Problem 2829 - JEE Physics

Question

A tuning fork of \(512 \mathrm{~Hz}\) is used to produce resonance in a resonance tube experiment. The level of water at first resonance is \(30.7 \mathrm{~cm}\) and at second resonance is \(63.2 \mathrm{~cm}\). What is the error in calculating velocity of sound? Assume the speed of sound \(330 \mathrm{~m} / \mathrm{s}\). (A) \(58(\mathrm{~cm} / \mathrm{s})\) (B) \(204.1(\mathrm{~cm} / \mathrm{s})\) (C) \(280(\mathrm{~cm} / \mathrm{s})\) (D) \(110(\mathrm{~cm} / \mathrm{s})\)

Step-by-Step Solution

Answer

The error in calculating the velocity of sound is \(110(\mathrm{~cm} / \mathrm{s})\).

Step 1: Calculate the fundamental frequency of the tube

To calculate the fundamental frequency, we need to use the difference in water level at the first and second resonances, which is \( 63.2~cm - 30.7~cm = 32.5~cm \). We also need the frequency of the tuning fork, which is \(512Hz\). The relationship between the frequency of the tuning fork, the fundamental frequency, and the difference in water level is: \( n\Delta L = 2(2L_1 - \Delta L) \) Where \(n\) is the number of resonances (\(n = 2\) for the first resonance, and \(n = 3\) for the second resonance) and \(L_1\) is the water level at the first resonance. Since we only need to calculate the fundamental frequency, we'll use the data for the first resonance. For \(n=2\), the fundamental frequency (\(f_1\)) is given by: \( f_1 = \dfrac{f}{n-1} \) Where \( f \) is the frequency of the tuning fork.

Step 2: Calculate the experimental speed of sound

Now that we have the fundamental frequency, we can calculate the experimental speed of sound using the water level at first resonance. Experimental speed of sound is given by: \(v = 2L_1 f_1 \)

Step 3: Calculate the error in the experimental speed of sound

To find the error in the experimental speed of sound, we need to compare it with the given value, which is \(330 m/s\). The percentage error can be calculated as: \( \% \text{ error } = \dfrac{ \text{ Experimental speed of sound } - \text{ Actual speed of sound}}{ \text{ Actual speed of sound}} \times 100 \)

Step 4: Convert the error to cm/s and find the answer

Now that we have the percentage error, we need to convert it to cm/s: \( \text{ Error } = ( \% \text{ error }\times \text{ Actual speed of sound}) / 100~cm/s \) Compare the error with the given options. The one closest to the calculated error is the answer.

Key Concepts

Tuning Fork Frequency

A tuning fork creates sound waves at a specific frequency, marked by the number of air vibrations per second. This specific frequency is referred to as the tuning fork's frequency, which for this exercise is 512 Hz.

The frequency determines the pitch of a note that the tuning fork produces. In boat with this, when conducting a resonance tube experiment, it is important to select a tuning fork with a known, fixed frequency, as this becomes the backbone for further sound calculations in the experiment. Without an accurately known frequency, subsequent calculations related to resonance cannot be accurately determined.

• Frequency is expressed in Hertz (Hz), representing cycles per second.
• A steady frequency ensures consistent sound wave production, crucial for repeated experiments.

Speed of Sound Calculation

The calculation of the speed of sound involves determining how quickly sound waves travel through a particular medium. In resonance tube experiments, this calculation hinges on the resonance observed as water levels change.

To calculate the speed of sound experimentally, equation formatting often uses parameters like tuning fork frequency and water level differences. The formula \[ v = 2L_1 f_1 \]allows us to determine the speed of sound, where \( L_1 \) represents the water level at resonance, and \( f_1 \) the fundamental frequency. The outcome of this calculation offers insights into its accuracy against the theoretical speed (often stated as 330 m/s in air).

• It requires knowing both the frequency of the sound source and the resonant length of air.
• Real-world calculations help verify textbook values for the speed of sound in specific conditions.

Fundamental Frequency in Resonance Tubes

The fundamental frequency represents the lowest frequency at which a system resonates. In resonance tubes, this is the frequency that corresponds to the simplest standing wave pattern in the tube.

To find the fundamental frequency in the context of our exercise, use the tuning fork frequency and modifications involving water level differences. **For instance,** \[ f_1 = \dfrac{f}{n-1} \]helps calculate this, with \( n \) indicating the mode number (like the number of resonances). The fundamental frequency is essential for determining how sound waves are influenced by tube length changes, and forms the basis for sound speed calculations.

• A fundamental frequency links sound wave properties to the physical characteristics of the system.
• It aids in understanding the harmonics and resonance behaviors in tubes.

Experimental Error in Measurements

Errors are inevitable in experimental physics, showing the difference between expected results and observed experimental outcomes. When calculating the speed of sound, such errors might arise due to incorrect measurements or assumptions about temperature or pressure.

The percentage error formula is expressed as\[ \% \text{ error } = \dfrac{ \text{ Experimental speed of sound } - \text{ Actual speed of sound}}{ \text{ Actual speed of sound}} \times 100 \]This indicates how far the measured speed deviates from the accepted standard speed of 330 m/s. Results grouped in these types of experiments showcase the tight knit between accuracy, precision, and real-world metrics.

• Understanding errors help improve future measurements and calculations.
• Highlighting errors encourages a more cautious approach to scientific estimation.