A tube is open only at one end. A certain harmonic produced by the tube has a frequency of 500 Hz. The next higher harmonic has a frequency of 611 Hz. The speed of sound in air is 343 m/s.
(a) What is the integer n that describes the harmonic whose frequency is 500 Hz?
(b) What is the length of the tube?
m
To solve this problem, we can use the formula for the wavelength of a harmonic in a tube with one open end:
λ = 2L / n
where λ is the wavelength, L is the length of the tube, and n is the harmonic number.
(a) To find the harmonic number n for a frequency of 500 Hz, we need to rearrange the formula:
n = 2L / λ
Since the speed of sound in air is 343 m/s and the frequency is 500 Hz, we can find the wavelength as:
λ = speed of sound / frequency
= 343 m/s / 500 Hz
= 0.686 m
Substituting the values we know into the harmonic number formula:
n = 2L / λ
n = 2L / 0.686
To find the integer value of n, we need to solve the equation. Rearranging it:
L = (0.686 * n) / 2
Now we can try different integer values of n and calculate the corresponding length L until we find a value that matches the given frequency of 500 Hz.
(b) To find the length of the tube, we can use the value of n from part (a) and substitute it back into the formula:
L = (0.686 * n) / 2
By substituting the value of n, you can calculate the length of the tube.
To find the integer n that describes the harmonic whose frequency is 500 Hz, we can use the equation for the frequency of a harmonic in a tube open at one end:
f = (2n - 1) * v / (4L)
Where:
f = frequency of the harmonic (500 Hz)
n = integer describing the harmonic
v = speed of sound in air (343 m/s)
L = length of the tube (unknown)
Solving for n:
n = ((4L * f) / v + 1) / 2
Substituting the given values:
n = ((4L * 500) / 343 + 1) / 2
n = (2000L / 343 + 1) / 2
n = (2000L + 343) / (2 * 343)
n = (2000L + 343) / 686
Since n has to be an integer, we can manipulate the equation to find a suitable value for L:
2000L + 343 = 686n
Now, let's try some values for n and see if we get an integer value for L:
For n = 1,
2000L + 343 = 686(1)
2000L + 343 = 686
2000L = 343
L = 343 / 2000 = 0.1715 m
For n = 2,
2000L + 343 = 686(2)
2000L + 343 = 1372
2000L = 1372 - 343 = 1029
L = 1029 / 2000 = 0.5145 m
For n = 3,
2000L + 343 = 686(3)
2000L + 343 = 2058
2000L = 2058 - 343 = 1715
L = 1715 / 2000 = 0.8575 m
It seems that for n = 1, we get a suitable value for L:
(a) n = 1
(b) L = 0.1715 m