Solution:
Sample problem: A piano string is 1.1 m long and has a mass of
9 g. How much tension must the string be under if it is to vibrate at a
fundamental frequency of 131 Hz? What are the frequencies of the first few
harmonics?
Solution:
Sample problem: In an experiment on standing waves, a string 90 cm
long is attached to the prong of an electrically driven tuning fork that
oscillates perpendicular to the length of the string at a frequency of 65 Hz.
The mass of the string is 0.055 kg. What is the mass of the suspended weight if
the string vibrates in four loops?
Solution: The key here is that the weight of the suspended mass equals
the tension, and it is the tension which controls the velocity of the wave on
the string.
Sample problem: One of the harmonics on a 1.3-m-long string has a
frequency of 15.6 Hz. The next higher harmonic has a frequency of 23.4 Hz. What
is the fundamental frequency? What is the speed of the wave?
Solution:
Sample problem: A hollow tube is partly submerged in water, as shown. If
the smallest value of L for which a peak occurs is 9 cm, what is the frequency
of the tuning fork and the value of L for next two resonant modes?
Solution:
Sample problem: A tuning fork is set into vibration above a vertical open
tube filled with water. The water level is allowed to drop slowly. As it does
so, the air in the tube above the water level is heard to resonate with the
tuning fork when the distance from the tube opening to the water level is 17.5
cm, 53.5 cm, 89.5 cm, and 125.5 cm. If the speed of sound is 343 m/s, what is
the frequency of the tuning fork? What is the end correction (i.e., the distance
from the end of the tube to the node or antinode just outside the tube)?
Solution: At resonance, the displacement antinode lies a little beyond the
open end of the tube. This so-called end correction is typically around .6*R
(R=radius) for a tube with a circular cross-section.
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