Askiitians Tutor Team
Last Activity: 4 Months ago
To determine the total number of resonances heard as the glass tube is gradually raised out of the water, we need to understand how sound waves behave in a tube that is open at both ends. The fundamental frequency and its harmonics will play a crucial role in this scenario.
Understanding Resonance in Open Tubes
In an open tube, the resonant frequencies can be described by the formula:
Where:
- f_n is the frequency of the nth harmonic.
- n is the harmonic number (1, 2, 3,...).
- v is the speed of sound in air (330 m/s in this case).
- L is the length of the tube (1.5 m).
Calculating the Fundamental Frequency
First, let's find the fundamental frequency (n=1):
- Using the formula: f_1 = 1(v/2L)
- Substituting the values: f_1 = 1(330 m/s / (2 * 1.5 m))
- This simplifies to: f_1 = 110 Hz
Now, since the tuning fork vibrates at 660 Hz, we can find the harmonics that will resonate in the tube.
Finding the Harmonics
Next, we need to determine which harmonics of the tube can match the frequency of the tuning fork:
- For n=1: f_1 = 110 Hz
- For n=2: f_2 = 220 Hz
- For n=3: f_3 = 330 Hz
- For n=4: f_4 = 440 Hz
- For n=5: f_5 = 550 Hz
- For n=6: f_6 = 660 Hz
Here, we see that the 6th harmonic (n=6) matches the frequency of the tuning fork at 660 Hz. This means that the first resonance occurs when the tube is fully submerged and the 6th harmonic is achieved.
Counting the Resonances
As the tube is raised out of the water, the effective length of the tube increases, allowing for additional resonances. Each time the tube is raised by a quarter wavelength, a new resonance occurs. The wavelength for the 6th harmonic can be calculated as follows:
- Wavelength (λ) = v / f = 330 m/s / 660 Hz = 0.5 m
Since the tube is 1.5 m long, we can find how many quarter wavelengths fit into this length:
- Length of tube = 1.5 m
- Quarter wavelength = λ/4 = 0.5 m / 4 = 0.125 m
- Number of quarter wavelengths in 1.5 m = 1.5 m / 0.125 m = 12
As the tube is raised, each quarter wavelength corresponds to a new resonance. Therefore, the total number of resonances heard before the tube is completely out of the water is:
- 12 resonances (including the initial resonance when fully submerged).
Final Thoughts
In summary, as the glass tube is gradually raised from the water, a total of 12 resonances will be heard, culminating in the 6th harmonic matching the frequency of the tuning fork at 660 Hz. This illustrates the fascinating relationship between sound waves and the physical properties of the medium they travel through.
