A variable length air column (closed on one end) is placed just below a vibrating wire of mass 5.00g, which is fixed at both ends. The length of the air column is gradually increased from zero until the first position of resonance (first harmonic) is observed at 34 cm. The wire is 120 cm long and is vibrating in its third harmonic. If the speed of sound in air is 340 m/s, what is the tension in the wire?
To determine the tension in the wire, we need to use the equation for the frequency of a vibrating string:
f = (1/2L) * sqrt(T/μ)
Where:
f is the frequency of the wave in Hz,
L is the length of the string in meters,
T is the tension in the wire in Newtons, and
μ is the linear mass density of the string in kg/m.
Given that the wire is vibrating in its third harmonic, the frequency of the wave is 3 times the fundamental frequency. So, we have:
f = 3 * (v/λ)
Where:
v is the speed of sound in m/s, and
λ is the wavelength of the sound wave in meters.
To find the tension T, we need to determine the wavelength of the sound wave produced by the vibrating wire.
The wavelength λ of a sound wave can be calculated using the equation:
λ = 4L/n
Where:
L is the length of the air column in meters, and
n is the harmonic number.
In this case, the air column is vibrating in its first harmonic, so n = 1.
Given that the length of the air column is 34 cm (0.34 m), we have:
λ = 4 * 0.34 m / 1 = 1.36 m
Now, we can substitute the values into the equation for frequency and solve for T:
3 * (v/λ) = (1/2L) * sqrt(T/μ)
Rearranging the equation for T, we have:
T = (9/4) * μ * (v/λ)^2
Since we don't have the information about the linear mass density of the wire, we can't directly calculate the tension.