PROBLEMS
1. Define resonance in terms of energy.
2. At what frequency in Hz will a mass of 2.0 kg oscillate on a spring with a spring constant of 300 N/m?
3. What is the function of position versus time, x(t), for the mass on the spring in the previous problem if at t=0 it is passing equilibrium position (x=0) and has a velocity of 3.0 m/s? Please describe the amplitude in terms of mentioned parameters.
4. Is it easier or harder to excite a relatively large vibration in a system by driving it slightly below its resonant frequency if the system is more heavily damped?
5. Is collection of lost sound waves from natural or man-made phenomena a viable solution to our energy demands as a civilization?
6. A human ear drum has a diameter of 7 mm. What is the force on it in newtons at the threshold of human hearing (the quietest sound human ears can typically detect)? Realize that this is an oscillating force at the frequency of the sound.
7. Why do we use a musical scale that has 12 steps to complete the octave?
8. The most common musical interval played in chords besides the octave (combinations of notes) is the perfect 5th. We saw in lecture that you NEVER play just a single note on an instrument, but rather generate many higher order harmonics as well, even when the intention is to play just one note. Calculate in terms of the root note's frequency f0, the frequency of 9 harmonics above that root. Then calculate 9 harmonics of the note that is a perfect 5th above the root. What do those lists of frequencies have in common? This is another way to understand harmony. You may want to use a spreadsheet program on this one to avoid tedious calculation.
9. Are standing waves really just single waves standing still?
10. What happens if you try to make a sine wave on a string if the wavelength is a bit longer than the string's length L? Describe this in terms of phase and/or phasors.
11. What frequencies of resonance can exist in a pipe of length 0.50 m with only one side open?
12. What frequencies of resonance can exist in a pipe of length 0.50 m with both sides open?
13. In the act of tuning a guitar string to a frequency of 220 Hz by using a tuning fork, what sound will be heard when the string is slightly out of tune making a pitch of 222 Hz? Be specific about the tone's frequency and the beats heard.
14. What pitch of sound should we hear while standing near the side of a racetrack with a race car exhaust making a pitch of 500 Hz while the car is traveling 70 m/s directly at you?
15. What pitch should you hear as it passes right by in front of you? When it's traveling directly away?
16. An ice cream truck playing a 1000 Hz pitch drives slowly through a neighborhood at 3 m/s in the +x direction. With respect to the center of the intersection down the block, the position of the truck is (50,0) m. Your position is (60,5)m. What pitch do you hear? What about at a position of (40,5)?
17. The half angle of the mach cone in the photo of the jet in this chapter measures 62 degrees. If the speed of sound is 300 m/s at the temperature of air where the jet is flying, how fast is the jet flying?
18. Astronomers use the Doppler effect for light to measure the rate at which celestial objects are moving with respect to us. They use the terms "blue-shift" for light that appears higher frequency due to a Doppler shift and "red-shift" for lower. For speeds less than around v/c=0.1 we can use the equation we derived for the Doppler shift of sound for a moving source. How fast would a distant galaxy be moving away from us if the light it gives off is Doppler shifted by 5.0%?
Answers
1. Find answer in first two sections of chapter.
2. 1.95 Hz
3.
4. Easier. Damping makes it so that the driving frequency doesn't need to be as closely matched to the resonant frequency.
5. No. Sound has very little energy.
6.
7. To maximize the likelihood that two notes played in unison sound good together.
8. root harmonics: f, 2f, 3f, 4f, ... The perfect 5th is 3f/2. its harmonics are 2(3f/2), 3(3f/2), etc. Every third harmonic of the root is the same frequency as every second harmonic of the perfect 5th.
9. No. They are two traveling waves going in opposite directions that meet and give the illusion of a wave standing still. See text.
10. A doubly reflected wave will be out of phase with the wave that initiated the process. That wave will reflect twice more and be out of phase by twice as much. Ultimately, represented as phasors, we'd have lots of them all with a slight rotation with respect to one another such that they cancel.
11.
12.
13. A pitch of 221 Hz with a beat frequency of 2 Hz.
14. 628 Hz
15. 500 Hz in front of you and 415 Hz as it drives directly away.
16. 1008 Hz at (x,y)=(60,5) and 992.2 Hz at (40,5).
17. 340 m/s.
18.