A variable-length air column is placed just below a vibrating wire that is fixed at the both ends. The length of air column open at one end is gradually increased from zero until the first position of resonance is observed at 20 cm. The wire is 115.4 cm long and is vibrating in its third harmonic.
Find the speed of transverse waves in the wire if the speed of sound in air is 340 m/s. Answer in units of m/s.
To find the speed of transverse waves in the wire, we can use the formula:
Speed of transverse waves = (2F * L) / m
Where:
F is the frequency of the vibrating wire
L is the length of the wire
m is the mode of vibration
Given:
The length of the wire (L) = 115.4 cm = 1.154 m
The mode of vibration (m) = 3rd harmonic
First, let's find the frequency (F) of the vibrating wire. The frequency of the fundamental mode can be calculated using the formula:
Fundamental frequency = (v / 2L)
Where:
v is the speed of sound in air
Given:
The speed of sound in air (v) = 340 m/s
The length of the wire (L) = 115.4 cm = 1.154 m
Fundamental frequency = (340 / 2 * 1.154)
= 147.688 Hz
Now, we need to find the frequency of the 3rd harmonic. The frequency of the n-th harmonic can be calculated using the formula:
Frequency of nth harmonic = n * Fundamental frequency
Frequency of 3rd harmonic = 3 * 147.688 Hz
= 443.064 Hz
Now, let's find the speed of transverse waves in the wire using the formula mentioned earlier:
Speed of transverse waves = (2 * 443.064 * 1.154) / 3
= 840.9848 m/s
Therefore, the speed of transverse waves in the wire is approximately 840.9848 m/s.
To find the speed of transverse waves in the wire, we can use the formula:
v = (λ)(f)
where:
v is the speed of the wave
λ is the wavelength of the wave
f is the frequency of the wave
We are given that the wire is vibrating in its third harmonic. The frequency of the wave in the wire can be determined by considering the fundamental frequency (1st harmonic) and applying the concept of harmonics.
For a fixed wire, the frequency of the nth harmonic can be calculated using the formula:
f_n = (n)(v/2L)
where:
f_n is the frequency of the nth harmonic
n is the harmonic number
v is the speed of the wave
L is the length of the wire
In this case, the wire is vibrating in its third harmonic (n = 3), and the length of the wire is given as 115.4 cm. We can plug in these values to calculate the frequency of the wave in the wire:
f_3 = (3)(v/2L)
Next, we need to find the wavelength of the wave. The length of the air column open at one end is gradually increased until the first position of resonance is observed at 20 cm. For a tube open at one end, the relationship between the length of the tube and the wavelength of the sound wave is given by:
λ = (4L/n)
where:
λ is the wavelength of the sound wave
L is the length of the air column
n is the harmonic number
In this case, the length of the air column is given as 20 cm, and we are considering the first position of resonance (n = 1). Substituting these values into the formula:
λ = (4L/n) = (4 * 20 cm)/(1)
Now that we have the wavelength (λ) and frequency (f_3), we can calculate the speed of the transverse waves in the wire using the formula:
v = (λ)(f_3)
Substituting the values we have:
v = (20 cm)(3)(v/2L)
Simplifying the equation, we can solve for v:
v = (60/115.4) * v
To get the answer in m/s, since the speed of sound is given in m/s, we need to convert the centimeters to meters:
1 cm = 0.01 m
Substituting the conversion value:
v = (60/115.4) * v * (1 cm/0.01 m)
Simplifying the equation, we have:
v = (600/1154) * v
Now we can solve for v by isolating it on one side:
v - (600/1154) * v = 0
Combining like terms, we have:
(1154/1154 - 600/1154) * v = 0
(554/1154) * v = 0
Dividing both sides by (554/1154):
v = 0
Therefore, the speed of transverse waves in the wire, based on the given information, is 0 m/s.
Note: It's important to double-check the provided information and calculations to ensure accuracy, as the obtained result may be affected by any potential errors or inaccuracies.
open one end, acoustic length of tube must be 1/4 lambad
sound wavelength=.80meters
frequency=340/.8 hz
now, you know the length of the wire is 3/2 lambda
frequency(lambda)=speed
solve for speed on the wire.