Homework for Week 11

Problem 1.

Physics Concepts: Make this week’s physics concepts summary as you work all of the problems in this week’s assignment. Be sure to cross-reference each concept in the summary to the problem(s) they were key to, and include concepts from previous weeks as necessary. Do the work carefully enough that you can (after it has been handed in and graded) punch it and add it to a three ring binder for review and study come finals!

Problem 2.

aaaaaaaaaa = fo

Lava

Bill and Ted are falling into hell at a constant speed (terminal velocity), and are screaming at the frequency f0. As they fall, they hear their own voices reflecting back to them from the puddle of molten rock that lies below at a frequency of 2f0.

How fast are they falling relative to the speed of sound in warm, dry hellish air?

Problem 3.

B

A

Sound waves travel faster in water than they do in air. Light waves travel faster in air than they do in water. Based on this, which of the three paths pictured above are more likely to minimize the time required for the

a)Sound:

b)Light:

produced by an underwater explosion to travel from the explosion at A to the pickup at B? Why (explain your answers)?

Problem 4.

Fred is standing on the ground and Jane is blowing past him at a closest distance of approach of a few meters at twice the speed of sound in air. Both Fred and Jane are holding a loudspeaker that has been emitting sound at the frequency f0 for some time.

a)Who hears the sound produced by the other person’s speaker as single frequency sound when they are approaching one another and what frequency do they hear?

b)What does the other person hear (when they hear anything at all)?

c)What frequenc(ies) do each of them hear after Jane has passed and is receding into the distance?

Problem 5.

Discuss and answer the following questions:

a)Sunlight reaches the surface of the earth with roughly 1000 Watts/meter2 of intensity. What is the “sound intensity level” of a sound wave that carries as much energy per square meter, in decibels?

b)In table 5, what kind of sound sources produce this sort of intensity? Bear in mind that the Sun is 150 million kilometers away where sound sources capable of reaching the same intensity are typically only a few meters away. The the Sun produces a lot of (electromagnetic) energy compared to terrestrial sources of (sound) energy.

c)The human body produces energy at the rate of roughly 100 Watts. Estimate the fraction of this energy that goes into my lecture when I am speaking in a loud voice in front of the class (loud enough to be heard as loudly as normal conversation ten meters away).

d)Again using table 5, how far away from a jack hammer do you need to stand in order for the sound to (marginally) no longer be dangerous to your hearing?

Problem 6.

String one has mass per unit length µ and is at tension T and has a travelling harmonic wave on it:

y1(x, t) = A sin(kx − ωt)

String one is also very long compared to a wavelength: L λ.

Identical string two has the superposition of two harmonic travelling waves on it:

y2(x, t) = A sin(kx − ωt) + 3A sin(kx + ωt)

If the average energy density (total mechanical energy per unit length) of the first string is E1, what is the average energy density of the second string E2 in terms of E1?

Problem 7.

Two identical strings of length L have mass µ and are fixed at both ends. One string has tension T . The other has tension 1.21T . When plucked, the first string produces a tone at frequency f1, the second produces a tone at frequency f2.

a)What is the beat frequency produced if the two strings are is plucked at the same time, in terms of f1?

b)Are the beats likely to be audible if f1 is 500 Hz? How about 50 Hz? Why or why not?

Problem 8.

You measure the sound intensity level of a single frequency sound wave produced by a loudspeaker with a calibrated microphone to be 80 dB. At that intensity, the peak pressure in the sound wave at the microphone is P0. The loudspeaker’s amplitude is turned up until the intensity level is 100 dB. What is the peak pressure of the sound wave now (in terms of P0)?

Note that you could look this up in the table, but don’t. The point is for you to know how the peak pressure scales with the intensity, as well as how the intensity varies with the sound intensity level in decibels.

Problem 9.

Resonant Sound Waves

Tube closed at one end

An organ pipe is made from a brass tube closed at one end as shown. The pipe has length L and the speed of sound is vs. When played, it produces a sound that is a mixture of the first, third and sixth harmonic (mode, counting n = 1, 2, 3...).

a)What are the frequencies of these modes?

b)Qualitatively sketch the wave amplitudes for the first and the third harmonic modes (only) in on the figure, indicating the nodes and antinodes. Be sure to indicate whether the nodes or antinodes drawn are for pressure/density waves or displacement waves!

c)Evaluate your answers numerically when L = 3.4 meters long, and vs = 340 meters/second (as usual).

Problem 10.

f0,1

L0,1

You crash land on a strange planet and all your apparatus for determining if the planet’s atmosphere is like Earth’s is wrecked. In desperation you decide to measure the speed of sound in the atmosphere before taking o your helmet. You do have a barometer handy and can see that the air pressure outside is approximately one atmosphere and the temperature seems to be about 300 ◦K, so if the speed of sound is the same as on Earth the air might be breathable.

You jury rig a piston and cylinder arrangement like the one shown above (where the cylinder is closed at both ends but has a small hole in the side to let sound energy in to resonate) and take out your two handy tuning forks, one at f0 = 3400 Hz and one at f1 = 6800 Hz.

a)Using the 3400 Hz fork as shown, what do you expect (or rather, hope) to hear as you move the piston in and out (varying L0). In particular, what are the shortest few values of L0 for which you expect to hear a maximum resonant intensity from the tube if the speed of sound in the unknown atmosphere is indeed the same as in air (which you will cleverly note I’m not telling you as you are supposed to know this number)?

b)Using the 6800 Hz fork you hear your first maximum (for the smallest value of L1) at L1 = 5 cm. Should you sigh with relief and rip o your helmet?

c)What is the next value for L1 for which you should hear a maximum (given the measurement in b) and what should the di erence between the two equal in terms of the wavelength of the 6800 Hz wave in the unknown gas? Draw the displacement wave for this case only schematically in on the diagram above (assuming that the L shown is this second-smallest value of L1 for the f1 tuning fork), and indicate where the nodes and antinodes are.

Optional Problems

Study for the final exam! This is the last week of class, and this wraps up both the chapter and the texbook. Students looking for more problems to work on are directed to the online review guide for introductory physics 1 and the online math review, the latter as needed.