The 2nd harmonic of a string of length 60 cm and linear mass density 1.1 g/m has the same frequency as the 5th resonance mode of a closed pipe of length 0.8 m. Find the tension in the string.
To find the tension in the string, we need to use the equation for the frequency of a standing wave on a string:
\( f = \frac{v}{2L} \)
where \( f \) is the frequency, \( v \) is the wave speed, and \( L \) is the length of the string. The wave speed is given by:
\( v = \sqrt{\frac{T}{\mu}} \)
where \( T \) is the tension in the string and \( \mu \) is the linear mass density of the string.
We are told that the second harmonic of the string has the same frequency as the 5th resonance mode of the closed pipe. The frequency of the 5th resonance mode of a closed pipe is given by:
\( f = \frac{5v}{4L} \)
where \( f \) is the frequency of the resonance mode and \( L \) is the length of the pipe.
Setting these two expressions for frequency equal to each other, we can solve for the tension:
\( \frac{v}{2L} = \frac{5v}{4L} \)
Simplifying and solving for \( T \), we get:
\( T = \frac{4}{5} \mu v^2 \)
Now we can substitute the given values to find the tension. We are given the length of the string \( L = 60 \, \mathrm{cm} = 0.6 \, \mathrm{m} \), and the linear mass density \( \mu = 1.1 \, \mathrm{g/m} = 0.0011 \, \mathrm{kg/m} \). We also know that the frequency of the 5th resonance mode of the closed pipe is the same as the frequency of the second harmonic of the string.
By rearranging the equation for wave speed, we can substitute for \( v \):
\( v = \sqrt{\frac{T}{\mu}} \)
Squaring both sides and solving for \( T \), we get:
\( T = v^2 \mu \)
Since the frequency of the 5th resonance mode of the closed pipe is the same as the frequency of the second harmonic of the string, we can equate the expressions for wave speed:
\( v = \frac{5fL}{4} \)
Substituting this expression for \( v \) into the equation for tension, we get:
\( T = \left(\frac{5fL}{4}\right)^2 \mu \)
Finally, substituting the given values for \( f \) and \( L \), we can calculate the tension in the string.