A 350-Hz tuning fork is held above a closed pipe. Find the spacings between the resonances when the air temperature is 20°C.
To find the spacings between the resonances in a closed pipe, we need to consider the concept of standing waves. In a closed pipe, only odd harmonics (3rd, 5th, 7th, etc.) can be produced.
The formula to calculate the frequency of the nth harmonic is given by:
Fn = (2n - 1) * (v/4L)
Where:
Fn is the frequency of the nth harmonic,
n is the harmonic number,
v is the velocity of sound in air, and
L is the length of the pipe.
To find the spacing between the resonances, we need to subtract the frequency of the previous nth resonance from the frequency of the nth resonance.
In this case, since we know the frequency of the tuning fork is 350 Hz, we can determine the different harmonic frequencies using the formula above.
Since the temperature is 20°C, we also need to consider the effect of temperature on the velocity of sound. The velocity of sound in air can be calculated using the formula:
v = v0 * sqrt(T/T0)
Where:
v is the velocity of sound at the given temperature,
v0 is the velocity of sound at 0°C (which is approximately 331.4 m/s),
T is the temperature in Kelvin (20°C + 273.15 = 293.15 K), and
T0 is the reference temperature in Kelvin (0°C + 273.15 = 273.15 K).
By solving the equation using the given values, we can find the velocity of sound at 20°C.
Once we have the velocity of sound, we can calculate the frequencies of the respective harmonics (3rd, 5th, 7th, etc.) using the formula mentioned earlier.
To find the spacings between the resonances, subtract the frequency of the previous resonance from the frequency of the current resonance.
By following these steps, you should be able to find the spacings between the resonances when the air temperature is 20°C.
Resonance spacing ��
�
2
�so using ���
v
f
�
the resonance spacing is
�
�
2
���
2
v
f
���
(2
3
)(
4
4
3
40
m
H
/s
z) ��0.39 m