature rises to 50.0 °C without changing the temperature of the glass?

15. A glass flask whose volume is 1000.00 cm3 at 0.0 °C is completely filled with mercury at this temperature. When flask and mercury are warmed to 55.0 °C, 8.95 cm3 of mercury overflow. Compute the coefficient of volume expansion of the glass.

16. Ensuring a tight fit. Aluminum rivets used in airplane construction are made slightly larger than the rivet holes and cooled by “dry ice” (solid CO2) before being driven. If the diameter of a hole is 4.500 mm, what should be the diameter of a rivet at 23.0 °C, if its diameter is to equal that of the hole when the rivet is cooled to –78.0 °C, the temperature of dry ice? Assume that the expansion coefficient remains constant at the value given in Table 14.1.

17. The markings on an aluminum ruler and a brass ruler begin at the left end; when the rulers are at 0.00 °C, they are perfectly aligned. How far apart will the 20.0 cm marks be on the two rulers at 100.0 °C if the left-hand ends are kept precisely aligned?

18. (a) How much heat is required to raise the temperature of 0.250 kg of water from 20.0 °C to 30.0 °C? (b) If this amount of heat is added to an equal mass of mercury that is initially at 20.0 °C, what is its final temperature?

19. One of the moving parts of an engine contains 1.60 kg of aluminum and 0.300 kg of iron and is designed to operate at 210 °C. How much heat is required to raise its temperature from 20.0 °C to 210 °C.

20. In an effort to stay awake for an all-night study session, a student makes a cup of coffee by first placing a 200.0 W electric immersion heater in 0.320 kg of water. (a) How much heat must be added to the water to raise its temperature from 20.0 °C to 80.0 °C? (b) How much time is required if all of the heater’s power goes into heating the water?

21. Heat loss during breathing. In very cold weather, a significant mechanism for heat loss by the human body is energy expended in warming the air taken into the lungs with each breath. (a) On a cold winter day when the temperature is –20 °C, what is the amount of heat needed to warm to internal body temperature (37 °C) the 0.50 L of air exchanged with each breath? Assume that the specific heat capacity of air is1020 J/(kg·K) and that 1.0 L of air has a mass of 1.3 g. (b) How much heat is lost per  hour if the respiration rate is 20 breaths per minute?

22. A nail driven into a board increases in temperature. If 60% of the kinetic energy delivered by a 1.80 kg hammer with a speed of 7.80 m/s is transformed into heat that flows into the nail and does not flow out, what is the increase in temperature of an 8.00 g aluminum nail after it is struck 10 times?

23. You are given a sample of metal and asked to determine its specific heat. You weigh the sample and find that its weight is 28.4 N. You carefully add 1.25 × 104 J of heat energy to the sample and find that its temperature rises 18.0 °C. What is the sample’s specific heat?

24. A 25,000-kg subway train initially traveling at 5.5 m/s slows to a stop in a station and then stays there long enough for its brakes to cool. The station’s dimensions are 65.0 m long by 20.0 m wide by 12.0 m high. Assuming all the work done by the brakes in stopping the train is transferred as heat uniformly to all the air in the station, by how much does the air temperature in the station rise? Take the density of the air to be 1.20 kg/m3 and its specific heat to be 1020 J/(kg·K).

25. You add 8950 J of heat to 3.00 mol of iron. (a) What is the temperature increase of the iron? (b) If this same amount of heat is added to 3.00 kg of iron, what is the iron’s temperature increase? (c) Explain the difference in your results for parts (a) and (b).

26. From a height of 35.0 m, a 1.25 kg bird dives (from rest) into a small fish tank containing 50.0 kg of water. What is the maximum rise in temperature of the water if the bird gives it all of its mechanical energy?

27. A 15.0 g bullet traveling horizontally at passes through a tank containing 13.5 kg of water and emerges with a speed of 534 m/s. What is the maximum temperature increase that the water could have as a result of this event?

28. Maintaining body temperature. While running, a 70 kg student generates thermal energy at a rate of 1200 W. To maintain a constant body temperature of 37 °C, this energy must be removed by perspiration or other mechanisms. If these mechanisms failed and the heat could not flow out of the student’s body, for what amount of time could a student run before irreversible body damage occurred?  (Protein structures in the body are damaged irreversibly if the body temperature rises to 44 °C or above. The specific heat capacity of a typical human body is 3480 J/(kg·K), slightly less than that of water. The difference is due to the presence of protein, fat, and minerals, which have lower specific heat capacities.)

29. A technician measures the specific heat capacity of an unidentified liquid by immersing an electrical resistor in it. Electrical energy is converted to heat, which is then transferred to the liquid for 120 s at a constant rate of 65.0 W. The mass of the liquid is 0.780 kg, and its temperature increases from 18.55 °C to 22.54 °C (a) Find the average specific heat capacity of the liquid in this temperature range. Assume that negligible heat is transferred to the container that holds the liquid and that no heat is lost to the surroundings. (b) Suppose that in this experiment heat transfer from the liquid to the container or its surroundings cannot be ignored. Is the result calculated in part (a) an overestimate or an underestimate of the average specific heat capacity? Explain.

30. Much of the energy of falling water in a waterfall is converted into heat. If all the mechanical energy is converted into heat that stays in the water, how much of a rise in temperature occurs in a 100 m waterfall?

31. Consult Table 14.4. (a) How much heat is required to melt 0.150 kg of lead at 327.3 °C (b) How much heat would be needed to evaporate this lead at 1750 °C (c) If the total heat added from parts (a) and (b) were put into ice at 0.00 °C how many grams of the ice would it melt?

32. A blacksmith cools a 1.20-kg chunk of iron, initially at a temperature of 650.0 °C by trickling 15.0 °C water over it. All the water boils away, and the iron ends up at a temperature of 120.0 °C. How much water did the blacksmith trickle over the iron?

33. Treatment for a stroke. One suggested treatment for a person who has suffered a stroke is to immerse the patient in an ice-water bath at 0 °C to lower the body temperature, which prevents damage to the brain. In one set of tests, patients were cooled until their internal temperature reached 32.0 °C. To treat a 70.0 kg patient, what is the minimum amount of ice (at 0 °C) that you need in the bath so that its temperature remains at 0 °C. The specific heat capacity of the human body is 3480 J/(kg·K) and recall that normal body temperature is 37.0 °C.

34. A container holds 0.550 kg of ice at –15.0 °C. The mass of the container can be ignored. Heat is supplied to the container at the constant rate of 800.0 J/min for 500.0 min. (a) After how many minutes does the ice start to melt? (b) After how many minutes, from the time when the heating is first started, does the temperature begin to rise above 0.00 °C (c) Plot a curve showing the temperature as a function of the time elapsed.

35. An asteroid with a diameter of 10 km and a mass of 2.60 × 1015 kg impacts the earth at a speed of 32.0 km/s, landing in the Pacific Ocean. If 1.00% of the asteroid’s kinetic energy goes to boiling the ocean water (assume an initial water temperature of 10.0 °C), what mass of water will be boiled away by the collision? (For comparison, the mass of water contained in Lake Superior is about 2 × 1015 kg)

36. Evaporative cooling. The evaporation of sweat is an important mechanism for temperature control in some warmblooded animals. (a) What mass of water must evaporate from the skin of a 70.0 kg man to cool his body 1.00 °C? The heat of vaporization of water at body temperature (37 °C) is 2.42 × 106 J/kg. The specific heat capacity of a typical human body is 3480 J/(kg·K) (b) What volume of water must the man drink to replenish the evaporated water? Compare this result with the volume of a soft-drink can, which is 355 cm3.

37. An ice-cube tray contains 0.350 kg of water at 18.0 °C. How much heat must be removed from the water to cool it to 0.00 °C and freeze it? Express your answer in joules and in calories.

38. How much heat is required to convert 12.0 g of ice at –10.0 °C to steam at 100.0 °C? Express your answer in joules and in calories.

39. Steam burns vs. water burns. What is the amount of heat entering your skin when it receives the  heat released (a) by 25.0 g of steam initially at 100.0 °C that cools to 34.0 °C (b) by 25.0 g of water initially at 100.0 °C that cools to 34.0 °C (c) What do these results tell you about the relative severity of steam and hot-water burns?

40. Bicycling on a warm day. If the air temperature is the same as the temperature of your skin (about 30 °C) your body cannot get rid of heat by transferring it to the air. In that case, it gets rid of the heat by evaporating water (sweat). During bicycling, a typical 70 kg person’s body produces energy at a rate of about 500 W due to metabolism, 80% of which is converted to heat. (a) How many kilograms of water must the person’s body evaporate in an hour to get rid of this heat? The heat of vaporization of water at body temperature is 2.42 × 106 J/kg (b) The evaporated water must, of course, be replenished, or the person will dehydrate. How many 750 mL bottles of water must the bicyclist drink per hour to replenish the lost water?

41. Overheating. (a) By how much would the body temperature of the bicyclist in the previous problem increase in an hour if he were unable to get rid of the excess heat? (b) Is this temperature increase large enough to be serious? To find out, how high a fever would it be equivalent to, in °F

42. You have 750 g of water at 10.0 °C in a large insulated beaker. How much boiling water at 100 °C must you add to this beaker so that the final temperature of the mixture will be 75 °C?

43. A 0.500 kg chunk of an unknown metal that has been in boiling  water  for  several  minutes  is  quickly  dropped  into  an insulating Styrofoam™ beaker containing 1.00 kg of water at room temperature (20.0 °C). After waiting and gently stirring for 5.00 minutes, you observe that the water’s temperature has reached a constant value of 22.0 °C. (a) Assuming that the Styrofoam™ absorbs a negligibly small amount of heat and that no heat was lost to the surroundings, what is the specific heat capacity of the metal? (b) Which is more useful for storing energy from heat, this metal or an equal weight of water? Explain. (c) What if the heat absorbed by the Styrofoam™ actually is not negligible. How would the specific heat capacity you calculated in part (a) be in error? Would it be too large, too small, or still correct? Explain your reasoning.

44. A copper pot with a mass of 0.500 kg contains 0.170 kg of water, and both are at a temperature of 20.0 °C.  A 0.250 kg block of iron at 85.0 °C is dropped into the pot. Find the final temperature of the system, assuming no heat loss to the surroundings.

45. In a physics lab experiment, a student immersed 200 one-cent coins (each having a mass of 3.00 g) in boiling water. After they reached thermal equilibrium, she quickly fished them out and dropped them into 0.240 kg of water at 20.0 °C in an insulated container of negligible mass. What was the final temperature of the coins?

46. A laboratory technician drops an 85.0 g solid sample of unknown material at a temperature of 100.0 °C into a calorimeter. The calorimeter can is made of 0.150 kg of copper and contains 0.200 kg of water, and both the can and water are initially at 19 °C. The final temperature of the system is measured to be 26.1 °C. Compute the specific heat capacity of the sample.

47. A 4.00 kg silver ingot is taken from a furnace, where its temperature is 750 °C and placed on a very large block of ice at 0.00 °C. Assuming that all the heat given up by the silver is used to melt the ice and that not all the ice melts, how much ice is melted?

48. An insulated beaker with negligible mass contains 0.250 kg of water at a temperature of 75.0 °C. How many kilograms of ice at a temperature of –20.0 °C must be dropped in the water so that the final temperature of the system will be 30.0 °C?

49. A StyrofoamTM bucket of negligible mass contains 1.75 kg of water and 0.450 kg of ice. More ice, from a refrigerator at –15.0 °C is added to the mixture in the bucket, and when thermal equilibrium has been reached, the total mass of ice in the bucket is 0.778 kg. Assuming no heat exchange with the surroundings, what mass of ice was added?

50. A slab of a thermal insulator with a cross-sectional area of 100 cm2 is 3.00 cm thick. Its thermal   conductivity is 0.075 W/(m·K). If the temperature difference between opposite faces is 80 °C how much heat flows through the slab in 1 day?

51. You are asked to design a cylindrical steel rod 50.0 cm long, with a circular cross section, that will conduct 150.0 J/s from a furnace at 400.0 °C to a container of boiling water under 1 atmosphere of pressure. What must the rod’s diameter be?

52. Conduction through the skin. The blood plays an important role in removing heat from the body by bringing this heat directly to the surface where it can radiate away. Nevertheless, this heat must still travel through the skin before it can radiate away. We shall assume that the blood is brought to the bottom layer of skin at a temperature of 37 °C and that the outer surface of the skin is at 30.0 °C. Skin varies in thickness from 0.50 mm to a few millimeters on the palms and soles, so we shall assume an average thickness of 0.75 mm. A 165 lb, 6 ft person has a surface area of about 2.0 m2 and loses heat at a net rate of 75 W while resting. On the basis of our assumptions, what is the thermal conductivity of this person’s skin?

53. A pot with a steel bottom 8.50 mm thick rests on a hot stove. The area of the bottom of the pot is 0.150 m2. The water inside the pot is at 100.0 °C, and 0.390 kg are evaporated every 3.00 min. Find the temperature of the lower surface of the pot, which is in contact with the stove.

54. A carpenter builds an exterior house wall with a layer of wood 3.0 cm thick on the outside and a layer of Styrofoam™ insulation 2.2 cm thick on the inside wall surface. The wood has a thermal conductivity of 0.080 W/(m·K) and the Styrofoam™  has a thermal conductivity of 0.010 W/(m·K). The interior surface temperature is 19.0 °C and the exterior surface temperature is –10.0 °C. (a) What is the temperature at the plane where the wood meets the Styrofoam™? (b) What is the rate of heat flow per square meter through this wall?

55. A picture window has dimensions of 1.40 m × 2.50 m and is made of glass 5.20 mm thick. On a winter day, the outside temperature is –20.0 °C while the inside temperature is a comfortable 19.56 °C.  (a) At what rate is heat being lost through the window by conduction?  (b) At what rate would heat be lost through the window if you covered it with a 0.750-mm-thick layer of paper (thermal conductivity 0.0500 W/(m·K)?

56. One end of an insulated metal rod is maintained at 100 °C while the other end is maintained at 0 °C by an ice–water mixture. The rod is 60.0 cm long and has a cross-sectional area of 1.25 cm2. The heat conducted by the rod melts 8.50 g of ice in 10.0 min. Find the thermal conductivity k of the metal.

57. Mammal insulation. Animals in cold climates often depend on two layers of insulation: a layer of body fat [of thermal conductivity 0.20 W/(m·K)] surrounded by a layer of air trapped inside fur or down. We can model a black bear (Ursus americanus) as a sphere 1.5 m in diameter having a layer of fat 4.0 cm thick. (Actually, the thickness varies with the season, but we are interested in hibernation, when the fat layer is thickest.) In studies of bear hibernation, it was found that the outer surface layer of the fur is at 2.7 °C during hibernation, while the inner surface of the fat layer is at 31.0 °C (a) What is the temperature at the fat–inner fur boundary, and (b) how thick should the air layer (contained within the fur) be so that the bear loses heat at a rate of 50.0 W?

58. A box-shaped wood stove has dimensions of 0.75 m × 1.2 m × 0.40 m, an emissivity of 0.85, and a surface temperature of 205 °C. Calculate its rate of radiation into the surrounding space.

59. Radiation by the body. The amount of heat radiated by the body depends on its surface temperature and area. Typically, this temperature is about 30 °C (although it can vary). The surface area depends on the person’s height and weight. An empirical formula for the surface area of a person’s body is
A(in m2) = (0.202)M0.425h0.725, where M is the person’s mass (in kilograms) and h is his or her height (in meters). (a) What would be the surface area of a 165 lb (75 kg), 6 ft (1.83 m) person? (b) How much heat would the person radiate away per second at a skin temperature of (At the low temperatures of room-temperature objects, nearly all the heat radiated is infrared radiation, for which the emissivity is essentially 1, regardless of the amount of skin pigment.) (c) How much net heat would radiation remove from the person’s body if the air temperature is 20 °C. (d) Take measurements on your own body to test the validity of the area formula. Treat yourself as a sphere and several cylinders.

60. How large is the sun? By measuring the spectrum of wavelengths of light from our sun, we know that its surface temperature is 5800 K. By measuring the rate at which we receive its energy on earth, we know that it is radiating a total of 3.92 × 1026 J/s and behaves nearly like an ideal blackbody. Use this information to calculate the diameter of our sun.

61. Basal metabolic rate. The basal metabolic rate is the rate at which energy is produced in the body when a person is at rest. A 75 kg (165 lb) person of height 1.83 m (6 ft) would have a body surface area of approximately 2.0 m2 (a) What is the net amount of heat this person could radiate per second into a room at 18 °C (about 65 °F) if his skin’s surface temperature is 30 °C (At such temperatures, nearly all the heat is infrared radiation, for which the body’s emissivity is 1.0, regardless of the amount of pigment.)  (b)  Normally, 80% of the energy produced by metabolism goes into heat, while the rest goes into things like pumping blood and repairing cells. Also normally, a person at rest can get rid of this excess heat just through radiation. Use your answer to part (a) to find this person’s basal metabolic rate.

62. The emissivity of tungsten is 0.35. A tungsten sphere with a radius of 1.50 cm is suspended within a large evacuated enclosure whose walls are at 290 K. What power input is required to maintain the sphere at a temperature of 3000 K if heat conduction along the supports is negligible?

63. Size of a lightbulb filament. The operating temperature of a tungsten filament in an incandescent lightbulb is 2450 K, and its emissivity is 0.35. Find the surface area of the filament of a 150 W bulb if all the electrical energy consumed by the bulb is radiated by the filament as light. (In reality, only a small fraction of the radiation appears as visible light.

64. A spherical pot of hot coffee contains 0.75 L of liquid (essentially water) at an initial temperature of 95.0 °C. The pot has an emissivity of 0.60, and the surroundings are at a temperature of 20.0 °C. Calculate the coffee’s rate of heat loss by radiation.

65. An 8.50 kg block of ice at 0 °C is sliding on a rough horizontal icehouse floor (also at 0 °C) at 15.0 m/s. Assume that half of any heat generated goes into the floor and the rest goes into the ice. (a) How much ice melts after the speed of the ice has been reduced to 10.0 m/s? (b) What is the maximum amount of ice that will melt?

66. Use Fig. 14.9 to find the approximate coefficient of volume expansion of water at 2.0 °C and at 8.0 °C.

67. Global warming. As the earth warms, sea level will rise due to melting of the polar ice and thermal expansion of the oceans.  Estimates of the expected temperature increase vary, but 3.5 °C by the end of the century has been plausibly suggested. If we assume that the temperature of the oceans also increases by this amount, how much will sea level rise by the year  2100  due  only  to  the  thermal  expansion  of  the  water?  Assume, reasonably, that the ocean basins do not expand appreciably. The average depth of the ocean is 4000 m, and the coefficient of volume expansion of water at 20 °C is 0.207 × 10–3 (°C)–1.

68. A Foucault pendulum consists of a brass sphere with a diameter of 35.0 cm suspended from a steel cable 10.5 m long (both measurements made at 20.0 °C). Due to a design oversight, the swinging sphere clears the floor by a distance of only 2.00 mm when the temperature is 20.0 °C. At what temperature will the sphere begin to brush the floor?

69. On-demand water heaters. Conventional hot-water heaters consist of a tank of water maintained at a fixed temperature. The hot water is to be used when needed. The drawback is that energy is wasted because the tank loses heat when it is not in use, and you can run out of hot water if you use too much. Some utility companies are encouraging the use of on demand water heaters (also known as flash heaters), which consist of heating units to heat the water as you use it. No water tank is involved, so no heat is wasted. A typical household shower flow rate is with the tap water 2.5 gal/min (9.46 L/min) being heated from 50 °F (10 °C) to 120 °F (49 °C) by the on-demand heater. What rate of heat input (either electrical or from gas) is required to operate such a unit, assuming that all the heat goes into the water?

70. Burning fat by exercise. Each pound of fat contains 3500 food calories. When the body metabolizes food, 80% of this energy goes to heat. Suppose you decide to run without stopping, an activity that produces 1290 W of metabolic power for a typical person. (a) For how many hours must you run to burn up 1 lb of fat? Is this a realistic exercise plan? (b) If you followed your planned exercise program, how much heat would your body produce when you burn up a pound of fat? (c) If you needed to get rid of all of this excess heat by evaporating water (i.e., sweating), how many liters would you need to evaporate? The heat of vaporization of water at body temperature is 2.42 × 106 J/kg.

71. Shivering. You have no doubt noticed that you usually shiver when you get out of the shower. Shivering is the body’s way of generating heat to restore its internal temperature to the normal 37 °C and it produces approximately 290 W of heat power per square meter of body area. A 68 kg (150 lb), 1.78 m (5 foot, 10 inch) person has approximately 1.8 m2 of surface area. How long would this person have to shiver to raise his or her body temperature by 1.0 °C assuming that none of this heat is lost by the body? The specific heat capacity of the body is about 3500 J/(kg·K).

72. A steel ring with a 2.5000-in. inside diameter at 20.0 °C is to be warmed and slipped over a brass shaft with a 2.5020-in. outside diameter at 20.0 °C. (a) To what temperature should the ring be warmed? (b) If the ring and the shaft together are cooled by some means such as liquid air, at what temperature will the ring just slip off the shaft?

73. Pasta time! You are making pesto for your pasta and have a cylindrical measuring cup 10.0 cm high made of ordinary glass (β = 2.7 × 10–5 (°C) –1) and that is filled with olive oil (β = 6.8 × 10–4 (°C) –1) to a height of 1.00 mm below the top of the cup. Initially, the cup and oil are at a kitchen temperature of 22.0 °C. You get a phone call and forget about the olive oil, which you inadvertently leave on the hot stove. The cup and oil heat up slowly and have a common temperature. At what temperature will the olive oil start to spill out of the cup?

74. A copper calorimeter can with mass 0.100 kg contains 0.160 kg of water and 0.018 kg of ice in thermal equilibrium at atmospheric pressure. If 0.750 kg of lead at a temperature of 255 °C is dropped into the can, what is the final temperature of the system if no heat is lost to the surroundings?

75. A piece of ice at 0 °C falls from rest into a lake whose temperature is 0 °C and 1.00% of the ice melts. Compute the minimum height from which the ice has fallen.

76. Hot air in a physics lecture. (a) A typical student listening attentively to a physics lecture has a heat output of 100 W. How much heat energy does a class of 90 physics students release into a lecture hall over the course of a 50 min lecture? (b) Assume that all the heat energy in part (a) is transferred to the 3200 m3 of air in the room. The air has a specific heat capacity of 1020 J/(kg·K) and a density of  1.20 kg/m3. If none of the heat escapes and the air-conditioning system is off, how much will the temperature of the air in the room rise during the 50 min lecture? (c) If the class is taking an exam, the heat output per student rises to 280 W. What is the temperature rise during 50 min in this case?

77. “The Ship of the Desert.” Camels require very little water because they are able to tolerate relatively large changes in their body temperature. While humans keep their body temperatures constant to within one or two Celsius degrees, a dehydrated camel permits its body temperature to drop to 34.0 °C overnight and rise to 40.0 °C during the day. To see how effective this mechanism is for saving water, calculate how many liters of water a 400-kg camel would have to drink if  it attempted to keep its body temperature at a constant 34.0 °C by evaporation of sweat during the day (12 hours)  instead  of letting it rise to 40.0 °C

78. A worker pours 1.250 kg of molten lead at a temperature of 323.3 °C into 0.5000 kg of water at a temperature of 75.0 °C in an insulated bucket of negligible mass. Assuming no heat loss to the surroundings, calculate the mass of lead and water remaining in the bucket when the materials have reached thermal equilibrium.

79. Time for a lake to freeze over. When the air temperature is below 0 °C the water at the surface of a lake freezes to form a sheet of ice. If the upper surface of an ice sheet 25.0 cm thick is at –10.0 °C and the bottom surface is at 0.00 °C calculate the time it will take to add 2.0 mm to the thickness of this sheet.

80. Jogging in the heat of the day. You have probably seen people jogging in extremely hot weather and wondered “Why?” As we shall see, there are good reasons not to do this! When  jogging strenuously, an average runner of mass 68 kg and surface area 1.85 m2 produces  energy  at  a  rate  of  up to1300 W, 80% of which is converted to heat. The jogger radiates heat, but actually absorbs more from the hot air than he radiates away. At such high levels of activity, the skin’s temperature can be elevated to around 33 °C instead of the usual 30 °C (We shall neglect conduction, which would bring even more heat into his body.) The only way for the body to get rid of this extra heat is by evaporating water (sweating). (a) How much heat per second is produced just by the act of jogging? (b) How much net heat per second does the runner gain just from radiation if the air temperature is 40.0 °C. (c) What is the total amount of excess heat this runner’s body must get rid of per second? (d) How much water must the jogger’s body evaporate every minute due to his activity? The heat of vaporization of water at body temperature is 2.42 × 106 J/kg (e) How many 750 mL bottles of water must he drink after (or preferably before!) jogging for a half hour? Recall that a liter of water has a mass of 1.0 kg.

81. Overheating while jogging. (a) If the jogger in the previous problem were not able to get rid of the excess heat, by how much would his body temperature increase above the normal 37 °C in a half hour of jogging? The specific heat capacity for a human is about 3500 J/(kg·K) (b) How high a fever (in °F) would this temperature increase be equivalent to? Is the increase large enough to be of concern?

82. A thirsty nurse cools a 2.00 L bottle of a soft drink (mostly water) by pouring it into a large aluminum mug of mass 0.257 kg and adding 0.120 kg of ice initially at –15.0 °C. If the soft drink and mug are initially at 20 °C , what is the final temperature of the system, assuming no heat losses?

83. One experimental method of measuring an insulating material’s thermal conductivity is to construct a box of the material and measure the power input to an electric heater inside the box that maintains the interior at a measured temperature above the outside surface. Suppose that in such an apparatus a power input of 180 W is required to keep the interior surface of the box 65.0 °C (about 120 °F) above the temperature of the outer surface. The total area of the box is 2.18 m2 and the wall thickness is 3.90 cm. Find the thermal conductivity of the material in SI units.

84. The icecaps of Greenland and Antarctica contain about 1.75% of the total water (by mass) on the earth’s surface; the oceans contain about 97.5%, and the other 0.75% is mainly groundwater.  Suppose the icecaps, currently at an average temperature of about –30 °C, somehow slid into the ocean and melted. What would be the resulting temperature decrease of the ocean? Assume that the average temperature of ocean water is currently 5.00 °C.

85. The effect of urbanization on plant growth. A study published in July 2004 indicated that temperature increases in urban areas in the eastern United States are causing plants to bud up to 7 days early compared with plants in rural areas just a few miles away, thereby disrupting biological cycles. Average temperatures in the urban areas were up to 3.5 °C higher than in the rural areas. By what percent will the radiated heat per square meter increase due to such a temperature difference if the rural temperature was 0 °C on the average?

86. Basal metabolic rate. The energy output of an animal engaged in an activity is called the basal metabolic rate (BMR) and is a measure of the conversion of food energy into other forms of energy. A simple calorimeter to measure the BMR consists of an insulated box with a thermometer to measure the temperature of the air. The air has a density of 1.29 kg/m3 and a specific heat capacity of 1029 J/(kg·K). A 50.0 g hamster is placed in a calorimeter that contains 0.0500 m3 of air at room temperature. (a) When the hamster is running in a wheel, the temperature of the air in the calorimeter rises 1.8 °C per hour. How much heat does the running hamster generate in an hour? (Assume that all this heat goes into the air in the calorimeter. Neglect the heat that goes into the walls of the box and into the thermometer, and assume that no heat is lost to the surroundings.) (b) Assuming that the  hamster  converts  seed into heat with an efficiency of 10% and that hamster seed has a food energy value of 24 J/g, how many grams of seed must the hamster eat per hour to supply the energy found in part (a)?

87. A thermos for liquid helium. A physicist uses a cylindrical metal can 0.250 m high and 0.090 m in diameter to store liquid helium at 4.22 K; at that temperature the heat of vaporization of helium is 2.09 × 104 J/kg. Completely surrounding the metal can are walls maintained at the temperature of liquid nitrogen, 77.3 K, with vacuum between the can and the surrounding walls. How much helium is lost per hour? The emissivity of the metal can is 0.200. The only heat transfer between the metal can and the surrounding walls is by radiation.

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