Physics 102 Homework set 7 Solutions L. Weinstein Exercises: ---------- 20.18: The pitch of the tapped glass decreases as it is filled with water. As the mass of the system increases, its natural frequency decreases. (Note that this is a very different situation than where you fill a container of water and listen to the pitch as it fills. In that case, the air inside the container is vibrating and the pitch increases as the air volume decreases. This is the difference between tapping a bottle of water and blowing over the rim.) 20.20: v = f * lambda. when you double the frequency, the speed is unchanged. Therefore the wavelength halves. 20.24: The ground tremor from a distant explosion can be heard before the sound because a) both are sound (ie: pressure) waves and b) the sound waves travel faster through the ground than through the air. 20.28: If the distance to the sound triples, the sound intensity will decrease by a factor of 3 squared which is 9. This is because the energy of the sound wave spreads out to cover the area of the sphere. If you are a distance R away from the source of the sound, then the area covered is 4 pi R^2. Tripling R increases the area by a factor of 9. 20.32: The marchers try to stay in step by listening to the footsteps of the marchers ahead of them. Since it takes time for the sound to reach them, they will be out of step by that amount. If the parade is three hundred meters (about 1000 ft) long, then they will be out of step by an entire second. 20.36: The table functions as a 'sounding board'. The tuning fork by itself cannot move much air and therefore is not loud. The tuning fork can make the table vibrate and the table, with a large surface area, can move a large amount of air, making a much louder sound. b) If the sound is louder, then the tuning fork will not vibrate for as long a time. This is because the tuning fork has a certain energy of vibration. The sound carries energy away. When the energy is gone, the tuning fork stops vibrating. If the sound is louder, then it carries more energy away and the tuning fork will stop vibrating sooner. 20.48: The string should be loosened. When the tuner heard 3 beats per second, it indicated that the frequency of the sound from the piano wire differed by 3 Hz from that of the tuning fork. After tightening the string, the difference increased to 5 Hz. Therefore, the tuner should loosen the string to decrease the difference. (Tightening the string increases the pitch and loosening it decreases the pitch.) 21.2: Increasing the pitch increases the frequency. Pitch is principally governed by frequency. 21.8: Pitch is principally governed by frequency. 21.12: Different notes are produced on a guitar by fingering. When you place your finger on a string, you change its effective length. 21.18: A vibrating guitar string on a workbench does not have a sounding board and therefore it is not as loud. A vibrating guitar string on a guitar forces the guitar body to vibrate. The vibrating guitar body moves a lot of air and therefore produces louder sounds. 21.19: Longer. This is the exact same situation as exercise 20.36b. (Note: you are not allowed to use the solutions manual for odd numbered problems. Using the printed solutions deprives you of the opportunity to practice and learn.) Problems: ------------------------------------------- 20.4: The sound travels for 0.05 s to get from the bat to the wall (and another 0.05 seconds back to the bat). At a speed of 340 m/s, this corresponds to a distance d = vt = 17 m. 20.10: The beat frequencies correspond to the difference of the two frequencies. You can get beat frequencies of 2, 3 and 5 Hz. (2 Hz = 261 Hz - 259 Hz; 3 Hz = 259 Hz - 256 Hz; 5 Hz = 261 Hz - 256 Hz.) Estimation: ------------------------------------ We need to estimate a) the total number of Americans b) the proportion of Americans that play an instrument c) the number of hours per week that they practice d) the available time during which they practice. There are 3*10^8 of us. About 10% of high school students play an instrument. (It's certainly more than 1% and less than 50%. I figure that there are about 100 students in a large 2000 student HS that are in the band and orchestra, plus a bunch more that play in other groups.) About half of those will keep playing as adults (maybe fewer). They will practice about 2 hours per week (this averages those who play a lot with those who don't play much at all). It will certainly be more than 1 hour per week and less than 10. They can practice in the evening or on weekends. Subtracting work, eating and sleeping, there are about 50 hours per week available for practice. Thus, the chance that a musician is practicing at a particular time (eg: 8:00 PM Thursday) = number of practice hours / number of available hours = 2 / 50 = 0.04 = 4%. Thus: the number of Americans playing a musical instrument at 8:00 PM Thursday is N = (number of Americans) * (percent that play) * (probability of practicing at a particular time) or N = 3*10^8 * (0.05) * (0.04) = 6*10^5 or 600,000. Rather a lot. That's about the population of Virginia Beach and Norfolk put together. Answers should be within a factor of ten or so (ie: between 10^4 and 10^7).