11
Vibrations Chapter and Waves
8.
The tire swing approximates a simple pendulum. With a stopwatch, you can measure the period T of gT 2 the tire swing, and then solve Equation 11-11a for the length, L . 42 To make the water slosh, you must shake the water (and the pan) at the natural frequency for water waves in the pan. The water then is in resonance, or in a standing wave pattern, and the amplitude of oscillation gets large. That natural frequency is determined by the size of the pan smaller pans will slosh at higher frequencies, corresponding to shorter wavelengths for the standing waves. The period of the shaking must be the same as the time it takes a water wave to make a round trip in the pan.
9.
10. Some examples of resonance: Pushing a child on a playground swing you always push at the frequency of the swing. Seeing a stop sign oscillating back and forth on a windy day. When singing in the shower, certain notes will sound much louder than others. Utility lines along the roadside can have a large amplitude due to the wind. Rubbing your finger on a wineglass and making it sing. Blowing across the top of a bottle. A rattle in a car (see Question 11). 11. A rattle in a car is very often a resonance phenomenon. The car itself vibrates in many pieces, because there are many periodic motions occurring in the car wheels rotating, pistons moving up and down, valves opening and closing, transmission gears spinning, driveshaft spinning, etc. There are also vibrations caused by irregularities in the road surface as the car is driven, such as hitting a hole in the road. If there is a loose part, and its natural frequency is close to one of the frequencies already occurring in the cars normal operation, then that part will have a larger than usual amplitude of oscillation, and it will rattle. This is why some rattles only occur at certain speeds when driving. 12. The frequency a of simple periodic wave is equal to the frequency of its source. The wave is created by the source moving the wave medium that is in contact with the source. If you have one end of a taut string in your hand, and you move your hand with a frequency of 2 Hz, then the end of the string in your hand will be moving at 2 Hz, because it is in contact with your hand. Then those parts of the medium that you are moving exert forces on adjacent parts of the medium and cause them to oscillate. Since those two portions of the medium stay in contact with each other, they also must be moving with the same frequency. That can be repeated all along the medium, and so the entire wave throughout the medium has the same frequency as the source. 13. The speed of the transverse wave is measuring how fast the wave disturbance moves along the cord. For a uniform cord, that speed is constant, and depends on the tension in the cord and the mass density of the cord. The speed of a tiny piece of the cord is measuring how fast the piece of cord moves perpendicularly to the cord, as the disturbance passes by. That speed is not constant if a sinusoidal wave is traveling on the cord, the speed of each piece of the cord will be given by the speed relationship of a simple harmonic oscillator (Equation 11-9), which depends on the amplitude of the wave, the frequency of the wave, and the specific time of observation. 14. From Equation 11-19b, the fundamental frequency of oscillation for a string with both ends fixed is
f1
v 2L
. The speed of waves on the string is given by Equation 11-13, v
FT mL
. Combining
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