turning fork of frequency 340Hz is vibrated just above a cylindrical tube of height 1.2m. If water is slowly poured into the tube, at what maximum height will resonance occur.
[speed of sound in air = 340ms-1]
A. 0.95m
B. 0.60m
C. 0.50m
D. 0.45m
The speed of sound in air is given as 340 m/s. The frequency of the fork is also given as 340 Hz.
Resonance occurs when the natural frequency of the fork matches the natural frequency of the air column in the tube.
The natural frequency of an air column in a cylindrical tube is given by the formula:
f = (n * v) / (2L)
where f is the frequency, n is the harmonic number (1 for the first harmonic, 2 for the second harmonic, etc.), v is the speed of sound in air, and L is the length of the air column.
In this case, we are given that the frequency of the fork is 340 Hz, so we can set up the equation:
340 = (n * 340) / (2L)
Simplifying, we have:
2L = n
L = n/2
Since the first harmonic corresponds to n = 1, the length of the air column at resonance is L = 1/2 = 0.5 m.
Therefore, the correct answer is C. 0.50 m.
To find the maximum height at which resonance will occur, we need to consider the conditions for resonance in a closed cylindrical tube.
The resonance condition for a closed cylindrical tube is given by:
L = (n * λ) / 2
Where:
L is the length of the tube
n is the mode of vibration (1, 2, 3, ...)
λ is the wavelength of the sound wave
In this case, the frequency of the fork is 340 Hz, so the wavelength (λ) can be calculated using the formula:
v = f * λ
Where:
v is the speed of sound in air (340 m/s)
f is the frequency of the fork (340 Hz)
By rearranging the formula, we can find the wavelength:
λ = v / f
Substituting the values, we get:
λ = 340 m/s / 340 Hz = 1 m
Substituting the values in the resonance condition equation, we have:
L = (n * λ) / 2
Since we are looking for the maximum height, we will consider the first mode of vibration (n = 1).
L = (1 * 1 m) / 2 = 0.5 m
Therefore, the correct option is C. 0.50 m, which is the maximum height at which resonance will occur.