Problem 2832 - JEE Physics

Question

In a resonance tube experiment, the first resonance is obtained for \(10 \mathrm{~cm}\) of air column and the second for 32 \(\mathrm{cm}\). The end correction for this apparatus is equal to \(\ldots \ldots \ldots\) (A) \(1.9 \mathrm{~cm}\) (B) \(0.5 \mathrm{~cm}\) (C) \(2 \mathrm{~cm}\) (D) \(1.0 \mathrm{~cm}\)

Step-by-Step Solution

Answer

The end correction for this apparatus is \(1 \mathrm{~cm}\), which corresponds to option (D).

Step 1: Understand the resonance condition for a tube with one open end and one closed end

The formula for the resonance condition in a tube with one open end and one closed end is: \( L_n = (2n - 1)\frac{\lambda}{4} \) where: - \(L_n\) is the length of the air column for the nth resonance - \(n\) is the order of the resonance (1 for the first resonance, 2 for the second resonance, etc.) - \(\lambda\) is the wavelength of the sound wave

Step 2: Find the wavelength from the given resonance lengths

We are given the lengths of the air column for the first and second resonances. We can treat these as two distinct resonances with an unknown end correction, denoted by \(e\): First resonance: \(L_1 = 10\text{ cm} + e\) Second resonance: \(L_2 = 32\text{ cm} + e\) Using the resonance condition formula for each resonance, we get: \(L_1 = \frac{\lambda}{4} \) \(L_2 = \frac{3\lambda}{4} \) Now we can substitute the expressions for \(L_1\) and \(L_2\) in terms of \(e\) into these equations: \(10\text{ cm} + e = \frac{\lambda}{4} \) \(32\text{ cm} + e = \frac{3\lambda}{4} \)

Step 3: Solve for end correction e

We can now solve these two equations simultaneously to find the end correction \(e\). First, we will eliminate \(\lambda\) by multiplying the first equation by 3 and then subtracting it from the second equation: \((32\text{ cm} + e) - 3(10\text{ cm} + e) = \frac{3\lambda}{4} - \frac{3\lambda}{4}\) Solve for \(e\): \(32\text{ cm} - 30\text{ cm} = 2e \) \(2\text{ cm} = 2e \) Divide both sides by 2 to find \(e\): \(e = 1\text{ cm} \) So, the end correction for this apparatus is found to be 1 cm, which corresponds to option (D).

Key Concepts

Resonance Tube Experiment

The resonance tube experiment is a fascinating way to explore the behavior of sound waves in air columns. In this experiment, we observe the resonance phenomenon in tubes, with one end typically open and the other closed. The purpose is to determine the length of an air column that resonates with a particular frequency of sound. When the tube's length is adjusted to a resonance condition, the sound produced inside the tube becomes notably louder. This happens because the sound waves inside the tube constructively interfere with each other, amplifying the sound.

The experiment involves changing the length of the air column and finding the points at which resonance occurs. These resonance points help us gather information about the speed of sound in the air and any end correction factor. Understanding these principles helps us calculate various parameters, such as the wavelength of sound waves, involved in open and closed tube resonances.

Open and Closed Tube Resonance

Open and closed tube resonance is a key concept in the study of acoustics. In a tube with one open and one closed end, resonance occurs at specific lengths of the air column. This is because sound waves reflect from the closed end, creating standing waves that result in vibration.

During resonance, the closed end of the tube acts as a node, where there is no movement, while the open end becomes an antinode, where the maximum movement occurs. This arrangement of nodes and antinodes forms a standing wave pattern. For a tube with one closed end, resonance satisfies the condition given by:\[ L_n = (2n - 1)\frac{\lambda}{4} \]where \(L_n\) denotes the length of the air column at the nth resonance, \(n\) represents the order of resonance, and \(\lambda\) is the wavelength of the sound wave.

The end correction factor, which accounts for the fact that the wave doesn't perfectly end at the geometric end of the tube, must be added to the length of the air column. This correction is essential for precise calculations in wavelength and speed of sound.

Wavelength Calculation in Sound Experiments

Calculating the wavelength in sound experiments is crucial for understanding how sound waves behave in different environments. In the context of a resonance tube experiment, determining the wavelength involves analyzing the different resonance lengths. These are typically the lengths of the air columns that produce a noticeable increase in sound intensity.

For a tube with one closed and one open end, the first resonance condition occurs when the length of the air column is one-quarter of the wavelength, \(\frac{\lambda}{4}\). The second resonance corresponds to three-quarters of the wavelength, \(\frac{3\lambda}{4}\).
To find the wavelength \(\lambda\), we can use the resonance tube conditions:

• For the first resonance, the length \(L_1 = 10\,\text{cm} + e\)
• For the second resonance, the length \(L_2 = 32\,\text{cm} + e\)

By solving these equations, we eliminate \(\lambda\) and find the end correction \(e\), which was calculated to be 1 cm in the original exercise. Once the end correction is known, accurate wavelength calculations can be performed by plugging back into the resonance length formulae.