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A nylon guitar string is fixed between two lab posts 2.25 m apart. The string has a linear mass density of μ = 6.60 g/m and is placed under a tension of 170.00 N. The string is placed next to a tube, open at both ends, of length L. The string is plucked and the tube resonates at the n = 3 mode. The speed of sound is 343 m/s. What is the length of the tube?

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Answer:

14.4259 m

Explanation:

F = Force on string = 170 N

[tex]\mu[/tex] = Linear density = 6.6 g/m

L = Length of string = 2.25 m

n = Mode = 3

[tex]L_t[/tex] = Length of tube

v = Speed of sound in air = 343 m/s

Speed of wave is given by

[tex]u=\sqrt{\frac{F}{\mu}}\\\Rightarrow u=\sqrt{\frac{170}{6.6\times 10^{-3}}}\\\Rightarrow u=160.492\ m/s[/tex]

Frequency of wave of the string is given by

[tex]f_s=\frac{u}{2L}\\\Rightarrow f_s=\frac{160.492}{2\times 2.25}\\\Rightarrow f_s=35.665\ Hz[/tex]

Frequency of wave of the tube is given by

[tex]f_t=\frac{nv}{2L_t}\\\Rightarrow L_t=\frac{nv}{2f_t}\\\Rightarrow L_t=\frac{3\times 343}{2\times 35.665}\\\Rightarrow L_t=14.4259\ m[/tex]

The length of the tube is 14.4259 m