Physics Solutions Manual

Chapter 15 continued
Challenge Problem
page 417
1. Determine the tension, FT , in a violin string of mass m and length L that will
play the fundamental note at the same frequency as a closed pipe also of
length L. Express your answer in terms of m, L, and the speed of sound in
air, v. The equation for the speed of a wave on a string is u F T
. where FT is
the tension string and is the mass per unit length of the string.
The wavelength of the fundamental in a closed pipe is equal to 4L, so
v
the frequency is f .The wavelength of the fundamental on a string is
4L
u
equal to 2L, so the frequency of the string is f ,where u is the speed
2L
of the wave on the string, u F T .The mass per unit length of the string
m/L. Squaring the frequencies and setting them equal gives
v2
u2
F
16L2
4L2
4L2
T FTL
FT
4L2
m 4Lm
Finally, rearranging for the tension gives FT mv2
4L
2. What is the tension in a string of mass 1.0 g and 40.0 cm long that plays the
same note as a closed pipe of the same length?
For a string of mass 1.0 g and length 0.400 m, the tension is
FT mv2
(0.0010 kg)(343 m/s)
74 N
4L
2
4(0.400 m)
344 Solutions Manual Physics: Principles and Problems
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