Physics 102 Homework set 7 Solutions Chapter 20: Exercises: 14: This is NOT the same problem as blowing over the lip of a bottle. (Note that last year's solution to this problem is incorrect.) In the blowing-bottle case, you are making the column of air in the container vibrate at its resonant frequency. As you fill the container with water, the air column height decreases so that the resonant wavelength decreases so that the resonant frequency increases so that the pitch increases. In this case, you are making the water in the cup and the walls of the cup vibrate. The higher the water level in the cup, the lower the note. Try it yourself. 16: The speed does not depend on the frequency or the wavelength. Thus the speed will not change. Since wavelength = speed/frequency, if you double the frequency, the wavelength will halve. 20: The earth tremor and the sound wave are both pressure waves. Pressure waves travel faster in the ground than in the air because the ground is a more rigid medium. Thus, the tremor will travel at faster speed and reach your feet before the sound wave reaches you ears. 24: a) sound cannot travel in a vacuum. There is no air in outer space. b) if sound could travel in a vacuum, it would still travel much slower than light so that you would see the explosion long before you heard it. 40: Two frequencies played together generate a 'beat frequency', a sound wave at the difference between the two frequencies. We did this in class where the beat frequency was a few hertz (so we heard the sound getting louder and softer a few times each second). If the beat frequency is high (eg: a few hundred or a few thousand hertz), then we would hear that as a separate tone. In this case, we would hear a sound at 102 kHz - 100 kHz = 2000 Hz. Problems: 2: wavelength = speed / frequency = 1530 m/s / 7 1/s = 233 m 6: For the echo to return in 8 hours, the sound has to travel out for 4 hours, bounce off of the mountain, and travel back for 4 hours. Obviously, this would have to be an unrealistically LOUD sound. The mountain is thus 4 hours away at the speed of sound. Thus: d = 4 h * (60 min/h) * (60 s/min) * 340 m/s = 4.9 * 10^6 m = 4.9 * 10^3 km or almost 5000 km (3000 miles) away. That's the distance from here to California. ------------------------- Chapter 21: Exercises: 2: wavelength = speed / frequency = 340 m/s / 20,000 1/s = 17 mm wavelength = speed / frequency = 340 m/s / 20 1/s = 17 m 8: The wavelength is 60 cm. You can do this one of two ways: a) the distance from one node to the next node of a standing wave is 1/2 wavelength. In this case there is 30 cm from node to node. b) the distance from crest to crest in the wave is 15 cm + 30 cm + 15 cm = 60 cm. 12: This is actually a question about energy conservation. First we need to discuss what a sounding board does. Sound is made by a vibrating object. A guitar string by itself cannot make much sound because it is too skinny to move much air. When the guitar string is mechanically connected to a sounding board, then the vibrations of the string cause the sounding board to vibrate. The sounding board is big enough that when it vibrates, it can make a lot of air vibrate (ie: make a louder sound). Now we need to look at the flow of energy. The starting energy for both systems is the same: you pluck the guitar string. This energy is given off in the form of sound waves (since waves carry energy). If no energy is given off (ie: if there is no sound emitted) and there is no friction, then the guitar string will vibrate forever. The louder the sound that is emitted, the faster the energy is radiated (given away). When the guitar string has given away all of its energy, it will have stopped vibrating. Thus, when the guitar string is attached to a sounding board, it transfers its energy more rapidly to the surrounding air in the form of sound waves and therefore stops vibrating in a much shorter time. 18: The vertical axis on the plots shows the pressure (or amplitude) of the sound wave. The horizontal axis shows the time. The left plot has more waves per time so it has a higher base frequency, and hence a higher pitch. 19: The right plot has a larger pressure variation (greater difference between high and low on the plot) so it is louder. -------------------------- Estimation: On average, how many people in the United States are playing a musical instrument at 8:00 PM on a Thursday evening? We need a few numbers to answer this question. First, the population of the US is about 3*10^8 people (300 million or 300,000,000). Now we need to estimate the proportion of people who play a musical instrument. It is obviously less than 100%. There are a few ways to estimate this. You can think about your friends and figure out how many play an instrument. This is an OK technique. Unfortunately, it is also probably biased. If you play, then your friends are more likely to play and vice versa. One other method (and there are many more) is to consider how many people played in my HS band and orchestra and double that to include the other musicians. My HS had about 40 people in the band and about 900 students so this gives a ratio of 2*40/900 = about 1/10. BTW: This number is also biased since a lot of HS students stop playing instruments as adults. On the other hand, some adults start. I'm not sure the proportion is 10% but I am very sure that it is less than 100% and more than 1% and that's good enough. Now we need the probability that they are actually practicing. I'll assume that each person practices 3 hours per week. That will average out the serious musicians who practice 2 hours per day and the non-serious ones who almost never practice. There are 168 hours in a week. I sleep 8 hours/night or 56 hours a week and I am at work another 40-50 hours/week. This leaves 70 hours in which I could practice. That means that the probability that I am practicing at any specific time outside of work or sleep is 3 hours / 70 hours = about 1/20. (I guessed that number of 3 hours. I am sure that it is less than 30 hours and more than 20 minutes [0.3 hours] and that is good enough.) Therefore the total number of Americans practicing their musical instrument at 8:00 PM on Thursday evening is about 3*10^8 * 1/10 * 1/20 = about 10^6 Just think, while you are practicing your instrument, 1 million other Americans are practicing with you! (Of course, that's also about how many people in the world are picking their nose at that time too.)