Question

A glass tube of $1\text{ m}$ length is filled with water. The water can be drained out slowly from the bottom of the tube. If a vibrating tuning fork of frequency $500\text{ Hz}$ is brought at the upper end of the tube, then the total number of resonances obtained are [Velocity of sound in air is $320\text{ m/s}$]

• A. 3
• B. 4
• C. 1
• D. 2

Hint:

To quickly find the max number of modes without listing them out, use the bounding inequality $(2N - 1)\frac{\lambda}{4} \le L_{\text{max}}$. Here, $(2N - 1) \times 16 \le 100 \implies 2N - 1 \le 6.25 \implies 2N \le 7.25 \implies N \le 3.625$. The largest integer value is $N = 3$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We have a cylindrical glass tube of maximum length $L_{\text{max}} = 1\text{ m} = 100\text{ cm}$. As water drains out from the bottom, it creates an air column of increasing length $L$ that is closed at the water boundary and open at the top. We need to find the total number of distinct resonance states that can be formed as the air column expands up to $100\text{ cm}$.

Step 2: Key Formula or Approach:
1. First, compute the acoustic wavelength $\lambda$ using the wave speed equation: $$\lambda = \frac{v}{f}$$ 2. For an air column closed at one end, resonance occurs when the length of the air column matches odd multiples of a quarter-wavelength: $$L_N = (2N - 1)\frac{\lambda}{4} \quad \text{where } N = 1, 2, 3, \dots$$ We count how many unique values of $L_N$ satisfy the physical length boundary constraint $L_N \le 100\text{ cm}$.

Step 3: Detailed Explanation:
First, calculate the wavelength $\lambda$ of the acoustic wave produced by the tuning fork: $$\lambda = \frac{320\text{ m/s}}{500\text{ Hz}} = 0.64\text{ m} = 64\text{ cm}$$ Now, compute the base quarter-wavelength value: $$\frac{\lambda}{4} = \frac{64\text{ cm}}{4} = 16\text{ cm}$$ Let's find the resonant lengths corresponding to successive odd multipliers: * For $N = 1$ (First Resonance / Fundamental Mode): $$L_1 = 1 \times 16\text{ cm} = 16\text{ cm}$$ * For $N = 2$ (Second Resonance / Third Harmonic): $$L_2 = 3 \times 16\text{ cm} = 48\text{ cm}$$ * For $N = 3$ (Third Resonance / Fifth Harmonic): $$L_3 = 5 \times 16\text{ cm} = 80\text{ cm}$$ * For $N = 4$ (Fourth Resonance / Seventh Harmonic): $$L_4 = 7 \times 16\text{ cm} = 112\text{ cm}$$ Since the maximum height of the glass column is $100\text{ cm}$, the fourth resonant length ($112\text{ cm}$) exceeds the physical dimensions of the tube. Only the column lengths $16\text{ cm}$, $48\text{ cm}$, and $80\text{ cm}$ can be achieved, giving a total of 3 resonances.

Step 4: Final Answer:
The total number of resonances obtained is 3, which corresponds to option (A).