Oscillation: Musical Tube and String

[Bryon]

Homework Statement

A brass tube of mass 22 kg and length 1.3 m is closed at one end. A wire of mass 9.7 g and length 0.39 meters is stretched near the open end of the tube. When the wire is plucked, it oscillates at its fundamental frequency. By resonance, it sets the air column in the tube oscillating at the column's fundamental frequency.

What is the frequency of oscillation of the air in the tube ?

M(tuba) = 22Kg
L = 1.3m
m(wire) = 9.7g
l = 0.39m

Homework Equations

f = v/(wavelength in tube)

The Attempt at a Solution

I think that the tuba will fit a wavelength 1.5 times the length of the tuba because it is open at one end. So, 1.5*1.3 = 1.95 which is the wavelenth in the tuba.

f = 343/1.95 = 175.897Hz

This answer is apparently incorrect. Any ideas?

 

[lippyka]

The correct answer is 343/2.7 = 126.667 Hz. The wavelength in the tube is determined by the speed of sound (343 m/s) divided by the frequency of oscillation. Since the frequency of oscillation is the same as the fundamental frequency of the wire, the wavelength in the tube is equal to the length of the wire. Therefore, the frequency of oscillation of the air in the tube can be calculated by dividing the speed of sound (343 m/s) by the length of the wire (0.39 m).

 

[kerimek]

I would approach this problem by first calculating the fundamental frequency of the wire, which is given by the equation f = 1/2L * √(T/m), where L is the length of the wire, T is the tension, and m is the mass per unit length. Plugging in the values given, we get:

f = 1/2(0.39) * √(9.8/0.0097) = 264.75 Hz

Next, we can use the concept of resonance to determine the frequency of oscillation of the air in the tube. Since the wire is plucked at its fundamental frequency, it will set up standing waves in the air column with the same frequency. In order for this to happen, the length of the air column must be equal to a multiple of half the wavelength of the standing wave. In other words, the length of the air column must be equal to half the wavelength of the standing wave in order for resonance to occur.

Thus, we can set up the equation L = λ/2, where L is the length of the air column and λ is the wavelength of the standing wave. Solving for λ, we get:

λ = 2L = 2(1.3) = 2.6 m

Finally, we can use the equation f = v/λ to calculate the frequency of oscillation of the air in the tube, where v is the speed of sound. Plugging in the values, we get:

f = 343/2.6 = 131.92 Hz

Therefore, the frequency of oscillation of the air in the tube is approximately 131.92 Hz. It is important to note that this is only an approximation, as the actual frequency may be slightly different due to factors such as the thickness and material of the tube walls.